Malone Algebra written by Joseph Malone Table of Contents Preface 3 Math Vocabulary and Concepts 5 Glossary 6 Operations 11 Arithmetic basics 13 Counting Basics 15 Property Laws 24 The Laws of Exponents 35 Polynomials 38 Graphing and functions 60 Function defined plainly 62 Determining whether an equation is a function or if an equation has an inverse 78 Operations&Compositions 88 Quadratics 111 Quadratic Formula 117 Graphing Quadratics 125 Why (x+h) moves graph left and why (x–h) moves graph right 126 Quartics and other higher power quadratics 130 Exponential functions f(x) = ax 132 Logarithmic Functions loga(x) 133 Combinatorics 136 Geometry 137 Trigonometry 138 Word Problems 139 Preface This book will cover all math from high school algebra to fourth year linear algebra. It will use mixed examples through out blending advanced concepts with foundational skills. It will start off will algebra then mix in calculus in the beginning with just enough geometry and trigonometry to cover all important topics necessary for passing assessment into a first year course at the university level. It also will cover the first two years of a masters degree. Since this is an exhaustive undertaking emphasis will be on building foundation through clear understanding of vocabulary and mathematical terms. All math with be cross linked as much as possible so that time can be spared and more ground covered. There will be thorough definition, illustrations, and processes shown for every operation in math. With an understanding of how to do math the idea is you will not actually have to due much. Will try to cover 4 years of high school math in one year and four years of college math in one year from a strong pure sense while also adding abstraction. Adding abstraction shows readers the order in which math progresses giving them early exposure. The more exposure they have the more likely they will seek advanced math or be able to create their own equations and formulas sooner. Formal logic and writing proofs will be covered. Early definitions will be given plainly so that children are not overwhelmed. I will try to breeze through each topic as fast as I can while leaving no question as to how I arrived at the logic I did. To bridge different math the early explanations are short so it does not interrupt the flow of instruction. I am putting many pieces of information that has been surmized from my own experience plus the internet and other books. If you feel you need more practice in a certain area buy a dedicated used algebra or calculus book from a thrift store for less that $20 each. In reality you do not need to spend time extraneously plodding over the same questions over and over. You need a concise view of all the mathematics and their relationship to each other. You will be able to create equations that explain your real life situations and be able to solve them will ever increasing sophistication. In addition you should gain confidence to explore other topics not covered in this book once you have completed it. See what math others are doing and you the internet to join their conversations and work on projects with them. Experiment with their formulas and write your own. There are two parts to math and they are the pure and the applied. Applied is the area that defines job skills and real life conditions. Pure is abstract in the sense that it only serves as a teaching model to give instruction on the legal operations that can be done within the confines of the agreed upon laws of math. Pure defines what math is and is not and applied is how you use it to enrich your life. Math as a tool is used to solve real world problems and if it is not doing that then it has failed its purpose and real for even being. Concepts will be introduced in a very nontraditional format but still achieve the same goal of teaching. With that being said here are things to consider: 1. I wrote this book for everyone regardless of skill or age. Only basic arithmetic skills like (+, -,, ÷) are required. Most people have already learned these skills in 4th-6th grade so I do not want to rehash basics that far back. If kids are reading this book it will make clear subjects and terms that they are not likely to see until much later or might have missed. A kid reading this book will improve not only their math but all subjects they face in school as well as their personal life because of the strong logic that it will build in them. Even if the material is too advanced they benefit from having a well referenced book. It will prepare them for college and beyond. 2. This book is split into three parts. It is at once a formal textbook, a series of notes from an upperclassman, and a conversation with a peer. As it begins slow skip parts that you do not need to deal with. As it is conversational raise questions that you have and we can devise a solution together. There is no such thing as a perfect book and I worked hard to write this at consider cost of my own time and money. I wrote this book because my passion for this project and helping others outweighed any other goals in life that had I pursued I would have been better of financially and socially. I put priority into this project above my own needs so that others would not have to go through the trouble of trial and error doing google or you tube searches to get bits or pieces of information they were looking for. This book truly does replace others attempts and it does not come with a subscription or or other reoccurring costs. Once you own this book it is yours to keep forever and use not only for yourself but for your children and friends as well. 3. Definitions will get more rigorous but start casually for ease of use. They will not be vague however. Certain parts that are basic may have examples that seem to be overly lengthy taking up a page or more. I only do that so that all my constraints are met to keep in theme of the book. I can not assume what people know or their educational exposure outside this book. This is to be a self contained reference except for the glaring omission of problems to solve. When a problem is given it is dealt with in the most complete manner possible so multiple teaching points are given. Other books do not do this. There were points where I was tempted to follow more professional texts that stated the process, gave an example, and then stated process once more and was done. I take a more examined look that goes past giving clues and deconstructs a problem as much as I can until there are no unknowns left about it. I only go onto the next instruction when I am satisfied that I have nothing more to explore or prove. This actually cuts time required to learn because nothing is left hidden or unexplained that will lead to confusion or lack of understanding later. This tends to happen when changing course to course and book to book. There are gaps. 4. Many books exist that teach the basics. I go beyond them by taking advanced concepts and making them easy to understand and then applying them in a real situation where applicable. By doing this I transition from textbook lecturer to enthusiastic individual willing to share his notes on a given subject or problem. This does away from the standard teacher student relationship by giving readers what an authoritive book won't, the clear approaches to process without quizing them or challenging them to find it. 5. The order of this book will be very simple algebra applied with higher level concepts like functions and Greek notation. I am not making easy problems harder, I am introducing higher concepts earlier using almost too easy problems. This let's readers see the full mechanics under the hood so they can move to any course work any where and know what to do. In this sense this is less a textbook but more a tutorial or how to guide. 6. After algebra I will show calculus and then linear algebra. After foundations and terminology is covered math will be used in a more direct sense. Students will be able to write their own formulas for over 189 real life problems. 7. I give exhaustive explanations for every math word and sign that exists so there is no gaps in knowledge. This benefits kids and adults alike. If you can read English you can use this book. I write and speak at a university level without leaving readers behind. If you do not know a word look it up and not only get better math skills but reading and writing as well. 8. Outside of process there are also fun concepts introduced to make math interesting. While technically not required for success at math the history, language and anecdotes makes for a greater experience. 9. Finally, the most advanced subjects are beyond the scope of this book. This is mostly for foundation building so that readers will be able to read the sequel to this book I plan on producing. These go beyond pure math into masters and doctorate degrees in engineering fields like electrical and programming as well as gravitational and quantum mechanics. These sciences are founded in math but also require specialized study of chemistry and other physics. I can not prime readers for that in one volume. I will touch on what I can but will need to split documents into separate volumes for portability and ease of use. I do not try to put all into one book and only combine the information I do for fear of it being separated and do not want that to happen. Look for the glossary of terms in the front of the book as well as having them fully covered through the text. I also included many reference tables and useful data sets that appear in the back and front for ease of use. Math Vocabulary and Concepts Basic math: 1. Arithmetic- The study of adding, subtracting, multiplying, dividing, powering, and radicalizing. 2. Algebra- The study of balancing equations using variables to solve for an unknown quantity using variables, graphing and functions. 3. Geometry- The study of shapes and their dimensions, number of sides, measure of their angles using proofs. 4. Trigonometry- The study of right angles and their measure. Using laws of trig to find unknown angles or distances. 5. Calculus- The study of rates of change using formulas for slope such as (y2 -y1) ÷ (x2 -x1). 6. Statistics- The study of odds and probability. This leads into further number theory. Advanced math: 1. Linear algebra- Studying higher level linear equations using matrices, determinants, and Gaussian elimination. 2. Discrete mathematics- Studying logic in the form of formal written proofs, Getting into set theory and set notation to list things in groups of numbers called sets. 3. vector algebra- Related to linear algebra, focuses on moving through vector spaces using multi dimensional plane geometry. 4. Tensor calculus 5. Lambda calculus- Formalizing mathematical concepts in a pure form for later use in emerging fields such as science and computer technology. 6. Differential equations – Higher levels of calculus concepts dealing with vector spaces. 7. Number theory – Statistics using tools like Bayesian theorems and computer modeling. It helps to minor in programming using R language and libraries. 8. Combinatorics – Set theory with application for computer algorithms. 9. Topology- Algorithms using ordered data in a set theory environment. Glossary Variable: Letter used to symbolize unspecified numbers. Used in algebra. If x-1 =7 then x =8. Since x can be anything it varies depending on the numbers around it. It is called variable because it varies, the opposite of constants like 1 or 7 which do not vary, they have constant value. Algebraic expression: Using letters as variables in equations. An expression is a sum written without an equation sign. An equation has an equals sign like 2+2 = 4 while an expression is 2+2. If we use algebra then 2x – 1 = 4 simply writing “2x – 1” would be an expression. Factors: Numbers used to form a product. If 2∙3 =6 then 2 and 3 are factors of 6. Product: Result of numbers multiplied. Raised dot: This symbol · used for multiplication. Has seven other names including interpunct. Obelus: This symbol ÷ used for division. Also called multiplication sign. We just use forward slash /. Fraction bar: The line between two numbers to indicate a fractional sum such as this ⅓ . Can also be horizontal but that is mostly used for solving equations as a fill in for an obelus since fractions are the quotient of division. Forward slash can be used in place of a horizontal bar if a computer does not have access to it. Alt 0188= ¼ alt 0189= ½ alt 0190= ¾. Quotient: The resulting number from a division operation. The dividend is operated on by the divisor resulting in a quotient. 100 divided by 4 equals 25. 100 is the dividend, 4 is the divisor, 25 is the quotient. Ratio: A ratio is sometimes but not always a fraction comparing two amounts. In chemistry and science we call this a solution or a proportion. Ratios tell either the amount of different elements added to each other at what rate or the rate at which something is changing. Ratios are written using a colon as 10:1 properly and spoke,” Ten to one”. This differs from writing them as a fraction since in this case we want to know the two quantities being compared. Using both numbers tells us that one element is 10x greater than the other. A ratio given as a fraction would entice us to simply the fraction or write it as a decimal. The ratio 3:2 becomes 1.5, causing the second element to be lost and only the first one remaining. Context of subject matter determines whether we want to keep information or not however colon formula is often preferential. Base: The number to be expanded using exponents. 107 has 10 as the base and 7 as the exponent, or power. Power: Another name for exponent. 22 is “2 squared”, 23 is “2 cubed”, 24 is “2 to the fourth, 2 raised to a power of four, 2 expanded to a power of four”. Raising to a power of 2 is called squaring it, raising to a power of 3 is cubing, raising to 4 and beyond is raising to a power of. Squared: Any number times itself is said to be squared. 2·2=4 4 is the square of 2. Cubed: Any number raised to a power of three is cubed. 27 is the cube of 3. A base times itself twice. 33= 27. Raised to a power of: Taking a base and raising it to a power of 4 or higher. See previous example of “power”. Gross: 122= 144. This is called a “gross”. Great gross: 123=1728 Volume: The amount of space inside a three dimensional shape serving as a container for some medium such as air or liquid. Used to measure the size of something using mathematical constants. Formula is lwh= volume when l= length, w= width and h= height. Used in geometry and related fields. Area: The space occupying a two dimensional plane measured by multiplying length by width. Formula is given as lw= area when l= length and w= width. Used in geometry and related fields. Formula for perimeter: The length of the borders of a two dimensional shape or plane. Formula varies based on the number of sides a two dimensional plane has. 1. Triangle = x+y+z. Measure all three sides and add. 2. Square = 4x. Measure one side and multiply by 4. 3. Parallelogram = 2(x+y). Measure sides then distribute out and add. 4. Trapezoid = x+2y+z. Measure sides then add. 5. Circle= dπ. Multiply diameter by π or multiply radius by 2 and them multiply by pi. Also called circumference. 6. Oval track= πh+2l. Circumfrence of circle plus perimeter of rectangle. Example using 60 for height and 120 for length, round to appropriate number of digits: Perimeter of track: π·60+2(120) → π60+240 → 188.49+240 → 428.496 meters 7. πr2. Measure the widest part of the circle to find the diameter then divide it in half to find the radius. Take the radius and square it then multiply by π. For entire area of track: lw+ πr2 → 60·120+ π302 → 7200+π900 → ≈ 10,027.43 You only have to find area of the whole circle not two individual halves, (area of rectangle +area of circle). Operators/Operations: There are six operators + – ÷ ∙ ^ √ used to perform arithmetic. Operators are the symbols used to indicate the operation to be performed. Operations the method of increasing or decreasing a quantity or number. Operation signs are called addition (+), subtraction (-), times or multiply (× or ·), divide (÷), power (^ or write as exponents using superscripts), and radical (√). There are six main operations, each corresponding to it's own unique symbol telling how to proceed with determining a solution to a given problem. More symbols can and do exist for the same operations so there is some redundancy. These other symbols exist as a convenience so that they make reading an equation easier to do. Often the (×) is replaced in algebra with the center dot (·) so that the × sign is not confused with the variable “x”. Most if not all math symbols (called operator signs or math signs) come from earlier and archaic sources. Many times these symbols have been repurposed for many different uses. They have variant sizing, patterns, and different names. Sometimes there are different names for the different versions. Mostly used for annotating literature these older names and versions are not likely appropriate for math or proofreading anymore. They are obsolete and many people would not understand them or their purpose making them only acceptable for private use and not public consumption. You are not expected to know these or learn them. You are however expected to learn the Greek letters and symbols commonly used in math, physics, and engineering. Using archaic and obsolete symbolism will make notes messier and hard to read so their use is discouraged. In addition to learning math operators and signs it is also useful to learn keyboard shortcuts to put them in computer text software programs. These signs do not appear on modern keyboards so you but to insert them using alt shortcuts. Alt shortcuts only work on keyboards with a 10 key numerical pad so you need to invest in one if you want to be productive. Learning to insert them onto your documents and websites is handy. They can even be used in search bars and comments. To embed them in a webpage you have to use html codes and they are different but same idea as alt shortcuts. Warning: Some courses have different meanings for use! Statement: A line of text or something said giving some description to the conditions of the facts at hand. We use these facts to form our equations so that we can solve them. Statements are often used in word problems when explaining the equation in written English rather than using algebraic expressions. Note however that algebraic expressions and equations themselves are forms of statements as well. In closing a statement must be true or false but not both. Open sentences: Mathematical expressions containing variables whether written in script or as an equation that must be solved before you know if the statement it is making is true or false. Solution: A solution is a number that solves our equation. Solving the open sentence: When we replace variables with numbers the sentence is said to be solved. Equation: Equations are expressions containing an equals sign. If it has an equal sign it is an equation. A set or string of terms with operations that requires working to derive a solution. In Algebra equations are balanced so that two sides, a left and right, or separated by an equation sign (=). To keep the equation true you have to perform equal actions on either side or your equation will become unbalanced and therefore false. Set: Sets group numbers or objects together so that they are listed using parentheses to differentiate them from other data. Using parentheses like this is called set notation and is mainly a study of set theory. Set theory uses an elongated version (∈ ) of the Greek letter epsilon (ε) to denote members of a set. Members of a set are also called elements. This is used in computer science for object orientated programming (oop), writing strings, arrays, and other places where programming code deals with nested items. Sets also show up in algebra when dealing with simple linear equations and plotting coordinates on a linear graph. Groups of points, called ordered pairs [(0,1)], can be collected and placed inside set notation to display what valves are relevant or appearing for a given function or chart. For example if we have the points (0,1), (1,2), (2,3),(…), we would nest them like {(0,1); (1,2); (2,3);(...). The ε means that each pair is a “member” of this set. Sets are also used in number theory and many other places in math. In spreadsheets the Greek letter sigma (Σ) is used to add. We call this summation or the summing of values inside the nested parentheticals. For example if we had Σ={1,2,3,} it would read as “the summation of {1,2,3,}” and that means that we add and get “6”. Whether actually adding or just knowing that the values inside are to be treated as a singular value this is how sigma is used. Note: Greek letters vary in meaning from branch of one mathematics to another. Uppercase epsilon looks just like the English letter E. To avoid confusion exaggerated versions are used to illustrate the meaning better. This is why ∈ is used instead of E or the regular Greek letters. Uppercase epsilon is usually for elements of a set and lower case epsilon (ε) is for denoting a small quantity. There are many uses for the Greek letters and I have included as many of them and their alt codes that I know but sometimes there will be contradictions in terms so you will have to substitute whatever you can and just improvise. If you are using eccentric notation that a reader is likely to be unfamiliar with make that clear before using ambiguous terms. It really only matters that you and your intended audience can understand what notation you are using not anybody else. Subset: A set with a set. Infinity (∞) is the set of all numbers. All even numbers are a subset of ∞. All even numbers ending with zero are a subset of that set. All even numbers ending with zero under 100 are a deeper nested subset still. A subset would be a level below a superset. For example: ∈ {+∞ (2,4,6...(10,20,30...100,110,120...(10,20...90)))}. This shows that within positive infinity (as denoted by the +) a subset exists that contains the set of infinite positive even numbers (as denoted by the ellipsis (…)). Then exists all positive even tens, finally there exists a set of even tens that ends at 90. Superset: A set containing subsets. A superset would be a level above a subset. Null: Ø. This means that a set contains no numbers. ∈ {Ø} means that there are no numbers in this set, not even zero as a placeholder. Literally {} instead of {0}. If null then does not exist abbreviated as dne. Difference between null and zero is if a function mapped 0,1,2 to elements a,b,c then 0 = a; if f(x) ≠ a, b, or c then f(x) ≠ null. Null is like saying n/a for not applicable meaning you exceeded a domain causing a null status. If selecting items from menu required ordering #0, #1, or #2 then Ø set is no selection at all. A = apple= 0 B = banana = 1 C = cherries = 2 f(0) = apple f(1) = banana f(2) = cherries f() = Ø nothing. Empty set: A set with no elements is called empty or null. Replacement set: The solution to an open sentence occurs when all variables have been accounted for using numerals. The replacement set for the variables is the solution to the expression. In factoring polynomials you often use replacement sets to find the factors that make up a trinomial. Negation: Negation means reversing an operation or taking the opposite of something. Subtraction negates or is the negation of addition. By multiplying by a -1 we can negate terms of equations (indeed the entire equation itself) to be the opposite of sign of what it already is. This is useful simplifying and reducing equations to make them cleaner looking or easier to solve. Properties: Properties are rules that dictate how order of operations should behave and what is permissible when reordering terms to restructure an equation. Properties show an how the numbers behave and interact with each other on a foundational level as opposed to simply solving for an answer. Understanding and having innate sense of properties shows an advanced mastery of mathematics and goes beyond superficial plugging in of values to get a result. You know how the numbers behave conceptually rather then mechanically. Additive identity property: For any number a, a+0 = a. This states that 0 does not change the value. Multiplicative identity property: For any number a, a·1 = 1·a. Multiplicative property of zero: For any number a, a·0 = 0·a. This states that the order of terms is irrelevant and moving them around does not change the answer. [In addition any number times zero is zero. Further diving by zero is undefined. It is undefined since dividing by zero does not give us non zero factor. Additive inverse: The number of the same numerical value but opposite quality value. Two additive inverses sum to zero. Multiplicative inverse: The reciprocal of a number or function. The product of two reciprocals is 1. Ex. ⅕∙5 = 1. Properties of equality: The following properties are true for any a, b, or c. Reflexive: a = a Symmetric: if a = b then b = a. This just restates commutation law or uses double substitution. Transitive: if a = b and b = c then a = c. Substitutive: if a = b then a may be replaced by b. Distributive property: For any numbers a, b, c: a (b + c) = ab + ac and (b + c) a = ba + ca a (b – c) = ab – bc and (b – c) a = ba – ca To remove parenthetical terms you must use multiplication by the factor outside the parentheses, in this case “a”. If there is no factor then multiply by “1” to cancel the parentheses. If there is a minus sign infront of the parenthetical then you must multiply every term inside by –1. This inverts all signs of the terms (sign flipping) changing them to there additive inverses: 5(x+2) = 5x+10 –5(x+2) = –5x – 2 –(5x+3x–4) becomes –5x–3x+4. Commutative property: This states that the order of terms is irrelevant and moving them around does not change the answer for addition and multiplication. 2∙3/4 is same as 3∙2/4 , 2∙¾ , 3∙ ½. Associative property: For any numbers a, b, c (a + b) + c = a + (b + c) and (ab) c = a (bc). It says that if you change the grouping for + or ÷ the result is the same. Term: Is a number, variable, product, quotient, or mixed number. Terms are the groups of numbers separated by operands. Each term may be composed of multiple parts but is considered a single value. A trinomial has three terms each separated by a plus or minus sign. Simplest form: An expression without like terms or parentheses. The final answer when solving equations. Like terms: Terms that contain the same variables raised to the same power. 5x2 and 4x2 are like terms but 2x3 and 5x are not. 5xy5z and y5xz are like but 5x5yz and 5xy5z2 are not. Coefficient: The numeral factor in a mixed number. In 5x the coefficient is 5 because it is a constant number. Constant: A number that is not a variable. Numbers 0-9 are constants because they are not alphabetic. In 5x the coefficient 5 is the constant and x is the variable. All variables have letters to denote their changing values. All constants except irrationals have numbers to denote their value is always the same. Some constants like Euler's number “ e ”, the imaginary number “ i ”, and π use letters to represent their value but this is because they are very long and only used as a convenience. They are considered constants because their value is known and it does not change. The difference between a variable and a constant is 5 is constantly 5 while x varies. Pi: The mathematical constant approximated by 3.141592 or 22/7 or 355/113 to denote the symbol π. Pi is used to calculate circumference and area of circles using the formulas dπ for circumference and πr2 for area. There are different approximations for pi but the symbol always remains the same. Domain: The set containing x values of a graph. What choices exist for x as an input for a function. Range: The set containing x values of a graph. What choices exist for x as an output for a function. Graph: The line or curve of a set of plotted points on a piece of graph paper. Literally the image itself. Function: A ratio of x:y used to write equations in graph form. Written as f(x) = equation. Inverse function: The graph of a function that has inverted (swapped) x and y values. Written as f-1(x) = equation. Slope: The tangent ratio of a line given by the formula M= y2 – y1/x2 – x1. Slope is how angled a line is to the x-axis. The formula simplifies to M = y/x. The M is chosen because it looks like a mountain slope making it easier to recognize. Tangent is a trig function that is the height of a triangle divided by it's base. Tangent =M =sine/cosine. Linear Equation: A line graph given by the formula y = Mx+b. This is heavily used in algebra and beyond. Quadratic Equation: A parabola graph. Quadratics are second degree polynomials while lines are 1 degree. Cubic Equation: A graph having 3 degrees. Think of phase “giving someone the third degree”. Cartesian grid: A sheet of graph paper with numbered lines called axes for plotting points. Any rectangular grid for graphing. x-axis: The horizontal number line in a rectangular coordinate system. y-axis: The vertical number line in a rectangular coordinate system. x-value: The horizontal distance from the y-axis. This is the x in an ordered pair (x,y). y-value: The vertical distance from the x-axis. This is the y in an ordered pair (x,y). Coordinate: The location of a point given by distance from the origin of two axes. Pronounced “ax ease”Axes is plural of axis. Also called “ordered pairs”. Point: In geometry a location having zero area. A point is given by (x,y) using x and y values. In arguments a point is a logical conclusion. Two or more points that co-linear creates a “line of thought”. If points are not colinear then the person is “going off on a tangent”. A tangent line touches another line forming a 90° vertex so that a tangent is going off track from the original direction at the fastest rate possible (∞). If two lines of thought intersect at a vertex that point called the “solution” to a “system of equations”. In conflict two opposing lines of thought that intersect have a solution that represents compromise both parties can and will agree to. A parabola is a curved line forming an elliptical vertex given x2. There are different symbolic interpretations but either building a base for your case using x2 to represent area in 2 dimensions or approaching a problem from two different angles which share symmetry and meet at the vertex as a conclusion. The former is easy to apply the latter represents teamwork or independently coming to the same conclusion from different points or using proof to confirm a suspected idea. A cubic is 3d so is a solid block of evidence. Giving someone the third degree involves invasive questioning to gather as much evidence as possible. Depending on the quantity and quality of the evidence a verdict is reached. Side note a vertex represents a corner of 90° – θ where theta can be 0°<θ<90°. Verdict and vertex share Latin roots. Ver-dict is “90° diction” of a decision meaning two points of view of complete opposite rotation meet at a single conclusion: yes or no, guilty or innocent. Start at the meeting point and work backwards. You have two rays forming a 90° angle and extend to – ∞ for x and ∞ for y. These are complete opposite axes at there furthest points, one a positive and the other a negative corresponding to yes~innocent or no~guilty. Ver-tex is lesser important but refers to the crossing of lines to make letters of the alphabet stemming from Phoenician cuneiform the precursor to Latin. Versed is 90° and tex is text meaning writing as in context which means “with text” or the other writing that goes with previous writing to explain it. Context is always the definition as it defines or illustrates the picture, which is a graph made from points. On a different note a song has “verses” which is “rounds” of singing representing a rotation in degrees from end of line to end of line making an arc angle till angles added to 360°, the Greeks represented music mathematically in their cults, too much to surmise here. Note here is a double ontaundre (french for tangent because I am going off on a tangent completely changing points) because note can refer to music or prose) Compare verses with the homophone versus. Verses means lyrics referring to bullion writing which turns and twists down the page and versus is two competing objects, in math versus is the battle between the ratio x:y of a function. Ratios rotate like gear ratios or credit/debit ratios ie revolving credit and capitalizing versus expensing.) Explore more subjects like foreign language, engineering, music theory, and finance. You have a “overturned verdict” which is a complex rotation of degree 4 meaning an inverse function where x and y are swapped making no~guilty becomes yes~innocent and vise versa. A “snap verdict” where a line of thought P1→P2→P4 = A→B→D where you have a hole in your argument at point C. Holes in your argument (also called logic) are gaps or leaps of faith that lead to illogical rational and bad policy. Of course a hole in a function is given by dividing by 0 creating a domain error at that point. There are many other figures of speech using verdict that are logical in directly mathematical way because that is how legal terminology originated. Operations There are four operations in pre-algebra called addition, subtraction, multiplication, and division. Algebra adds two more operations called exponentiation, radicalization. Together these six operations are called arithmetic. Arithmetic means adding numbers. It is the beginning of all math. Algebra does more than add numbers, it takes numbers and organizes them by using variables, performing functions, graphing, and using formulas. The six operations have symbols called operators telling you what operation to do. Every operation has an opposite operation called its reciprocal operation. The word reciprocal and inverse almost mean the same thing but are used slightly differently depending on context. In general all math from arithmetic to calculus is just adding numbers some way. Here is a table to show some language and what symbol to use: Operation Symbol used English Reciprocal/Inverse Operation Addition + plus sign Sum+sum= total total – sum = Sum Subtraction – minus sign Minuend – subtrahend = difference difference + subtrahend = Minuend Multiplication ∙ dot Factor ∙ factor = product product÷factor = Factor Division ÷ obelus / slash dividend÷divisor = quotient quotient ∙ divisor = dividend Exponentiation “powers” ^ caret superscript root^exponent = product rootexponent = product base^exponent = power base^power = product exponent√product = root Radicalization √ radical sign exponent√product = root rootexponent = product Every operation has an inverse operation that takes the answer and works backwards to get the starting number. This can be used to check your work. Checking your work can be used to prove you are correct. Later in geometry you will be asked to show “formal proofs”. If we are adding numbers like 2+3=5 we use plus sign “+”. The numbers being added are called addends or sums, the number is called the sum or total. The opposite of addition is called subtraction. Addition and subtraction are reciprocal operations of each other. Here is a short proof: 2+2 =4 because 4 – 2 = 2. Teachers call this “proving your work”. We used the reciprocal operation to work backwards starting with the previous answer 4 and the using the opposite of “+” sign and instead use “–” sign taking 2 away instead of adding 2. This can be shown on a number line. A number line has numbers on it evenly spaced like a ruler. The little lines that point to the number are called “tick marks” or “graduations”. A number line showing numbers from 0 to 10 is shown below. Numbers never end so we can make a number line shorter or longer. We can even go all the way to infinity if we use a symbol for it. When we add we place a dot on the starting number and move the dot by the addend to the answer. The addend is the number we are adding. If we add 2+2=4 we place a dot at 2 and move 2 units to 4. We count until we are at the right place. Sometimes we draw a curve from the first dot to the second dot. There are many ways to use a number line. There is a red dot at 2 and 4. The curve shows we added “+2” units. A number can be negative or positive. If negative we use a minus sign like this –2. Negative numbers are called “signed numbers” because they have a minus sign. This shows adding in blue and subtracting in red. This shows a number line with negative numbers Adding is the inverse of subtraction. We subtract two numbers that are the same like 2–2 = 0 we get 0. This is also the same as adding a two numbers that have the same value but different signs like –2+2 = 0. When the numbers are the same value but have different value signs they cancel. In adding to cancel is to equal 0. When two sums cancel they are called additive inverses. – 4 and 4 are additive inverses. If we add them they cancel to equal 0. Note that positive numbers are blue and negative numbers are red but 0 is black. Zero is in the middle of all numbers so it is neutral being neither + or –. This concept is used in graphing later. If we are multiplying we are adding by groups. We do this to make equations shorter and faster to perform. If we are powering we are adding groups of groups of numbers. Addition is adding numbers: 2+3 = 5 Multiplication is adding groups of numbers: 2∙3 = 3+3 = 6 Powering is adding groups of groups of the same numbers: 23= 2∙2∙2 = 2+2+2+2 = 8 If we add we just count how many we have. When we multiply we count the number of groups we have and call it “groups”. Then we count how many per group and call it our “base”. Base number is the amount of each group. Base is our root number and also called quantity. Multiplication is “base times groups” or “quantity times amount per group”. If we keep adding the same number we should multiply: 2+2+2 = 3∙2 = 6 If we keeping multiplying the same number we should power: 2∙2∙2 = 2^3 or 23 = 8 Subtraction is adding negative numbers to positive or negative numbers: – 2+7 = 7 – 2 and – 2 – 7 = – 7 – 2. Division is subtracting numbers by groups: 35/7 = 5 because 35 –5 –5 –5 –5 –5 –5 –5 = 5. Rooting is dividing by groups: 3√27 = 3 because 27/3/3 = 3. Addition is commutative meaning they can be added any order and the result is the same: 3+7 = 10 and 7+3 = 10. Multiplication is also commutative: 3∙7 = 21 and 7∙3 = 21. We can have 3 groups of 7 or 7 groups of 3. Division is not commutative: (1÷2)÷3 ≠ 1÷(2÷3) because ⅙ ≠ 1.5 But multiplication by a reciprocal fraction is...we say“Two divided by three equals two divided by three.” If I have... 1∙2÷3 = 2÷3∙1 because 1∙2/3 = ⅔∙1 (3∙4)÷2 = 3∙(4÷2) because 12/2 = 3∙2. Changing the () is called association which is like commutation 3∙4/2 = 4÷2∙3 because 12/2 = 2∙3 Changing the order is called commutative I get the same results but use a different law to give me permission. Stating a law is called “justification”. We sTate laws so we can prove we did the problem right and got the right answer to avoid arguments. (1÷2)÷3 ≠ 1÷(2÷3) because ⅙ ≠ 1.5 Order matters for division. Division does not let you change order. I have no justification for saying (1÷2)÷3 = 1÷(2÷3) because there is no law giving me permission. Arithmetic basics The basics of arithmetic are the operations, the order of operations called Pemdas, and the property laws. The acronym Pemdas states the “order of operations”. Operations must be performed in a specific order or else answers will vary. The letters stand for Parentheses, exponents, multiply and divide, addition and subtraction. Parentheses are used to group numbers to show in what order steps should be taken. Equations are solved doing the operations from Pemdas in the order stated. Math is done starting with the most left number then moving right. Something inside parentheses should be done first followed by exponents then multiplication and division, with addition and subtraction at the end. Some equations have only one or all operator signs and following the correct order means that two different people will get the same answer. If there were not these rules nobody would agree on the answer. This is the same as how people agree on the order of numbers. If they didn't one person would think 7 or 4 can after 1 in counting. These rules are rules because everybody agreed on the order of numbers and the order of operations. What is “7 –4∙2” ? The correct answer is – 1 not 6 because multiplication comes before subtraction. (7 – 4∙2) First we multiply (7 – 4∙2) subtracting first is so wrong... 7 – 8 then subtract 3∙2 then multiplying... – 1 this is right 6 we get 6 which is not correct See how following a different order gives different answers? This is why it is important to follow correct process or your answer will be different than everybody else's, like your friends or the answers in a teacher's book. Please follow the order so that we all are in agreement. Multiplication and division can be done interchangeably as can addition with subtraction. For example: (2∙3÷4) multiply first (2∙3÷4) this time divide 6÷4 now divide 2∙¾ = 6/4 1.5 this is correct 1.5 the answers are the same (5+3–1) add 5+3 first (5+3–1) this time add 3–1 8 – 1 subtract 5+ 2 now add 7 this is right 7 the answers are the same. Why does a different order work this time? Because adding and subtracting are reciprocal, that is they share reciprocity. Changing this order doe not matter because they are inverses which means they act like almost the same operation. We see that subtracting a positive is the same as adding a negative like 3–1 = 3+(–1); and that dividing a whole number is the same as multiplying by a fraction like 10/2 = 10∙½. 4+√9 Because – and + are inverse operations order doesn't matter. Because ∙ and ÷ are inverse operations order doesn't matter. But we still have to do ∙ and ÷ before + and – . We state this rule as “Multiplication does not share reciprocity with addition”. For the example 10/2 = 10∙½ dividing by 2 is the same as multiplying by ½ because of fraction laws: 10÷2 = 5 10∙ ½ = 10/1 ∙½ = (10∙1)/(1∙2) = 10/2 = 5 both equal 5 In algebra rewriting an equation is called “algebraic manipulation”. We do this almost every problem. Follow the order from top to bottom as shown. Memorize Pemdas chart at right: A few points If a number is outside parentheses without an operator you multiply If a number is in front of a radical you multiply Rooting is done during “E” in Pemdas If “ – ” is outside parentheses you flip all sign inside parentheses For exponents radicals are obvious when to perform because they act like a grouping symbol. 3√4 –1 note that 3√4 = 3∙√4 3(2) – 1 note that √4 = 2 such that 3√4 = 3∙(2) a radical is really a () 6 – 1 note that – 1 was not under the radical or it would have been grouped leaving 3√3 5 Although the radical symbol “√” means to root a number, is really a parentheses in disguise. Taking the square root of 4 gave us 2. Since a 3 was in front of √4 we needed to write 3(2). Something like 3(2) means 3∙2. This is from point #1 “If a number is outside parentheses without an operator you multiply”. Point #2 is about “3√4”. Sometimes radicals have irrational roots so we leave them in radical form instead of writing them as a decimal. Irrational numbers never end so it is better to write a radical than a decimal: √2 ≈ 1.41421356. The ≈ sign is called approximation sign and means we estimated the number by removing the small parts that go on forever. 2√2 ≈ 2.828 so you can see we are missing some numbers since 2√2 ≈ 2∙1.41421356. Point #3 is about radicals and require an in depth treatment but for now: 2∙2 = 22 = 4 we call this 2 to the 2nd power or “two squared”. 3∙3 = 32 = 9 we call this is 3 the 2nd power or “three squared”. Squaring a number means to multiply it by itself. This is simple. Taking a square root means to divide a number, “the square”, by one of its factors to get the other factor. We call a factor of a square a square root. “2 is the square root of 4” and “3 is the square root of 9” is how we say our answers. We could also say “second root” if we wanted. Point #4 is about negative numbers: –1∙any number = –that number. This is true: – 1∙4 = –4 “negative one times four is negative four” This is true: –1(7) = –7 “negative one times seven is negative seven” This is true: –(52) = –52 “the additive inverse of fifty-two is negative fifty-two” This is true: –(2+3–4 ) = ( –2–3+4) “the additive inverse of 2+3–4 is –2–3+4” Additive inverse means change all plus signs to minus signs and all minus signs to plus signs. We are inverting the operations and this means to reverse them so we are taking its inverse. The operations were additive so we call this type of inverse “additive inverse”. The slang term for taking an additive inverse is “flipping signs”. Multiplying a number n by –1 creates the additive inverse for that n. We write a rule, called a formula, that uses the script letter n be mean any number. We write this rule as an equation and follow it by replacing n with an actual number we want to take the inverse of. The formula for additive inverse is: n∙ – 1= –n Counting Basics Counting basics introduces variables using set theory. From there we move on to property laws and better examples of algebra. Math after arithmetic is called “higher math”. In higher math we categorize numbers by type into groups called sets. The study of this is called set theory. “Counting basics” uses a simple introduction to set theory. Before 0 or negative numbers were recognized by western, math numbers began at 1 and counted upwards. We call this system the “natural numbers”. Natural numbers are counting numbers of the set (1,2,3...). When you see numbers in a set they are written in either set notation as a series. A series is a pattern of numbers separated by commas inside parentheses to group them together as a set. The set of counting numbers are whole positive numbers that start at 1. Remember that 0 is neither positive nor negative so it is not included. The set of whole numbers is: (1,2,3,4,5,6,7,8,9…) The three dots are called “an ellipsis mark” or “ellipsis”. It means that the numbers go on forever. If we continue the series the numbers go forever by adding +1 to then number before the ellipsis. If we start at 0 we have a different set called the “whole numbers”. Whole numbers are positive whole numbers. The set of natural numbers is: (0,1,2,3,4,5,6,7,8,9…) If we start and 0 and either add 1 forever or subtract one forever we have all the positive and negative whole numbers. This set is called “integers”. Integers are whole numbers that can be positive or negative. This is the set we use most often in math. Because the numbers go in both direction forever we put ellipsis in the beginning to show that the series continues backwards forever. We are interested only in set integers and reals. The set of integers uses two ellipsis: (…–4,–3,–2,–1,0,1,2,3,4…) The next set is fractions also called “quotients” because a fraction bar is a division bar. Fractions are made by taking a whole number and dividing it by a whole number to get a fractional number that is not whole. Fractions can be negative like –¼, mixed like 22½ or improper like 5/4. A fraction can be written like a fraction or a decimal or even a percent. The mixed fraction 22½ can be decimal like 22.5 or improper like 45/2 . There are proper fractions, improper fractions, mixed fractions, and decimal fractions. All can be negative and all can be converted back and forth from a fraction to a decimal or percentage. A proper fraction: two whole numbers divided using a “/” that is less than “±1”. An improper fraction: a mixed fraction written as a fraction instead of a decimal. A mixed fraction: an improper fraction written as a decimal. ¼ and –¼ are proper fractions because they are part of a number. 45/2 and 5/4 are improper because they are mixed. We could say “mixed improper fractions” but that is redundant. The set of fractions is: (all whole or negative fractions). There are fancy ways to write this that come up later. The next set is “irrationals”. These are square roots and other radicals. Names like irrationals, radicals, roots, or even surds imply a rooting operation; we call these radicals for the radical symbol √. There are the “real numbers” called the “reals” and they are all numbers except a unusual one called “complex numbers”. Finally the last set is call complex numbers which uses the letter i = √–1. Complex numbers are not used until last part of Algebra for graphing and conics or hyperbolas. There are important ways to write the sets but that requires a study of set theory. Sets can be written dozens of way. Some more useful than others depending on what you want to convey. Instead of defining sets using formal logic I want to present a more informal definition using blackboard letters. Blackboard letters are also called “double struck” letters because they have double lines. On blackboards they were written this way so that people did not mistake them for variables or other letters. The blackboard letters are the capital funny looking letters. *This is very hard even for professional adult mathematicians and is excluded from books. What I offer here is not required but shown to provide early exposure to understand how to think and not be alarmed by complex concepts. Look at the definitions and then read the post script. Every set is a superset of the subset before it: Set ℕ= Natural numbers. Starts at 1 ends at infinity. Set 𝕎 = Whole positive numbers The same set as ℤ but 0 is excluded. W-hole is “With hole” at 0 on graph. Set ℤ = Integers which called Zahlen in German. This and reals are used in Algebra. Set ℚ = Quotients is another name for fractions. Fractions are p/q where both p and q are integers. Set 𝕀= Irrationals: roots taken from non-index-able numbers like √2 or constants like π or e. Set ℝ = Reals are all numbers except imaginary ones like √–1 or any of the form a+bi. Set ℂ = Complex numbers are the set of imaginary ones like √–1 or any of the form a+bi. The colored letters match the colored word in either meaning and/or show why that letter is chosen. The blackboard letters are the first letters of the word they represent. These 7 groups are basic and all the groups that exist unless somebody makes up there own group and gives it a letter. Algebra uses the integers and reals. You are not expected to know this or memorize these sets so that Is why I am telling you. Focus on the red sets. Any set can be represented with a number line. Tools to find numbers, explain concepts or derive answers will use either a number line, a Cartesian graph, or a unit circle. With start with a number line to perform arithmetic functions. Then move on to graphing with uses two number lines, a horizontal number line called the x-axis, and a vertical number line called the y-axis. After Algebra concepts are mastered we introduce geometry with the use of the unit circle. The unit circle is a circle drawn on a Cartesian graph. From geometry we introduce trigonometry and graph triangles in the unit circle. Following basic trigonometry comes advanced definitions of trig functions shown on a Cartesian graph. These definitions help lead to pre-calculus and calculus topics. The algebra concepts we want to focus on will be variables, functions and formulas. In geometry the focus moves from to construction problems involving area and volume formulas. In Algebra1 we use integers, fractions, and radicals. This set of all these sets is called the reals. In Algebra2 begin to cover complex numbers and complex square roots of a polynomial. A set is a group of elements. A set can contain other sets. A set in a set is called a subset, meaning under the set above it, meaning one level beneath it. A superset is the set above a subset. A superset contains smaller sets. A subset in inside a superset. If we want to express infinity on a number line we use the symbol ∞. It can be –∞ for negative infinity or ∞ for positive infinity. If we wanted to graph all the numbers in existence: ∞ = 0+1+1+1…∞ we start at 0 and add 1 forever. It does not end so it not an actual finite number. –∞ = 0–1–1–1…–∞ we start at 0 and subtract 1 forever. It does not end so it not an actual finite number. Infinity is not a number but a concept. We think of something being as big as possible and say “to infinity”. All the negative numbers are red. Moving to the left represents subtraction. All the positive numbers are blue. Moving to the right represents addition. 0 is neither positive nor negative. This must be proven. If a number is positive we write it as a number. The number 4 is a positive number. If a number is negative we write it as a number with a negative sign. The number –4 is negative. Negative numbers are called “signed” numbers because they have a minus or negative sign in front of them. All numbers have a quality value and a quantitative value. The quality is whether it is positive (+) or negative (–). We use plus and minus signs to indicate quality. The quantitative is how much or its “quantity”. A quantity is a number value. A sign is a quality value. If we add move to the right. 2+3 =5 so we start at “2” and move three units to the right stopping at “5”. If we subtract we move to the left. 5 –3 = 2 so we start at “5” and move three units to the left stopping at “2”. People confuse the sign of the number with the sign of the operation. In both examples all numbers are positive quality so they have positive value. This places emphasis on position rather than simply figuring out the final total. The are special rules for dealing with negative numbers called “value”, “absolute value”, and “sign negation”. There is a special rule for subtracting negatives called “canceling”. Canceling or cancellation or negation mean similar things and there are different situations when these words are applied. We cancel an operation or cancel a value among other things. For subtraction remember this rule: “Two negatives cancel”. If 4–1 then 3, if 4 – –1 then 5. When we have two negative signs they cancel and turn into a single plus sign. 4 – –1 we are subtracting a negative number 4+1 the signs cancel making a plus sign 5 the answer is 5 There are exactly 8 combinations for whether you move left or right. The only important one is when you “subtract a negative”. Change “ ––1” to “+1”. In 4 – –1 there are two signs touching with no number between them. This is called “sign confliction”. Sign conflicts require rewriting both signs as one. If the signs match then add. If ++ or –– then add. If the signs do not match subtract. If +– or –+ then subtract. Sign conflicts only care about 1 thing: “Do I add or subtract?” which is “Do I move left or move right”. Whatever is before the conflict is irrelevant, only the two signs, operator sign and subtrahend sign, matter. A “minuend” is the thing to be subtracted from which the subtrahend is the thing that does the “subtracting”. They are also both called sums: sum – sum = minuend – subtrahend. Numbers are in ()'s Operators are in ()'s (positive)+(positive) (4)+(1) = 5 (positive)–(positive) (4)–(1) = 3 (positive)+(negative) (4)+(–1) =3 (positive)–(negative) (4)–(–1) =5 (negative)+(positive) (–4)+(1) =–3 (negative)–(positive) (–4)–(1) =–5 (negative)+(negative) (–4)+(–1) =–5 (negative)–(negative) (–4)–(–1) =–3 positive(+)positive 4(+)1 = 5 positive(–)positive 4(–)1 = 3 positive(+)negative 4(+)–1 = 3 positive(–)negative 4(–)–1 = 5 negative(+)positive –4(+)1 = –3 negative(–)positive –4(–)1 = –5 negative(+)negative –4(+)–1 = –5 negative(–)negative –4(–)–1 = –3 Do not worry about the chart or attempt to memorize. The focus is the blue options where we cancel negatives. Later during polynomials all terms whether (–) or (+) are put in () and added using a + sign. Learn to understand why signs cancel, the difference between quality and quantity, and learn not to confuse “the sign of the operation” with the “sign of the number”. A positive number like 1 is written without a sign and we still know it is “positive 1”. This a “number”. A negative number like –1 is written with a minus sign to tell us it is “negative 1”. This a “signed number”. You have either a positive or a negative which some people call a number or a signed number. 1+1 both numbers are positive and the operation sign is positive. We isolate the numbers from the operation sign like this: (1)+(1) see how the operator is excluded? 4 – 1 Both numbers are positive but some people and even teachers describe this as only 4 is positive. (4)–(1) this proves both are positive because the sign is excluded. In 4 –1 teachers think “1” is “–1” but it isn't. This confusion leads to students getting a bad intuition. There is a difference between “four minus one” and “four minus negative one”. “four minus one” = “4 – 1” “four minus negative one”= “4 – –1” If you ever have trouble with any equation try writing it as an English sentence. This is an important life skill[1]. Part of math is getting the answer the other part is communicating it. Four parts: Learning to write the correct equation in math symbols. Learning to solve the equation. Learning to articulate the problem in English writing. Learning to articulate the answer in English writing. The chart contained 8 different situations but highlighted in red was when two negatives canceled. Combinations are found using “combinatorics”. Combinatorics would know that there are 8 choices because we have two different places, two different operators, and two different qualities. This gives us 2∙2∙2 choices or 23 which is 8. It does not matter if you start with a negative or positive number and whether the final answer is negative or positive. The only consideration is if you are adding or subtracting from that number. This limits our choices: 1. 1+1 Adding two positives we move right there is no conflict so no signs to cancel. 2. 1–1 Subtracting a positive. There is no conflict so we move left. 3. 1+–1 Adding a negative creates conflict. The signs differ so subtract and move left on the number line. 4. 1––1 Subtracting a negative creates conflict. The signs same so move right on the number line. The rules above do not change if the sum before is negative. A sum implies a single number or subtotal. Only negative numbers have value signs in front so that is why they are called signed. If there is doubt group numbers with () leaving only one operator between two numbers. 1. 1+1 = (1)+(1) the last number is positive 1++1 →1+1 = 2 2. 1 – 1 = (1) – (1) the last number is positive 1–+1→ 1–1 = 0 3. 1+–1 = (1)+(–1) the last number is negative 1+–1→ 1–1 = 0 4. 1– –1= (1)–(–1) the last number is negative 1––1→ 1+1 = 2 These last two pages can be summed up in if signs same add and if they differ subtract but time was spent to demonstrates why and to prove whether a number is positive or negative not just what direction to move on the number line. This ends proof. ■ The black dot is used at the end of proofs so say “end of proof”. You do not need to use both. This proof is important because logic and Boolean operators do weird adding of 1 and – 1. We would never encounter a situation where 1 –+1 existed because we do not write positives with a plus sign. If we group (1)–(+1) the + sign cancels by itself leaving (1)–(1) which is 1–1. Proof of the neutrality of zero part 1 Before we can count we need to be sure of number value and operation quality at hand. We need to know certain things about 1 and 0 because they have special properties. This comes up later so it is a good idea to prove 0 is neutral. For this proof we need to understand variables. Variable means the number is unknown and we have to solve to find it. Example: 2+x = 17 then x= 15. Before pre-algebra students are taught to find the missing number of an equation like 2+[ ] = 17 and write it in the brackets or a hollow square box. In algebra we but a letter there instead: Variables 2+ [ ] = 17 we are asking “2 plus what number equals 17?” 2+ [x] = 17 for now we use the variable x as a placeholder for the true number 2+x = 17 we now erase the brackets and have our first algebra equation How do we solve something like 2+x = 17? There are two ways: we could keep guessing until we get the right number or we could use algebra. Algebra uses methods to solve without guessing and it quicker than guessing. Using algebra: Using guessing: 2+x=17 2+[ ] = 17 2+–2+x = 17–2 2+ 13 ≠ 17 (≠ means not equal) 2–2+x = 15 2+14 ≠ 17 0+x = 15 2+15= 17 this is right. x =15 this is right Both methods work but algebra is faster. This example used extra steps than it normally does. It is also written a certain way to show the importance of signs and the previous value sign lesson. Taking a guess is called an estimation or approximation. When we guessed we tried to pick a number that was close to the right number. We choose 13 which was a pretty good guess since we were only off by a few. When we have a good guess it is called an “educated guess”. If we had no idea what the number was and just started from 1 it would take 15 tries to get the number 15. This is called a bad guess or rough estimate. Guessing is important for working problems you have not seen before but as you learn more algebra you guess less and use something called “formulas” to know how to find the right answer without guessing. Eliminating guesswork can save a lot of time. Let's look at that again: 1. 2+x=17 This is how a problem will look. It is asking us to “solve for x”. 2. 2+–2+x = 17–2 We need to get “x” by itself. If we add 2 then we subtract 2 from both sides. 3. 2–2+x = 15 Remember that if signs differ change them to a single minus sign. 4. 0+x = 15 5. x =15 We state “ x equals 15” and then check by “plugging” it into the equation. 6. 2+15 = 17 We write a 15 instead of x this time 7. 17 = 17 Both sides are the same so it proves we had the right number In step #2 we had 2+–2 =0. This call be moved around to –2+2= 0. It looks better this way because it is easier to read. See how it does not have sign conflict anymore? Re-arranging things is called algebraic manipulation. When you add two numbers and they equal 0 it is called “canceling under addition”. To cancel under addition the numbers have a special name called additive inverses. An additive inverse is any number minus itself to equal zero like 7–7= 0 because – 7 and 7 are the same quantity value but opposite sign value. If we move the order we get – 7+7 =0. Either we have 7 and subtract 7 to get 0 or; We are missing 7 and add 7 to get 0. Both 7– 7 = 0 and – 7+7 = 0 because they cancel. The only difference is where the start on the number line: – 7+7 = Start at blue dot and add 7 or 7 –7 = Start at red dot and minus 7 adding two opposites is called additive inverses Proof of the neutrality of zero part 2 Anything to the left of 0 is negative and anything to the right of zero is positive. A number has to be either negative or positive to work. If not how do we know the value sign and which direction to travel? If 0 is in the middle it has to be either both positive and negative and the signs cancel, or it has to be neither sign. If it is neither one then we call it neutral. Example #1: To make purple you make half red and half blue. Is our color both red and blue or neither red and blue? If we say both than we have a 50% to 50% blended color. If we say neither we have a 100% purple color. Some say it is perspective but truth is: art says 50/50 ratio and math says neither color but a new color called purple. In the real world an additive inverse creates neutrality. Example #2: I vote in an election for two opposing candidates, the blue candidate and the red candidate. Did it vote for both or did I vote for neither? Technically I did vote twice, which is both, but voting for both creates cancellation so I end up voting for neither. I neither added a “+1” to either side. So I voted for neither which being neutral. Said differently if I do not vote at all I am neutral. Whether I vote 0 times or twice the net result is the same so the verdict is the same: I am neutral. Example #3: Two of your friends are arguing and ask you to take sides, “Who is right?” both exclaim. You say, “I think you are right, and I think you are right”. Right there you maintained neutrality by not committing to either side. Any rephrasing means you are neutral because you are on the fence. This is an extension. You can not say you are on either persons side because in reality you are on neither ones side. Example #4: A man walks into a 7-11 store and sees three flavors of Slurpee, red cherry on the left dispenser, blue raspberry on the right dispenser, and purple grape in the middle. He walks out with a purple Slurpee. Did he order half cherry half raspberry or did he order grape? This is a counter example that uses sly and guile to dupe, also called sophistry or a fake proof. The problem is that it uses a tactic from statistics to lie called “mis-correlation” or “false reporting”. If the drink is purple he must have bought the purple but the problem indicates he would have to have a drink that is 50/50 with a solid color on the bottom and a solid color on the top. In proving something all it takes is one counter example to disprove your assumption, also called hypothesis or bias. Since a counterexample has been brought we must refute it by proving it does not apply. To do that we have to investigate it by using deduction. He can not have purple unless he ordered grape. But ordering grape does not correlate to our data. Having grape means an equation that starts at zero and ends there with no operation, this does not apply in an additive inverse where must add and subtract. Adding cherry to raspberry does not create grape but only the color purple. Adding ½ cherry on top of ½ raspberry you can see the divide. Adding ½ raspberry on top of ½ cherry you can see the divide. To cancel we need to meet in the middle. This example is silly and nonsense. Adding two different flavors will not create a grape flavor nor will the colors blend like the first example from color theory. Picking grape from the middle does not show a left to right canceling operation so the data does not fit. We are given inconsistent data. If this last statement is incorrect then an inverse did occur and this example actually corresponds with our bias that the drink in neither red nor blue but a third option that is totally independent of the two choices and therefore 0 is neutral. I have a drink that is half red and half blue. You will see two different colors separately. Look at the number line, is it purple? Or is is half red and half blue? Maybe if the flavors melted it would be a murky purple but this is a stretch. You can make some assumptions but not too many or else it will corrupt the test experiment. Further this is a stretch which means bad data so we must throw it out of our data set. However if we keep it, it is the only example that implies that 0 is both – and +. Conclusion Either we have 0 is neutral and that is the bias we want as math students or we have 0 is ± and that is the bias we do not want unless we are art students. Neutrality works better in the application of political science and engineering or finance or any other math intensive career. A 50/50 approach works better in art, design, literature, film, communication and other soft skills careers due to its flexibility, visual appeal, and appreciation of the nature of duality. In essence art is emotion and emotion lies and corrupts data. We do not want corrupt data. TV and news talk shows corrupt data to consider their audience plus rating and the temptation to deliver confirmation bias wins over raw data. They filter and soften data. This is called “coloring” the news just as you would color the truth. It excels at entertainment not education. Entertainment sacrifices educational value for entertainment value while education does the opposite. Nobody is going to become smart watching TV and nobody is going to be entertained going to school. Each has their own purpose. Duality says 0 is both at the same time. The equation to back this up is x ±0 = x where 0+x = x and –0+x =x. Math says 0 is neither + nor – at any time. The equation to back this up is 0/–1 = 0 and 0/1 = 0. Changing signs does not change value so there is no sign to flip. The interpretation given is if you do not have a number you can not have a value for that number. If I point to empty air and say, “Is that dog half blue and red or is it purple?” anybody would respond “what dog? There is no dog.” and rightfully so. There is no dog so it can't be any color just as 0 has no quantitative value so it can not have a qualitative value. Our conclusion is 0 is not + nor – but neither. Proof of the neutrality of zero part 3 That was a long and exhaustive journey. We learned to think critically and examine a situation from both sides not just our own bias. Now we need to actually prove 0 is neutral. Anything before was anecdotal. Examples and theory is a form of weak proof called “naive”. Naive means assumption or bias. Naive is we get the right answer, we know how to get it, but we do not know why it is the answer, it just pops up. Rigor is the opposite of naive and provides stronger foundation for thinking skills. Rigorous proof understands the what and why of the answer. “>” and “<” mean “more than” and “less than”. We go to the grocery store to purchase a pound of candy. It costs $1 per pound and we need exactly 1 pound. An empty scale is balanced like an empty teeter totter. If we use a 1 pound counterweight the scale will be balanced when we have exactly 1 pound. We will use either 1 for weight or we will use 1lb as an abbreviation for pound. We do not know how much the candy weighs so we use “c” as a variable for weight. We put some candy on the scale and it is too light so it is up in the air. This means that 1lb>c. We add more candy but it is too much so now the counterweight is up in the air. This means 1lbm. Less easy (x –5)(x+2) 5. –x2+bx+c –x2+4x+5 –a then factor out – 1 and all signs flip. Less Easy –(x2 –4x–5) –(x –5)(x+1) 6. –x2–bx+c –3x2–6x+72 –a then factor out – 1 and all signs flip. Less Easy – 3(x2+2x – 24) –3(x –4)(x+6) 7. –x2+bx–c –4x2+12x–16 –a then factor out – 1 and all signs flip. Less Easy –4(x2+3x–4) –4(x –1)(x+4 8. –x2–bx–c –5x2 –15x–10 –a then factor out – 1 and all signs flip. Less Easy –5(x2+3x+2) –5(x+1)(x+2) x2+6x+9 x2+2xy+y2 (x+m)(x+m) (x+3)(x+3) This is a perfect square since both factors are the same3∙3=9 3+3 = 6 √x2+6x+9 = (x+3) when factors match you are actually taking the square root x2 –12x+27 this one has a negative bx term (x2–bx+c) (x–3)(x–9) we find factors of 27 that “add” to – 12 If bx is negative and c is positive both factors are positive. Two negatives cancel when you multiply: –3 ∙ –9 = 27. Why do negatives cancel in multiplication? What would it mean to have a negative group? 2∙3 = 3+3 = 6 two groups of 3 2∙–3 = –3–3 = –6 two groups of –3 –2∙–3 = –(–3) –(–3) = 6 negative two groups of – 3, If a positive group is something added a negative group is something subtracted. I added two groups into my basket I took away two groups from my basket I took away from the other basket so it goes into my basket Any concept of negative numbers involves debt. You change directions on a number line. The absolute value is unchanged the only determination is whether sign is positive or negative. Changing direction is called sign negation. Essentially it is a game of “takeaway”. “3 minus 3” does not equal or mean the same thing as “3 minus negative 3”. As you can see 3–3 ≠ 3–(–3). I blame poor language and instruction. 1. You can add a positive: (3+) + (3+) = 3 ++ 3 = 6 normal 3+3 2. You can subtract a positive: (3+) – (3+) = 3 –+ 3 = 0 normal 3–3 3. You can add a negative: (3+) + (3-) = 3 +– 3 = 0 weird 3+(–3) “adding a negative” 4. You can subtract a negative: (3+) – (3-) = 3 –– 3 = 6 weird 3–(–3) “subtracting a negative” This proved to the “ancients” the existence of negative numbers. This very much alarmed them. Numbers in of themselves have a value other than numerical we call “sign quality”. We write a (+) or (–) in superscript to the right of the number so it does not cause confusion with the sign of the operator. This is well known to science for hundreds of years. In modern science velocity of particle, its position, charge, or its “spin” is given by value signs. An “upspin” is (+) and a “downspin” is (–). You can have positive or negative charge and add or subtract them and there is a numerical quantity attached. Early cult mathematicians thought this +/– was hidden and therefore “evil”. They were shook. When you subtract the number is positive but the minus sign negates it causing subtraction to occur. Anytime you have ever subtracted both numbers were positive. You were removing a positive quantity as opposed to a negative quantity. N – m both are positive, N – (–m) “–m” is negative but you are removing a negative which is a positive move to the right on a number line. We call this latter move “two negatives cancel” but in reality if signs match add if they are opposite subtract. We notice the operator and the number to the right of it to determine which direction a number is traveling. Whether the first number is +/– is irrelevant because we do not care if we start at a negative of positive sum we only care if we are moving left or right from it. We have an operator and then a number that follows it. Both the operator and the number doing the operation have a quality sign. Quality is whether a number is (+) or (–). The compound operator “±” (plus or minus) is said to be different from the quality notation “+/–” (positive/negative) and while true is very strict. I personally do not see myself or anyone I know having a problem with using ± for quality and operation. Write () around numbers to determine quality 1. 4+4 becomes (4)+(4) 2. 4 –4 becomes (4) – (4) 3. 4+ –4 becomes (4) +(–4) which is 4–4 which is 4 –4+ which is a (-)(+) so we subtract 4. 4 –+4 becomes (4) –(4) 5. 4 ––4 becomes (4)+(4) x±3 gives two equations: x–3 ; x+3 while (+/–)x is stating that x can be positive or negative with no operation taking place. For particle science I get it but for pure math it is too far removed for practical use. The community wants to discourage the use of ± when referring to quality and leave it to operations only. They do not like when it is used to state “x can be negative or positive” but rather wish “x plus or minus” when giving the zeros of a quadratic. I see no difference and think that is a silly fight over syntax. Needing two qualities to move left or right on the number line is a matter of mechanics. Math could not operate without these mechanics. These mechanics have been known for hundreds of years. This argument of syntax makes me think a) they do not know what the hell they are talking about, b) scientists have poor background in math and language, or c) they were desperate to invent do to vanity or publish or perish so they invented cockamammie naming conventions. Personally it looks better and is a space saver to use ± for both operation and singular quality but some insist it is an abuse of “math and language”. Pure math only cares about mechanics and does not need permission from code snobs who seek to be the gatekeepers of mathematical expression. People are free to communicate how they wish. Difference of squares, cubics, and other formulas A difference of squares has the additive inverses for the constants in the binomial factors of a trinomial. (x+n)(x–n) = x2 –nx+nx+n2 = x2 – n2. We can see the middle terms cancel; “ –nx+nx = 0x”. These formulas will not really help with factoring but are suggested. Ignore until precalc or completing graphing. All product and factoring formulas are usually written with (a,b) instead of (x,n) or (x,y) so they usually look like: (a+b)2 = (a+b)(a+b) = a2+2ab+b2 This is the standard formula for a quadratic: ax2+bx+c (a–b)2 = (a–b)(a–b) = a2 –2ab+b2 This is the standard form for quadratic: ax2–bx+c a2–b2 = (a+b)(a –b) = a2 – b2 This is a “difference of two squares” a2+b2 = prime = simplest form This is a sum of two squares Multiplying by a third factor creates “cubics” which deal with volume instead of area. Cubics are a quadratic from above multiplied by another binomial: (x+3)(x+3)(x+2) = (x2+6x+9)(x+2) = Ax3+Bx2+Cx+D a3+b3 = (a+b)(a2–ab+b2) This is a “sum of two cubes” a3 –b3 = (a–b)(a2+ab+b2) This is a difference of two cubes (a+b)3 = (a+b)(a2+2ab+b2) This is standard form of a cubic: Ax3+Bx2+Cx+D (a+b)3 =a3+2a2b+ab2+ba2+2ab2+b3 note: b is a constant so highlighted terms combine (a–b)3 = (a–b)(a2–2ab+b2) This is standard form of a cubic: Ax3–Bx2+Cx+D We have standard formula we use for graphing because they are well defined: They only contain 1 variable, “x”. This makes them “standard form” short for “standard formula”. When we have odd formulas, also called functions, we have other variables which makes solving harder. In general to solve a polynomial expression you need to have only one undefined variable. This comes up in graphing linear equations where we can have two or more undefined variables. Factoring we have done so far: factoring a product during arithmetic like the prime factors of 42 are 2,3,7; factoring and factoring quadratics. We factor quadratics to prepare for graphing them so we can then move on to pre-calculus. We skipped factoring a binomial because it is trivial but will do so now as part of a quick review. prime number is a number that is only divisible by itself and 1. A composite number, called a product, is the result of multiplying two or more factors. A factor is a number that divides a product and yields an whole number called an integer. An integer solution is a solution that is not a fraction but a whole number. Factors can only be integers. While the statement 2.5∙2.5 = 6.25 is true 2.5 is not a factor of 6.25 because it is fractional. Integers only. The prime factorization of a number is the smallest factors that produce a product. The fundamental theorem of arithmetic states that every number has a unique prime factorization. When factoring pull out the greatest common factors (gcf) of a polynomial (then reduce to lcm). 3x+15 factor a binomial 3(x+5) 3 is the gcf of 3x and 15 x2y+2x factor a binomial x(xy+2) (4a3y+16af) factor a binomial 4a(a2y+4f) 3x5y2+9x4y2+6x3y2 factor a trinomial 3x3y2(x2+3x+2) 3x3y2 is the gcf, note this gave us a quadratic we can further factor into (x+1)(x+2) The prime factorization for: 3x5y2+9x4y2+6x3y2 =(x+1)(x+2)(3x3y2) If a polynomial only has variables that are all the same letter then that polynomial is set to be “written in terms of (variable name)”. Standard formulas are all written in terms of x making them easy to solve. If a polynomial is not written all in terms of one variable but instead many you can try to manipulate it into having one letter and then it will be easier to solve. Taking multi-variable equations and rewriting them all in term of x, or some other letter, is a strong skill and impressive. Here are mixed forms that will be useful to know how to factor later: (x2 –x) looks prime... x(x – 1) but wasn't. This is sneaky because things were hiding inside what looked already factored. x4 – 1 looks prime... (x2 –1)(x2 +1) looks fully factored... (x–1)(x+1)(x–1)(x+1) make sure you pull as many factors out as possible to get the least common multiple. x2+7x+10 standard quadratic form (x+2)(x+5) standard binomial factored form (x+2)(x+5) we multiply of factors to check work. If factors produce original poly then we did it right x2+5x+2x+10 This is FOIL x2+7x+10 we got original poly so it checks Application: We have a garden and scale it to build a pool. We take existing square garden “with area x2” and add 2 units to one side and 7 units to another side. This is a formula because I do not know how big everyone's garden is and so I do weird scaling using this formula by plugging in their measurements and finding new area for pool. This is the point of formulas, to have a equation that can be customized for each customers space. You want to know math before you get the job rather than learn as you go because it's harder when you are unprepared. I do not know actual application this is used for but know the possibilities are endless. I do not know what you are going to use it for because I do not know how you live. Show me your lifestyle and I will show you what formula to use to make your job easier. I do not understand Tom Hardy. He should be ashamed of himself. When you take Calculus the application are very clear, those formulas help you more. Calculus is written in the language of Algebra. Between Algebra and Calculus is some geometry for proofs and trig for functions. We want to get from algebra to calculus and start solving those problems because they solve everything in the real world like physics does but on steroids. Note that our quadratic was area for a garden. If we take Area/side1 = side2. All we did by factoring was divide the area by one side to get the other: x2+7x+10/(x+2) = x+5 x+5 (x+2)|x2+7x+10 x2+2x 5x+10 5x+10 when we do not have anything left to bring down we are down. The point of factoring is to build confidence to begin to do division with with polynomials. I do not really feel like giving weird examples of complicated factoring using polynomials I do not understand. I can teach how to factor them but division rules all and factoring is not very challenging. Polynomials that are ugly are just for practice not really application. There will be too many undefined variables and that is problematic. I will put all practice at end of book but really want to get to better math as quick as possible instead of wasting time on fluff. There are two more important types of factoring to cover and then we move on to division and radicals. That will end polynomials and we will begin graphing polynomials using functions. Factoring by grouping Sometimes FOIL fails because we can not get integer solutions for a trinomial. What are the factors of 42? 42/7 = 6 so 6 and 7. What are the factors of 6 and 7? the factors of 6 are 2 and 3 which are prime and 7 is also prime. We have 2,3,7 for the prime factorization of 42. Every numeral has a unique string of prime factors that produces it and only it. This is the beauty of the Fundamental Theorem of Arithmetic. Polynomials have a prime factorization too as we shall see in division. What is the factors of 7? we already said it is prime. But 3.5∙2=7 how come those are not factors? Because those are fractions and “‡ any number can be multiplied by endless combinations of fraction pairs because fractions create endless ratios ‡”. ‡Proof: 1. Pick a number and call it “product” 2. Divide it by a number called “divisor” 3. Call the result “quotient” and multiply by divisor to get original “product” 4. Now take quotient and divide by 2 to get new quotient and multiply by 2(divisor) to get original number 5. x = x/2∙2 = x/4 ∙4 = x/0.125∙8 = x/10∙10... these are not factors this is merely demonstrating reciprocity 6. You can create ∞ ratios and make them reciprocal with x/y ∙y ▬ Remember a ratio is slope: y/x = x:y We need integer solutions for polynomial or any other type of factoring or we get endless ratios for any number. In the case of a polynomial a non integer solution means division induced remainders which by definition are rational expressions not polynomial. This is why set theory was introduced in the beginning and I said we are mostly working with ℤ integers. I do not put everything in books only what is relevant. If it is important and other books lack it I add it, if other books have it and it is useless I toss it. Lean and mean and clean code. All math is code so we need to learn the language not cobble answers out. It is simple operations but obscured with funny notations and symbols. This book makes the language clear and shows why it matters and what we are training for. x2 +x–6 there are no factors of 6 that add to 1 so we need to use a method called”factoring by grouping” x2 –2x+3x–6 using ax2+bx+c, multiply ac then find factors of ac that add to b ac= 1∙–6, –2+3 =b = 1 (x2 –2x)+(3x–6) group using ()() x(x–2)+3(x–2) factor gcf out of both binom and make sure both pairs of () match (x+3)(x–2) keep one pair of () as the first factor and then stuff outside terms into a new () for the factor (x+3)(x–2) we are done. If you want to check foil back. x2+3x –2x – 6 foil x2+x –6 it checks When you factor 3x2+11x+6, you want to find whose product is 18 and whose sum is 11: 3x2+11x+6 if a =3, c=6 then ac = 18 ; if n=2, m=9 then 2+9 = 11 3x2+2x+9x+6 3 and 2 don't mix but 9 and 6 do. Order wrong so rearrange until you can factor both pairs (3x2+9x)+(2x+6) group into two factor-able pairs 3x(x+3)+2(x+3) use “+” to separate () then factor making sure both binomials match (3x+2)(x+3) this was clean so we don't need to check but... (3x+2)(x+3) check anyways for practice 3x2+9x+2x+6 9x+2x = 11x we are good, move on 2x2 + x – 21x 2∙ –21 = –42 –6∙7 = 42 1x is positive so the bigger factor 7 must be positive 2x2–6x+7x–21x (2x2–6x)+(7x–21x) 2x(x–3)+7(x–3) (2x+7)(x–3) 15x2 – 13xy + 2y2 this has two undefined variables just as a challenge. Y is normally a constant =c 15x2 –3xy –10xy +2y2 ac = 30 ac= –3∙–10 both have to be negative to sum to – 13 3x(5x–y)–2y(5x–y) pulling out as much stuff as I can... (3x–2y)(5x–y) checked it mentally... looks good x2+5xy+6y (x+2)(x+3) (x+2y)(x+3) 2a2x+3a2y–14ax–21ay+24x+36y mad scientist equation...looks nuts*this is solvable but worthless (2a2x + 3a2y) “+” (–14ax –21ay) + (24x +36y) group like terms (a2 with a2),(a with a),(leftovers with leftovers) group negative signs inside and then separate 90 with “+” a2(2x + 3y) –7a(2x +3y) + 12(2x +3y) pull out gcf. “+–7a” turns int “ –7a”. all three () match so move on (a2 –7a+12)(2x + 3y)(2x +3y)(2x +3y) collect outside loose terms into one () (a–3)(a–4)(2x + 3y)3 factor quadratic into two binom and then group duplicates with exponent x3 + x2 – x – 1 last one (x3 + x2 ) + ( – x – 1) this is important: I grouped with – inside as always, put + sign outside () x2(x + 1) – 1 (x + 1) factoring out a –1 makes –x–1 turn into x+1 (x2 – 1)(x + 1) (x + 1)(x – 1)(x + 1) this is what we want. Prime fully factored form. Smallest pieces. x3 + x2 – x – 1 if I do this (x3 + x2) – (x – 1) I make x positive but keep –1 negative. This is an illegal operation. 4–7–1→(4)– (7–1)→ (4) – (6) =2 changed –7 to 7 but left – 1 negative, result is 2 4–7–1→(4)+(–7–1)→ (4) +(–8) = 4 –8 = – 4 left both – 7 and –1 the same, inserted + sign resulting in – 4 2 ≠ –4 answer should be – 4 not 2 Whatever you do you have to keep parity. It is easier for me to put () around binomials keeping – sign inside and then adding a + sign so that all terms are being added. Then if the negative sign conflicts erase the + sign while keeping whatever was in the () balanced. Putting the – outside means you multiplied by – 1 to one of the components but not both causing a parity error. This unbalances the signs and causes wrong answers. (x ± y)2 = x2 ± 2xy +y2 perfect square trinomial (x+y)(x–y) = x2 – y2 difference of two squares (x+y)3 = x3+3x2y+3xy2+y3 perfect cube quadnomial (x – y)3 = x3 – 3x2y + 3xy2 – y3 perfect cube quadnomial That was tough. There are endless forms. I wanted a quick run through to get to graphing and functions. There is such a thing a extraneous training. If it isn't one of the main versions then we try to solver a variable and turn it into one but that is informal real life problems where someone has deviated horribly off the path and created their own proprietary formulas. There is no real reason to do this except sabotage. Read short story: After WW11 Russia was very antisemitic and abused newly freed Jews. They created the P.O.S. Or Pale of Settlement near Poland (which was first country Hitler invaded and occupied as it shares border with Russia and Germany. Jewish Professionals who lived in Russia's capital (when Russia was still the U.S.S.R. Or CCCP) Moscow were discriminated against. When they tried to get jobs as math teachers at the University they were given unfair tests with trick questions called “Jew Killers”. The questions were overly complicated but contained relativity simple solutions so that if they complained it would be demonstrated how easy to solve it was when in reality it wasn't. The goal was to prevent them from gaining jobs at a prestigious University, a way of saying,” we do not want you here”. I do not use bogus tricks or nonsense as it doesn't help or prepare students for anything meaningful. This is enough factoring. We will do more during rational expressions and when we learn division. Graphing and functions Graphing is done using coordinates similar to longitude and latitude. We use a rectangular grid with lines equally spaced. We use two number lines. The First one is called the x-axis and is identical to the number lines we have been using so far. The second one is vertical and called the y-axis. The grid has 4 quadrants, or regions numbered 1-4 starting in the upper right and rotating counterclockwise. Locations are given as a combination of an x value and a y value using this notation (x,y). An x value represents a horizontal position and a y value a vertical position. The center of the grid has a vertex where the two axes cross. This point is called the origin and its coordinates are (0,0). Any steps are measured in “units” away from this location. A coordinate of (1,0) would be 1 unit to the right and 0 units up. This would place us on the x-axis 1 unit to the right of the origin (0,0). The axes are neutral being in not any quadrant. If we moved from (1,0) up 1 unit we would be at (1,1). The quadrants have ± signs showing the quality of x and y for each region. In the 1st quadrant both values are positive: (+,+) both positive. Move right and up. In the 2nd quadrant both values are mixed: (–,+) here x is negative because we moved left from 0 In the 3rd quadrant both values are negative: (–,–) both x and y are negative because we moved left and down In the 4th quadrant both values are mixed: (+,–) here y is negative because we moved down Graphs use functions to plot points on the grid. There are pre-algebra graphs using stem and leaf and scatter plots. These are used for science data and financial data. There are geometry and trig graphs for drawing geometric shapes and sines waves. There are also calculus graphs for transforms, integrals and ODEs and PDEs, which are ordinary derivatives and partial derivatives. We are interested right now in functions. A function is an operation using variables to calculate many answers as points. The points connect and make lines called “graphs” or “functions”. The image or an equation can be called the function. Any arithmetic or higher equation can be written as a function. We can graph points using a graphing calculator or we can make a table with values and then graph the function by hand using graphing paper. The website Desmos has a very good graphing calculator while the free one on Windows PC is okay but clumsy. On page 36 of “Polynomial Operations” we had a function already stating: “I will double any offer plus $1 ”. If I wanted to graph this first I would make a table and then find points. For a function we need three things an equation, an input, and an output: f(x) = 2x+1 where “double any offer” is 2x and “plus $1” is +1 X = any number Y = total price point 0 1 (0,1) 1 3 (1,3) 2 5 (2,5) 3 7 (3,7) 4 9 (4,9) x is our input that we choose. When we chose a value for x it is no longer an unknown but some number we assign to it. Then we substitute that number by erasing x and putting it in its place. We do the operations and get a single number. That single number replaces the y variable. For a point (x,y) we have two unknowns both x and y. We create a function called f(x) = 2x+1. Since f(x) is really y we can write y = 2x+1. We are going to make a table, calculate some y values and then plot them on a graph. 2x+1 =y (x,y) Replace x and y with your input for x and your output (meaning answer) for y. 2(1)+1=y 3=y we have point (1,3) 2(2)+1 = 5 point (2,5) 2(3)+1 = 7 point (3,7) 2(4)+1 = 9 point (4,9) This function is called a linear equation since it has a degree of 1 and makes a 1 dimensional line when plotted. Linear equations Linear equations take a number then multiply it and either add or subtract from it. There are three ways to write a linear equation that are useful; standard form, y intercept form, and point slope form. Standard form: Ax+By =C Y intercept form: y = Mx+b Point slope form: y – y1 = M(x – x1) formula for slope: M = y2–y1/x2–x1 Standard form is when two unknown sums equal a known total. These are old problems relating to age or combinations like: graph of f(x) = 2x+1 points with intercepts (0,1) & (–½,0) “I have 70% hamburger and 93% hamburger how many pounds of each do I need to make a 5 pound mix of 85%?” “If I need make a pie for $1.75lb. how many apples do I buy at $2.63 a pound and at $1.25?”. Point slope is when you know one point on a graph and slope and want to use it to find the equation for the line. There are applications for this but off the top of my head I could not tell you. Intercept form is the easiest to use and is used to graph lines and calculate rates of change using slope. The graph above was drawn using this one. Slope is the angle to the x axis the make but more precisely it is a ratio of x:y written as y/x . The formula for slope M = (y2 – y1)/(x2 – x1) needs two points to calculate slope. Using the first two points from our table (0x1,1y1);(1x2,3y2) we can calculate slope: (3–1)/(1–0) = 2/1 = 2 = M. We can use any two points as long as we are consistent in subtracting. Mixing a points values with another will not work y2 – y1/x1 – x2 or y1 – y2/x2–x1. In other words we can use (y2 – y1)/(x2 – x1) or (y1 – y2)/(x1 – x2) but not (y2 – y1)/(x1 – x2) or (y1 – y2)/(x2 – x1) For the first point has subscript 1 as in (x1,y1) and second point has subscript 2 as in (x2,y2). A linear equation with multiply a number then add a sum to give an answer. This creates an ordered pair that we use to plot the points. The points go no forever since numbers never end giving us infinite solutions. From a glance we can see patterns on a large scale and make decisions, mostly business based, about our options and how to proceed in a given situation. Business uses charts to track sales or trends or to impress clients and win new contracts. The stocks market displays financial data using charts and graphs. Graph are very popular for business because you can see a lot of information at once. Function defined plainly y = x+1 then f (x) = x+1 y =f(x) F or f is does not mean variable for f(x). The “ ƒ ” is rarely stylized as lowercase Latin f that is a shorthand notation for “function”. Do NOT multiply the the left side of the equation. We abbreviate function as “func”. “Function of (x)” is stating that the output of (x) is given by the equation on the right side of the equals sign. Output is defined as the y value of an ordered pair. Input is what number we replace x with. An output is just the answer to an equation. We solve for x and then plug it in to solve for y. We solve for x by choosing one that will suit our needs. Doing this is called defining the variable. Giving the variable definition means we erase it and write a number instead so we can actually do math. There are methods to pick good numbers for x we will learn. The “x” is in brackets to indicate that it is a number in a series and not in parentheses as used in multiplication. As parentheses indicate multiplication in algebra you will see calculators use brackets for f[x] most times but in text it is f(x) = y because assumes using an infinite domain of (–∞,∞) as opposed to a finite one of [x1,x2] as calculators might have. When you see f(x) say “f of x” or “function of x”. Functions create ratios between variable usually x and y. We are comparing two different quantities. A function tries to make one variable bigger or smaller than the other. y= x+1 this means that x needs to add 1 to equal y. Then y is bigger. ƒ [x] = x+1 f(x) =y so we can use either label at will. The function is the equation we will solve for one (x,y) group, called an ordered pair, where x is called “input” and “y is called output”. Types of functions : linear, polynomial, exponential, logarithmic, trigonometric f(x) = Mx+b where m is slope, x is k is some constant for the y intercept f(x) = Ax2+Bx+C coefficients a,b,c… where A is leading coefficient and x is base f(x) = ax where a is base and x is exponent f(x) = loga(x) where a is base and x product f(x) = sin(x) x is input of a trig function, here sine is used F of x means “function of x”. Function is written on calculators as func. You chose a number for x then substitute the number you chose into your equation. Function notation is: f(x) = some equation. When you find a value for the right side of the equation your work looks like: ƒ [x]= answer = the number we use for y. This describes a Cartesian pair. Cartesian pairs “map” a ratio, also called slope, between x & y values. y= x+1 This is our function. We need to pick a value for x. We choose 0. y= 0+1 Substitution y= 1 Solved giving us point (0,1) This means that our Cartesian pair is (0,1). We have infinite pairs since we can choose any number for x and it will have a matching y value. The function y = x+1 will mean that for every part y is one more than x: This means that our pair is (1,2). Using a different x will produce a different y. Choosing for x is called “picking an input”. We input our number for x. Solving is called “receiving an output”. For y = x+1: If x = 1 then (1)+1 = 2 y = 2 If x is 1 and y is 2 our (x,y) pair is (1,2) If x = 2 then (2)+1 = 3 y = 3 If x is 1 and y is 2 our (x,y) pair is (2,3) If x = 3 then (3)+1 = 4 y = 4 If x is 1 and y is 2 our (x,y) pair is (3,4) If x = 4 then (4)+1 = 5 y = 5 If x is 1 and y is 2 our (x,y) pair is (4,5) The x-axis is a horizontal number line. The y-axis is a vertical number line. They intersect at a 90° angle at the point (0,0). The intersection point is called the origin. This is for all grids. Every unit is equally spaced from each other. The grid is numbered like this. We plot points and label them like this. Then we draw a line. A line in geometry is said to go on forever in both directions. A linear equation has a line that goes in both directions forever. Later in geometry the explanations get more technical. Using the function y = x+1 write a table showing the points on the graph with the blue line drawn. y =x+1 x y (x,y) –4 –3 (–4,–3) –3 –2 (–3,–2) –2 –1 (–2,–1) –1 0 (–1,0) 0 1 (0,1) 1 2 (1,2) 2 3 (2,3) 3 4 (3,4) 4 5 (4,5) Study the table and how it is setup. We write the equation at the top. This equation is called our function. We make columns for x and y. Making a column for (x.y) showing them together is optional but very helpful. The x column is a list of x values. x values can be any number we want. It depends on what parts of the line we want to study. For starters it is good to start at “x=0” and then add 1 every row. The y column is a list of y-values. Once we say what number we are using for x we plug that number in for x and solve for y. Once we solve for y we write the number. This creates an x,y pair which is called an “ordered pair” and uses this notation (x,y). I will demonstrate how to solve for y using the function f(x) = x+1 and for a polynomial: f(x) = x+1 write a function using “f(x)” notation f(–4) = –4+1 plug in any value for x. Anywhere the is “x” replace with the number you are using. f(–4) = –3 spoken as: “the function of –4 is – 3” or you could say “f of –4 is – 3” f(x) = x2+3x+7 this is a polynomial function f(–5) = (–5)2+3(–5)+7 plugging in a random number for x f(–5) = 25 –15+7 f(–5) = 17 (–5,17) this is a point that exists on the quadratic. y = 17 (–5,17) I used Microsoft calculator to graph both functions. The linear equation is a line and the quadratic makes a shape called a parabola. The point (–5,17) is found by moving –5 units on the x-axis and 17 units on the y-axis. This is easier to show on Desmos which I highly prefer and recommend. Try it yourself. Type in equations we have done or make your own. Or make your own formulas by creating variables and using different values for your variables. Start basic first then add complexity later. Eventually the graphs should show you answers to real life applications. Applications are also called “word problems”. Application is how you apply it you your life. Not because I tell you it does but because you tell yourself it does. At the end of the book I will give applications. I do not have any applications for these type of problems right now but studying them helps learn the next math and the math after that. This is a basic primer to gear students for calculus and statistics which will help you in your real life much more. Algebra problem will be “work, rate, and mixture” problems. Geometry problems will deal with physical dimensions like area and volume of solid shapes using algebra notation and formulas like functions. f(x) = x2+3x+7 this is a polynomial function. If we choose values for x then solve for y we find points that form a parabola shape. For a linear equation the points form a line. We can use decimals for x like 1.1 and 1.2 or 1.00001 and 1.00002. We can add as many zeros and create points that are infinitely close. This means that as locations the locations touch and when graphed make a solid line with no gaps. We can use fractions for x like ½, ⅓ or any other fraction. The only thing we can not do in a function is have an equation that somehow divides by zero. When this happens we have something called a domain error. Domain is the set of numbers x is allowed to be. If we had y = 2x then our function takes a number and multiplies it by 2. We can multiply any number by 2 so we say the domain of x is (–∞,∞). This means any number from negative infinity to positive infinity can be substituted for x. Once we have a number for x we call this “defining x”. Defining a variable is when you set it equal to a constant. We define x by rewriting our equation with a number instead of x. If x = 3 then y = 2x becomes y = 2∙3. y will then equal 6 because we defined x as 3. Domain is the set of possible x values. Range is the set of possible y values. There are equations that limit what x or y can be. For example If I had the function f(x) = 1/x I can not choose x to be 0 because I would get f(0) = 1/0 and there is a rule called “division by zero” that says division by zero is undefined. There is that word again “define”. Define means to know what number the variable represents. Define can also mean other things but in general the definition is the answer. If x=4 then x is defined as 4, since we have “something = 4”. So the answer to “something” is 4. Take 2+2 =4. 4 is the answer to 2+2. This is called an equivalence. We have an equals sign with two quantities on either side. Left side:right side is 2+2 = 4. In general “this = that”. The point is we can write a number different ways. This is a critical concept in all math but especially this point and beyond. You have to walk away with this understanding for any situation. We can write four as “4” or as “2+2”. We can write function of x as “f(x)” or “y”. If x = 4 then 4 = x. “The answer to four” is x and “the answer to x” is four. Very important. We if x=4 we are not really stating a problem to be solved, it is solved. It is the solution and therefore the answer. The solution = the answer. The goal is to teach that x = some number is not a problem but an equality of two groups. 4=4 this is called reflexive x=x this is called reflexive If x = 4 then they are both 4. x = 4 this is called substitution. This the basis for equivalence statements. When we write an “=” sign we are equating two things. On page 44 I said you could equate an equation. From page 43: “(j)(x+3) substitute j for x+2 (yes you can use a variable for a sum or even entire equation)” Note: j = x+2 is an entire equation and that x+2 is a mathematical expression called a binomial. For something to be an equation it needs an equal sign. Having the binomial x+2 without an equals sign makes it an “expression” rather than a full equation. We treat expressions like equations or sometimes call expressions equations but they are not. This is technical. The difference is whether a person is speaking English correctly which translates to whether they are doing math correctly. If you do not understand what the words or notation or symbols mean how can you do math? Example: The phrase “do your homework” can be made a sentence but is still a phrase. 1. “do your homework” a phrase 2. “Do your homework.” a sentence 3. “Do your homework or you're grounded.” a compound sentence 4. “Do your homework...” a cutoff sentence If we add a period it becomes a sentence. If we leave it off it isn't. While technically a phrase we can use it as a sentence. This is bad grammar and syntax but people will understand you. If you lack strong reading skills your work will suffer. Bad communication and comprehension follows. This goes beyond getting the correct answer but understanding how and why. Understanding what 2+2 equals is different than understanding why 2+2=4. 1. 2+2 a phrase 2. 2+2 = 4 an equation 3. 2+2 = 4 and x = 4 a compound equality 4. 2+2 = 4 and x = an incomplete compound inequality Starting from basic applications A linear equation performs multiplication. That is all it does. If someone says “linear equation” it is a fancy way There are three different way to write a linear equation. You can convert from one formula to another. Form is another way to say formula. A formula is the notation we are using. Notation is any formulas or symbols used in the equation. Every lesson we learn new notation and introduce it as it appears. Math is just symbols and higher math takes basic arithmetic and garbles it in code. All math is code and learning notation is the way to decode it. Standard form: Ax+By =C This is used to find the intercepts. All linear equations have y and x intercepts. Rare in graphing. Used for sums. Standard form takes two ingredients called A and B and combines them to make an ingredient called C. The literal interpretation is mixing ratios like a cocktail made for 5% alcohol and 10% alcohol mixed to make a drink that is 7% alcohol. These are real life problems not pure math and just find a number. The number has to represent something more than just a dot on a grid. Our hypothetical drink would have some ratio of x:y to make a 7% mixture. If we made a 1:2 ratio then we would have 1 part A and 2 parts B. This makes 3 parts not 2. Ratios are different from standard fractions. They require in depth instruction. First lets manipulate the formulas to better understand them and talk about ratios later. y intercept form: y = Mx+b This gives a form that shows the height of the point that intercepts the y-axis. This point is called the y-intercept. This is useful if you need an equation and use values that are close to the y-axis. This is used in graphing lines. Useful if you know or need slope and/or y-intercept. Most popular. This is popular early on for business equations that show multiples of some value like x amount of units equals y amount of dollars or science applications like tracking rainfall, population, or anything else that grows at a steady rate. Point slope form: y – y1 = M(x – x1) This is useful if you are using points far from the axis. This is used in calculus when calculating tangents of curves. Useful if you have two points or if you have one point and slope. Less popular. We can convert from one formula to another by using algebra. Algebra means rewriting from one form to another. The “algebra” is the notation which people call, “the language”. Ax+By = C standard x+y = c the coefficients are not important y = –x+c this is y-int form except it uses “c” and calls it “b” Whether you call it b or c or anything else the constant added to x will be you y value. Observe: y = –(x) +c let x = 0 y = –(0)+c there is no such thing as negative zero, –0 is 0 y = c our y value is 0 giving us point (x,0) where x is undefined If x = 0 then from the origin (0,0) we did not move left or right any units. Our x value is 0. If y = c then from the origin (0,0) we moved “c” units up or down. Whatever “c” is that will be the height of our point. y = –(x) +c let y = 0 0 = – (x)+c we have point (–x,c) which is the x-int this is x-int but we have no values except that we move to the left because x is (–) if we had numbers we would know what the x-int and y-int are Ax+By =C this is standard form for a linear equation By = –Ax +C this is ugly way to write y-int formula y = Mx +b we write this way is better. We use b for vertical rise but some use other letters note: By and Mx+b are different “b”. b is vertical rise and B is cof of y if it has 1 y = Mx+b y – b = Mx subtract b from both sides y – y1 = M(x –x1) rename b as y1 and x as the quantity x–x1 We did it. We rewrote all formulas using algebra. Huge success. This means that for a line using any of these forms or even the ugly ones we stated the equation for a unique line. If we put any of these forms into a graphing calculator they would all draw the same line. Note that we had – (0)+b earlier. Using commutation we change order to b – 0 = b, and 0 cancels. Also there is no such thing as “negative 0” as it is neutral so it is just 0. 5 – 4 = 1 five has a “hidden” plus sign because it is positive –4+5 =1 the –4 is really –+4. the signs do not match so it is negative 5–0 = 5 –0+5 = 5 we do not write “ – ” in front of 0 0+5 = 5 We went from standard to y intercept to point slope. What a coup de gracè. Using algebra we can rewrite any of the forms to any of the other forms. There are 3 choose 2 combinations so 6 possible conversions. That is too many to do but practice for yourself and your graphing skills will be stronger not to mention your algebra as well. Let's work with y = 2x+1. Let's try to convert it: y = 2x+1 intercept form y = 2x+1 intercept form y – 2x = 1 standard form y – 1 = 2(x – x1) point form y = 2x+1 if want to know where the line intercepts the x axis... 0 = 2x+1 then the height above the x axis will be 0. Use y=0 –1 = 2x subtract “b” from both sides –½ = x divide both sides by 2 (0,–½) this is the point for the x-intercept y = 2x+1 if want to know where the line intercepts the y axis... y = 2(0)+1 then the horizontal distance from y axis will be 0. Use x=0 y = 1 hmm... (0,–½) this is the point for the y-intercept 1. using 0 for x or y will give you the other variables intercept 2. intercept form gives you the intercept already with b as the vertical increase This is a such a simple function we can calculate points mentally. What are the y values for 0,1,2,3,4,5,6,7,8,9,10? [(0,1);(1,3);(2,5);(3,7);(4,9);(5,11);(6,13);(7,15);(8,17);(9,19);(10,21)] I notice a pattern. The outputs are all odd. If inputs are integers this function gives all outputs as odd numbers, why is this so? Let's do a different function y = 2x. Can you make a prediction about the results if we use integer inputs? Note: Brackets are not necessary but semicolons are. Semicolons are ugly but commas are hard to read. f(x) = 2x f(x) = 2x f(x) = 2x f(x) = 2x f(x) = 2x f(x) = 2x f(x) = 2x f(x) = 2x f(0) = 2∙(0) f(1) = 2∙(1) f(2) = 2∙(2) f(3) = 2∙(3) f(4) = 2∙(4) f(5) = 2∙(5) f(6) = 2∙(6) f(7)=2(7) y = 0 y = 2 y = 4 y = 6 y = 8 y = 10 y = 12 y = 14 They are all even! We learned an important fact about number theory using functions. 1. Any number times two will result in an even number~ the way to write with functions is y = 2x 2. Adding 1 to any even will result in an odd~ The way to write with function is y = 2x+1 We do functions to either solve a real world application or to study patterns. In graphing functions we can see a large amount of information quickly. This helps us see the bigger picture than just the answer to a single equation. A function shows the answers to many equations of a formula, infinitely many. Last page I asked why you thought the outputs were all odd. I care more about getting you to think creatively than making you do a bunch of complicated equations. People like to solve a problem and move on but do not take time to think about their answers or what they mean. The meaning will be related to a pattern of some type. Understanding patterns in life is important. While it is important to solve math quickly for work there are calculators for that. A calculator can get an answer faster than a person but it can not think and understand what that answer means. When I asked you to do a prediction that is called forecasting. The news has weather forecasting based on graphing weather patterns and also financial forecasting graph stock and market data. Seeing patterns in life is critical for making the right decisions at the right time. We use math for that to get money, self preservation and quality of life. If you keep making the same mistakes in life you are missing the patterns you need to notice or you are deliberately ignoring them. Both y=2x and y=2x+1 are shown graphed together. Graphing two functions at once is more powerful than separately as we can compare the two functions side by side. We look for patterns and try to extrapolate any data we can. What patterns if any do you see? When we have two functions graphed together we name them alphabetically f(x), g(x), h(x)... and so on. f(x) = 2x has y intercept 0 g(x) = 2x+1 has y intercept 1 For business the y intercept is often called “initial value”. In a way it says we should look at the graph by starting where x = 0. If you deposit money in an account that earns interest the initial value is your deposit. Over time you can graph how your money grows. The x value counts time in days, months, years, or in the case of the stock market even hours. The y value counts price in dollars. If you hear the term initial value think y intercept where before any time has passed the starting value is on the y axis somewhere. It takes practice to read and interpret graphs skillfully. Notice how the two graphs do not intercept each other. This is because they have the same slope. Two lines with the same slope are parallel. Solving systems of Equations When two functions cross each other we want to know at what point this happens. It represents a state of equilibrium where both function has the same ordered pair. Depending on application this can be a good thing to aim for or bad thing to avoid. This relates to point of diminishing returns, point of no return, and point of gained value. Systems use the standard format Ax+By = C. Claim: If we have two functions then we can find a point where they meet. If then 2x = 2x+1. Is this a true statement? no. 2x+0 = 2x+1 this doesn't look right 0 ≠ 1 0 does not equal 1 These two functions are said to lack reflexivity. If they were reflexive they would both equal thew same number at the end. Because they same same slope they have the same angle to the x axis, about 63.43°, so they with never cross. More of a lesson of geometry or trig but slope is the same thing as tangent and these have a tangent of 2. Using the function h(x) = x we create a new function that crosses both. We can see that f(x) = 2x and h(x) = x share the point (0,0) but do not know where h(x) crosses the g(x). We can solve using one of two methods. Colors blue;red;purple match the graphs to right. Notice how the h(x) intercepts f(x). What is the point this happens? Will this happen with every set of two functions? Use an algebraic method to find where h(x) intercepts the other two. We can solve by substitution or elimination. Below equations show x values in blue y values in red. Substitution f(x) =2x g(x)= 2x +1 y =2x use “x=y” for y y = 2x +1 (x)=2x looks wrong... (x) = 2x+1 looks wrong... x/2 = x how can half of x equal x? –x = 1 subtract – 2x from both sides f(x/2) = 2x need real number for input x = –1 multiply both sides by –1 to make x positive cant use variable as input g(–1) = 2(–1)+1 f(x/2) = x useless but true (x,2x) y= –1 (–1,–1) Elimination subtracts on equation from the other (2x = y) 2x+1 = y x = y – (x = y) – x+0 = y 0=0 x = 0 x+1=0 2(0)= y x = –1 0= 0 (x,y)=(0,0) 2(–1)+1 = y becomes –2+1 = y becomes –1 = y becomes (–1,–1) Using substitution did not work because we got a solution point of (x,2x) which was just our original equation y=2x rewritten. It take two “x” to equal one “y”, x+x =y, x+x = 2x, our ratio is 2x =1y. To use substitution we need a variable clearly defined. This is dealt with in arithmetic series explaining the difference between an explicit definition and an implicit definition. Using elimination we get (0,0) which we can see from the graph. (0,0) = (x,2x) because 0 = 2(0) and because 0/2 = 2∙0 which was x/2 =2x. x = 0 and y = 0. We can not use x/2 to plug back into our equation to solve because x/2 is not a constant. For a function to work it needs x to be explicitly defined. x = 0 is explicit, x/2 is implicit. Explicit will state a constant, implicit will imply a ratio. Recap 1. f(x) = 2x g(x) = 2x +1 h(x) = x 2. y = 2x change f(x) to y y = 2x +1 change g(x) = y y = x 3. (x)= 2x looks wrong... (x) = 2x+1 looks wrong... 4. x/2 = x how can half of x equal x? –x = 1 subtract – 2x from both sides 5. f(x/2) = 2x need real number for input x = –1 multiply both sides by –1 6. cant use variable as input g(–1) = 2(–1)+1 7. f(x/2) = x useless but true (x,2x) y= –1 (–1,–1) We had three functions: 1st f(x) = 2x 2nd g(x) = 2x+1 3rd g(x) = x f(x) and g(x) are parallel because they have the same slope of 2. g(x) does not have a slope of 2 so will intercept both of them. Intercept means to pass through. We want to find the points where lines intercept. This point is called “the solution” to a system of equations. The solution is a point that both lines share so it will have the same (x,y) values. Solving by substitution if fairly simple. First we have g(x) = x and f(x) = 2x. Right away we know the solution is (0,0). We physically see it on the graph drawn and there is no “b” value so the y-intercept formula (0,b) changes to (0,0). They both intercept each other at the same y-intercept. This happens a lot. However trying to solve through substitution fails because we do not have a b value to offset the equation. We have to change function notation to “y” because we are solving for y, not inputting any values for x. Use f(x) when you need to input a value, use y when you are solving for y. y= 2x → x =2x We said this looks wrong because how can x =x+x? They are different “x”. Both are 0. 0= 0+0. x/2 =x We tried to clear 2x by dividing by 2 on both sides but got no where. x= x+x changed to ½x + ½x = x. Instead of 1 = 1+1 we get ½ =½+½ .” f(x/2)=x We tried to solve for y using x/2 as an input but it failed. X/2 is not a constant. It is a rational expression. We need a constant to function. A rational expression is a fraction containing a variable namely “x”. Later when you see phrases like “a function must be well defined” it means “x must be a constant”. To define a function x must equal a constant. You can not input a variable for x. f(x/2) = x The only value that satisfies both y = x and y =x/2 is 0. Our answer is (x,2x)=(0,2∙0). That was fundamental and hard. Inputs must be reals, 0/x =0, and using subtraction works best with explicit values not implied ones. Explicit definitions will state a constant, implicit definitions will imply a ratio. y = 2x +1 solve for by changing g(x) = y (x) = 2x+1 substitute x for y since that was what was given from x =y equivalence. –x = 1 what you do one side you have to do on the other to keep equation balanced. x = –1 what you do one side you have to do on the other to keep equation balanced. g(–1) = 2(–1)+1 x = –1 so input it back into the original function g(x) y= –1 y= – 1 so our lines intercept at (–1,–1). This can be written as a string of polynomials. 1. f(x): (2x = y) – (x = y) → 2x–x = y–y → x =0 because 2x–x = x and y–y = 0. 2. g(x): (2x+1=y)–(x+0=y) → 2x+1–x+0 = y–y → x+1 = 0 → x = –1 3. h(x): x = y is pretty much famously given as 0=0 for both its x and y intercepts. This matched f(x). When a function does not have a vertical increase it will always have point (0,0) as it's y and x intercepts. This point (0,0) is always called the origin. With out + or – “b” a function always passes through the origin. Whether you are solving with substitution or subtraction you get x=constant. Substitute the constant back into the original function to solve for y. Abstract word problem might be more helpful: What two integers have a sum of 84 if one is three times the other? n+3n = 84 this is standard form ax+by = c 4n = 84 n = 21 3n = 63 21+63 = 84 What two consecutive integers have a sum 67? 2n+1 = 67 this was our function from above f(x) = 2x+1. 2x is changer to 2n 2n = 66 n = 33 n1 =33 n2 =34 use subscripts to indicate (1st number) + (2nd number) = (Total) 33+34 = 67 Any number times 2 is even ex. 2∙3= 6 17∙2 = 34 113∙2 = 226. As a rule: only way to get an odd is to 2n+1. We were supposed to write as x+(x+1)=67 but I skipped the associative step since this is so basic. What two consecutive odd numbers produce 1763? x(x+2)=1763 x2+2x= 1763 x2+2x –1763 = 0 use quadratic formula since number big or... √1763 ≈ 41.98 try x =41 and x+2 = 43 41∙43 = 1763 41 and 43 are the numbers The product of x = 9 and y is 41 less than the sum of x and y. Solve for y. 9y +41 = x+y 9y +32 =y 8y= – 32 y = –4 What two natural numbers produce 255 and sum to 32? m+n=32 and mn=255 →m =32–n 1+31 = 32 2+30 = 32 3+29 = 32 4+28 = 32 5+27 = 32 6+26 = 32 7+25 = 32 8+24 = 32 9+23 = 32 10+22 = 32 11+21 = 32 12+20 = 32 13+19 = 32 14+18 = 32 15+17 = 32 15∙17= 255 Use quadratic formula or guess 15 times Algebra is about formula not guessing I want to teach process and then get into problems solve n2 –32m+255 or – 255+32n – n2 , both are the same roots for n. 32±√1024 –1020 → 32±2/2 → 15 or 17 speed version shown of (–b±√b2 –4ac)/2a Function notation: f(x) = (x + 1) ·10 is the same thing as Intercept notation: y =10x + 10 The first one is function notation and the second one is y-intercept notation. Line graphs use these notations called formulas to find chart points. A chart point looks like (x,y). The x is the number of units moved to the right on the x axis and y is the number of units moved up the y axis. f(x) = (x+1) · 10 Using function notation we choose 5 for x. f(5) = (5 + 1) · 10 Inputting 5 for x. (Substitution) f(5) = (6) · 10 adding f(5) = 60 solving for f(5) There are no more steps to do so this is solved. Function of 5 equals 60. Our pair is (5,60). That means when x is 5 y is 60. As such f(5) = 60. Remember “the function of x is y”, because f(x) = y . Using y-intercept notation will give the same pair because algebraically it is the same as function notation. You use function notation when you want to find further pairs of (x,y), you use y-intercept notation when you want to find the value of y when x is zero or when want to find the slope. The y value is called the y-intercept if x is zero. Using zero for x will cancel out the x term leaving the constant term as the y-intercept. The coefficient of x is the slope. Slope is the rate of change. You can use slope and rate of change interchangeably. y =10x + 10 Using y-intercept notation y= 10 · (5) + 10 Using 5 for x y= 50 + 10 multiplying y= 60 solving Because y = 60 when x is 5 our pair is (5,60). This means that both formulas are equivalent. In this sense we can consider a formula to be some quantity of x to equal a quantity of y. Therefore a formula is a ratio of x to y. (5,60) is 5x for every 60y or 60y/5x. When we reduce we get a slope of 12/y. This called “ delta of 12y/5x”. In function notation formula: ƒ(x) = Δ12y/5x. Delta is Greek letter  for slope. It means “change”. It is saying “slope is 12y over 5x”. Also “the change in y over x”. We might hear someone say, “delta y over x” or in calculus, “Delta dy/dx” but we hardly use  for anything. It is not an operator so it does change anything. It is not a constant like π nor is it a variable. It just means slope and we use capital “M” for that. If M was being used as a variable someone might use delta to reference slope but slope is always given as y/x. Because ordered pairs come in set notation it is understandable the desire to use associate functions using parentheses. This is common practice but improper. Parentheses for f(x) ties the idea of (x,y) together to imply set notation not multiplication. Slope: M = (y2 – y1)/(x2 –x1) The formula for slope requires two points. Any two points work. If a line passes through the origin we can easily find slope. If it doesn't then using the y intercept is convenient. Any two points that are easy to subtract are convenient. Using y =10x + 10 has 10x so slope is 10. This means for every change 1 in x we have a change of 10 in y. Every point with integer values is one unit to the left and and ten units up. y =10x + 10 find y intercept M = 60-10/5-0 find slope using (0,10) and (5,60) y =10(0) + 10 M = 50/5 using 0 makes finding x easy y = 0+10 M = 10 any two points will have same slope y= 10 y=10 when x=0 (0,10) is y-intercept Slope is the ratio of y value/x value. We had point (5,60) and (0,10) for y intercept. It gave us M = 10. The y intercept form gets its name because the b value is the y value in (0,y). y-int = (0,y) make x=0 to cancel Mx leaving only b x-int = (x,0) make y=0 to cancel y then solve for x 1. y =5x+3 y=5x+3 2. y=0+3 0=5x+3 zeroing out the other variable 3. y=3 (0,3) –3/5 =x (–3/5,0) isolate variable For x we had to do more work. The third step was a compound operation. We subtracted 3 from both sides and then divided both sides by 5 to isolate x. We are mostly interested in y-intercept and not x-intercept. Most graphing is done in the first quadrant where all values are positive. Other quadrants contain negative values which might be hard to interpret or nonsense. Most measurements start at x =0. By definition this point will be the y intercept. For this reason we use the y intercept form a lot. We found slope using M = 60-10/5-0. If we picked the points in a different order we would get the same slope: (0,10) and (5,60) can be 10–60/0–5 = –50/–5 = 10. We can subtract point1 from point2 or subtract point2 from point2 but we can not use values from one point for another point. We can do math in the () as well. 1. (0–5,10–60) = (–5,–50) ⸫ y/x = –50/–5 which is 10 correct 2. (5 –0,60–10) = (5,50) ⸫ y/x = 50/5 which is 10 correct 3. (0–5,60–10) = (–5,50) ⸫ y/x = 50/–5 which is –10 do not mix values with wrong point 4. (5–0,10–60) = (5, –50) ⸫ y/x = –50/5 which is –10 do not mix values with wrong point In the first example both x and y are negative so they cancel leaving a positive 10. Example 2 also has slope 10. Example 3 and 4 are wrong because instead of using point (0,10) for intercept is got confused and used (0,60). This gave both wrong examples a slope of –10. You can have negative slope for a line but not like this. If line moves right and up then slope is positive. ↗ If line moves right and down then slope is negative.↘ x values are always plotted with increasing values to the right while y values can increase or decrease moving ↨. If y increases with x it is a direct variation. If y decreases with x it is a indirect variation. Also called inverse relationship. There are many times the word inverse comes up. The function “y= Mx” states y is a product of the factors M∙x. We multiply slope with x. This make slope “M” the coefficient of x. We had “y =10x+10” where Mx is 10x. In 10x, 10 is the coefficient of x. In general: If x = 0 then y is... If x = 1 then y is... If x = 2 then y is.. We make a table of x and y values and choose specific or random constants for x. An easy way to generate points is to use integers starting at 0 then “+1” every term. This creates many x to quickly solve many y mentally. Since every “step” of x is 1 unit any y≥1 has a more vertical slope. Any y≤1 has a more horizontal slope. The variables compete to see which one is greater or “gaining” This is useful when comparing two graphs or using exponential charts. Direct and Indirect variation 1. If M is positive you have a positive slope ↗ 0°