Preface

This book covers mathematics from high school algebra through fourth-year linear algebra. It blends foundational skills with advanced concepts, moving quickly while keeping explanations clear. We begin with algebra, introduce calculus early, and bring in just enough geometry and trigonometry to cover all essential topics. The goal is to prepare you not only for university placement exams but also for advanced study—up to the equivalent of a master’s level foundation.

Because this is an ambitious undertaking, emphasis is placed on building strong foundations through precise definitions, vocabulary, and cross-linked concepts. Rather than isolating topics, I weave them together so that you see mathematics as one connected whole. Every operation is thoroughly explained with definitions, illustrations, and step-by-step processes. The more you understand how math works, the less you will have to memorize or struggle with later.

My goal is to compress time: four years of high school math in one year, four years of undergraduate math in another—while also introducing abstraction. Exposure to advanced concepts early gives you a sense of how mathematics progresses and encourages creativity in building your own formulas and methods.

Formal logic and proof-writing will be included, but early definitions are written plainly so even younger readers are not overwhelmed. Explanations are intentionally short at first to preserve flow, then gradually grow in rigor. The book is based on my own experience, research, and synthesis of countless resources. If you ever feel you need more practice, I encourage you to pick up a used algebra or calculus book as a supplement—but my intention is to give you a self-contained reference.

In truth, you do not need endless repetition of the same problem. What you need is a clear overview of the mathematics and their relationships to each other. This book will equip you to create equations for real-life situations and solve them with increasing sophistication. You will also gain confidence to explore mathematics beyond this volume, engage with others online, and even contribute your own ideas.

Mathematics has two sides: pure and applied. Pure math defines the rules and logical framework; applied math uses those rules to solve real-world problems. If math is not enriching your life or helping solve problems, it has failed its purpose. Here, concepts are presented in a nontraditional but practical way, always with an eye toward usefulness.

Here are some things to keep in mind:

1. Audience – I wrote this book for everyone, regardless of age or skill level. Only arithmetic (+, −, ×, ÷) is assumed. Younger readers will encounter terms earlier than usual, giving them a head start. Adults will find gaps in their knowledge filled. Even if some material feels advanced, the book remains a valuable reference that prepares you for college and beyond.

2. Format – This book is part formal textbook, part upperclassman’s notes, and part conversation with a peer. Skip what you already know, pause where you need more clarity, and treat it like a guided dialogue. I wrote this at significant personal cost of time and resources because I believe access to complete, clear math instruction should not require endless subscriptions or piecemeal internet searches.

3. Style of Explanation – Definitions begin casually, then grow in rigor. Examples are sometimes long—not to waste time, but to ensure every step is illuminated. I refuse to leave gaps that might cause confusion later, especially when moving from one course or resource to another.

4. Beyond Basics – Many books stop at fundamentals. I take advanced concepts and make them understandable, then show how they apply to real-world situations. Instead of a detached teacher-student tone, I write as someone eager to share insights and clear approaches.

5. Order – We start with simple algebra and quickly incorporate higher-level concepts like functions and Greek notation. The problems are intentionally easy, so that the focus is on understanding the machinery behind the math. This allows you to walk into any course and recognize what to do.

6. Progression – After algebra, we move into calculus and then linear algebra. Once terminology and foundations are in place, math is applied directly to nearly 200 real-world problems.

7. Language – Every math term and symbol is explained thoroughly. If you can read English, you can use this book. Along the way, your reading and writing skills will improve alongside your math.

8. Enrichment – History, anecdotes, and side concepts are sprinkled in to make learning engaging, even when they aren’t strictly necessary for solving problems.

9. Scope – The most advanced fields are beyond this single volume. This book focuses on building foundations so that you can later approach engineering, physics, and other advanced sciences. A sequel will take on those challenges.

You’ll also find glossaries, reference tables, and data sets placed at the front and back of the book for convenience.

Pre-Algebra Main Takeaways

Pre-algebra is an advanced study of arithmetic to prepare for learning algebra and other higher math. The main things to study are:

• Variables
• Operations and their notation
• The use of the numberline
• Pemdas and order of operation
• Set Theory and set notation
• Properties, Laws, and Rules

Variables

Variables are letters that are used to represent unknown numbers. They have two main roles:

• A placeholder for missing values that make a statement true.
• Their use in writing formulas.

Intuition: for missing values

"What is two plus four?"

To answer the question above we could write out " 2 + 4 = ?" using a question mark for the missing answer because we do not know the answer.

If we were asked, "What number plus 3 equals 7?" We could use a question mark again and write:

\[? + 3 = 7\]

Using basic arithmetic we find that 2 + 4 = 6 and 4 + 3 = 7. In algebra instead of using question marks for missing numbers we use letters such as x, y, a, b, and c. We would write "2 + 4 = ?" as "2 + 4 = x". To solve we would perform something similar to this:

\[ \begin{align} 2 + 4 &= x \\ 6 &= x \end{align} \]

Notice that since 2 + 4 = 6 we re-write the problem under the original one and write 6 instead of "2 + 4". This is called "simplifying an equation". Algebra is based on solving problems by re-writing simpler versions of it. Since we are left with " 6 = x " we are finished. That is all there is to it.

When there are no more numbers to add we are done. What is left is one number on one side of the equal sign and one letter by itself on the other. If we have an answer like "6 = x" this is called "solving for x". To solve problems in algebra we often solve for x or another letter. We call the number we get "the solution for x".

How do we really know that 6 = x? What does that even mean?

To prove 6 = x we can use a method called substitution to check our work. To use substitution we "plug" the number we think is our answer in for x in the original equation:

In the end we got " 6 = 6 " which is true. If done correctly you should be left we two numbers; one on the left hand side of the equation and one on the right. If both numbers match then the answer is correct. If they do not match then the answer is wrong. If you have plus or minus signs and more and only two numbers you still have more work to do to simplify the problem.

Intuition: for formulas

A formula is a generic equation that sets the rules for finding answers to a group of related math problems. Area problems use the formulas "length x width", or "base times height":

"What is the area of a garden 4 feet by 8 feet?"

After solving equations do not forget to write the units of an answer. "Units" are what is being counted. For an area of 4 feet by 8 feet the unit label is "square feet" written as 32sqft. or 32ft².

Here we use the letters b,h, and A as variables because they are abbreviations for words that represent distances.

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. Here is a table to show some language and what symbol to use:

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.

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

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: 2√9=3 because 9÷3=3 while 3√27 = 3 because 27÷3÷3 = 3

Sometimes it is important to distinguish what number or symbol you are referring to instead of saying "The number", "squiggle line", or "that thing over there". This is especially important in complex equations that have multiple objects that may refer to the say name. In general math is a binary process where only two numbers are being used at a time. There is the initial value and the number performing the operation. Typically the left number is our "seed" or starting value and the right number is our operating number.

Explaination of Arithmetic notation.

Every equation has 5 parts and they are:

• The starting number
• The operator
• The second number
• The equals sign
• The answer

For addition subtraction multiplication division exponents, radicals:

• [ ] + [ ] = [ ]
• [ ] − [ ] = []
• [ ] × [ ] = [ ]
• [ ] ÷ [ ] = [ ]
• [ ] √ [ ] = [ ]

Which in English are called :

• [sum] + [sum] = [total]
• [sum] − [sum] = [difference]
• [1st factor] × [2nd factor] = [product]
• [numerator] ÷ [denominator] = [quotient]
• [exponent] √ [radicand] = [root]

• [augend] + [addend] = [total]
• [minuend] - [subtrahend] = [difference]
• [f₁] × [f₂] = [product]
• [dividend] ÷ [divisor] = [quotient]
• [index] √ [radicand] = [root]

In math "notation" means the format of how we write our equations. Notation often means the order and symbols used. This can also be called the "syntax" of math. Different operations use different notation. Different branches of math have different syntax.

• Addition: augend + addend = total. In 1 + 2 = 3, 2 is the addend.
• Subtraction: minuend - subtrahend = difference. In 10 - 1 = 9, 1 is the subtrahend.
• Multiplication: factor₁ ∙ factor₁ = product. In 2 ∙ 3 = 6, 2 is the first factor.
• Division: Dividend ÷ Divisor = quotient. In 10 ÷ 2 = 5, 2 is the divisor.
• Exponentiation: Base ^ Exponent = Product
• Radicalization: √Product = Base. In √9 = 3, 9 is the product and 3 is the base.

There are multiple names for the same number being used:

For any operation a value can be called a "number", "value", or a "sum". In addition both augend and addend can be called "addend" or simply "a sum", "sum" or "sums". The result of addition can be called a total or sum. Less common though valid is to call any number a "total", however "sum" is preferred. In any case both "sum" and "total" imply addition or the result of it. The operator for addition uses this symbol " + " and is called "plus sign" or "addition sign".

In subtraction we can say "ten minus one", "one subtracted from ten", "one less than ten", and so on because math allows non fixed use of English phrases due to the highly nuanced nature of the language. This is legal as long as the sentence uttered is logically true. The result of subtraction is called "the difference", "a difference", or even a "total" or "sum". The symbol for subtraction is " - " and is called the subtraction or minus sign.

In arithmetic the raised x symbol " × " is used for multiplication. It is commonly called the "times" symbol. We say "two times three equals six" or "two multiplied by three equals six". In algebra we use the raised dot symbol " ∙ " and language stays the same. Multiplication can also be implied by parentheses or brackets such as 2(3) = 6. Putting variables infront of a number also implies multiplication. When variables are introduced you will see that "2a" = "2 ∙ a".

In division we usually say "ten divided by two equals five". The format has the number to be divided on the left of the division operator followed the number that is doing the dividing. In this case 10 is the dividend (thing which is to be divided), 2 is the divisor (number which is doing the dividing), and 5 is the quotient (number which is the result of the division.) We will learn that there are many symbols for divison such as the obelus " ÷ ", the forward slash " / ", The horizontal fraction bar, the division box, and the pipe symbol " | ". The pipe symbol is very rare and mostly used in computing. It reverses the order of dividen and divisor such that 10 ÷ 2 = 5 becomes 2|10 = 5. This is spoken as " ten divided by two equals five" and "two divides ten = equals five". Pipe is not used in Malone Math courses but is in Malone Computer Science courses.

Fractions and ratios also imply division but they are not quite the same and will be discussed later. Mostly fractions call the dividend a "numerator" and the divisor a "dividend". This closely follows standard division and you may call a fraction a "quotient" such that 1/4 = 0.25 both 1/4 and 0.25 are quotients of "1 ÷ 4".

When a number is being mulitpled by itself over and over we call this a "power" or "exponent" operation. Exponentiation is used to show a number being mulitpled by itself many times. Therefore exponentiation is the way of writing "repeated multiplication of the same number". Example: 2×2×2 = 2³ = 8, 2×2×2×2 = 2⁴ = 16. 2 is the base or root number, the 3 and 4 are called the exponents, and the answers given as 8 and 16 are called the "products". Because the answer to a multiplication problem is called a product we use the same word for the result of an exponent operation. Exponents are also called "powers", which is a bit confusing since the word "power" can be used for the exponent, the answer, and the entire equation! To avoid confusion pick a naming convention and stick with it. Malone Math uses "base", "exponent", and "product" to avoid ambiguity of terms.

Radicals are the result of undoing an exponential equation. Just as division seems harder than multiplcation, radicals are harder than exponents. Since exponents are repeated multiplication, radicals are repeated division. 2×2×2×2 = 16 so 2⁴ = 16, and since 16 ÷ 2 ÷ 2 ÷ 2 = 2 then ⁴√16 = 2.

The Numberline

The numberline is the most important tool in math for counting. It is used to build other counting tools late like the Cartesian plane, unit circle and others. 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 can have as many or as few numbers as you want. Since we can have negative numbers they can be on the line as well. A number line showing numbers from –7 to 7 is shown below:

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”. When we work backwards from the answer to the starting number this is called “working backwards”.

We use the reciprocal operation to work backwards. Starting with the answer 4 use the opposite of “+” sign and instead use “–” sign. We subtract 2 away instead of add 2. This can be shown on a number line:

There is a red dot at 2 and blue dot at 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. Adding and subtraction are inverses of each other.

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:

Interval Notation

You can graph an inequality like you can graph an equation using a number line.3+5 > x is graphed as shown below. Solve the inequality by combining all constants placing a circle at the correct number. The correct number is 8 because 3+5 = 8. Note than is used a “more than sign” but 8>x is really x<8. It is a less than so it points to the left. We state the solution in terms of whether the variable is less or more not the sum.

The solution is all numbers less than “8”. We can write the answer using “interval notation”: (a, b) = {x| a< x< b} 3+5 > x = (–∞,8) in the form (a,b), this means The number is more than –∞ and less than 8. 3+5 ≥ x is rewritten as x ≤ 8. It uses the inequality sign “less than or equal to”. Use a dot instead of a circle.

You can have two inequalities at once called compound inequalities. If x is less than 8 and more than 2: 2 < x < 8, or 8 > x > 2, or (2,8). It is graphed as:

It uses two location marks, one at 2 and one at 8, both are circles because x can not be 2 or 8. x is any number in between 2 and 8. If x could be equal to 8 then we would use “≤”... 2 < x ≤ 8 has a dot at 8:

2 < x ≤ 8 is a line segment that shows a single distance. If we have 8 < x < 2 (also 2 > x > 8) then we have a gap:

If x can be 2 then we need to use a compound sign “less than and equal” ≤ or “more than or equal” ≥:

Now it has a dot at 2 instead of a circle. This can be written 2 ≤ x < 8 or 8 > x ≥ 2, normally signs point to the left. Signs point to the smaller number. We usually right the smaller number on the left and bigger number on the right but not always. This would be written as [2,8) because brackets include the number and parentheses exclude it. –∞ and ∞ always use parentheses. You can have [0,∞), which is all positive numbers, or (–∞,0), all negatives, but you could never write it as [–∞,0] or [0,∞] because infinity is not a number so can not be included since you can never reach it.

• Included values are graphed with a dot and use brackets.
• Excluded values are graphed with a circle and use parentheses.
• Infinities can not be included so always use parentheses.

Pemdas and the Order of Operations

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.

Why does a different order work this time? Because adding and subtracting are reciprocal, that is they share reciprocity. Changing this order does 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 such as 10/2 = 10∙½. To be clear 3–1 does not equal 1–3. This will give additive inverses 2, –2. What is implied is 3 – 1 = –1+3 = 2. This is clarified further under Counting Basics.

Because + and – are inverse operations order doesn't matter. The are considered the same level. Because ∙ and ÷ are inverse operations order doesn't matter. The are considered the same level. But we still have to do ∙ and ÷ before + and – . We state this rule as “Multiplication does not share reciprocity with addition”. The problems occur with three or more terms and you alternate level operations. Consider add and subtract as level one and multiply divide as level two. If different levels of operations are involved then you must follow Pemdas left to right.

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. Four important points:

1. If a number is outside parentheses without an operator you multiply
2. If a number is in front of a radical you multiply
3. Rooting is done during “E” in Pemdas
4. If “ – ” is outside parentheses you flip all signs inside parentheses

Point 1: 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

Point 2:

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. If a number is outside parentheses without an operator you multiply: 3√4 = 3∙√4 = 3(2) = 3∙2 = 6.

Point 3:

There are many rules for exponents. Although Pemdas does not explicitly state when to root it is done during exponents because rooting is the reciprocal operation of powering. In fact rooting uses exponents to determine which root to take be it the square, cube, or higher root. Here is one of the parentheses rules for roots:

3√4 3 = 3(2)3 = 3∙43 = 3(2∙2∙2) = 3(8) = 3∙8 = 24

First find the root. Then if an exponent is outside of the parentheses power the root before doing anything else.

Point 4:

–1∙any number = –that number.

This is true: – 1∙4 = –4 Multiplying by a negative inverts the sign of the number
This is true: –1(7) = –7 The parentheses here imply multiplication
This is true: –(52) = –52 A sign outside parentheses inverts the sign of the number
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 apply it by replacing n the actual number we want to take the inverse of. The formula for additive inverse is: n∙ –1 = –n.

Using letters to represent numbers is called variables. Variables are used to write rules called formulas. Formulas appear in algebra and every branch of math after that.

Counting Basics (variables and set theory)

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 or as a series. A series is a pattern of numbers separated by commas inside parentheses to group them together as a set. Set notation uses brackets [] and () to show what numbers are in a set. The numbers that are in the set are called the domain. Numbers that are not in the set are not in the domain.

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 √. A surd can be considered an irrational square root that has been operated on by another number such as √2 – 1 = 0.414213... . This isolates the irrational part from the integer. We call decimals either decimals or the mantissa.

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. Every set is a superset of the subset before it:

• Set â„•= Natural numbers. Starts at 1 ends at infinity.
• Set 𝕎 = Whole positive numbers The same set as ℤ but 0 is included. W-hole is “With hole” fills in 0 on graph.
• Set ℤ = Integers which called Zahlen in German. This and reals are used in Algebra.
• Set â„š = 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 Ï€ or e.
• Set ℝ = Reals are all numbers except imaginary ones like √–1 or any of the form a+bi.
• Set â„‚ = 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, conics sections, and sequences.

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 a concept not a number. 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 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”. 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 people think 1 is –1 but it isn't. This confusion leads to 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 Adding two positives we move right there is no conflict so no signs to cancel.
• 1–1 Subtracting a positive. There is no conflict so we move left.
• 1+–1 Adding a negative creates conflict. The signs differ so subtract and move left on the number line.
• 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) the last number is positive 1++1 →1+1 = 2
• 1 – 1 = (1) – (1) the last number is positive 1–+1→ 1–1 = 0
• 1+–1 = (1)+(–1) the last number is negative 1+–1→ 1–1 = 0
• 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 demonstrate why and to prove whether a number is positive or negative not just what direction to move on the number line. This ends proof. â– 

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.

[1] The part about writing your problem as an English sentence applies to non math as well. Before you can solve a life problem you need to define it first. Writing makes people smarter and focus more. It stimulates brain growth. Identify your problem, isolate it, then solve it. This works for all math and life problems. Write out your problems identifying what they really are, what causes them, and what steps you can do to solve them. Make a list of possible solutions then be honest and list the pros and cons for each solution. If you do not do this you will not ever change your predicament. If you are not honest it defeats the purpose. The truth does not care about your feelings and telling yourself what you want to hear is lazy and foolish. If you lie you are in denial. Feelings and emotion create bias that clouds judgment. Notice how your feelings prevent you from accepting what is right for you. Be fair and accept how you really feel but do not let that prevent you from making clear and sound decisions. Recognize bias and make sure you can function and treat people reasonably or else you are not really doing math but pseudo math. We use this in ethics and statistics to apply a fair culture of learning and working with the benefit of all members of society.

Inequalities

An equation is a statement that uses equals signs to compare two sums. “2+6 = 8” is an equation. Inequalities are statements that use inequality symbols instead of equation signs to compare two sums. “2<6” is an inequality.

There are four inequality symbols: <, >, ≤, and ≥.
They are read as “less than, greater than, less than or equal, greater than or equal”.

• Eight is less than fifty: 8<50.
• Fifty is more than eight: 50>8.
• Twice a number is at least eight: 2x≥8.
• Eight is at most twice a number: 8≤2x.
• All positive numbers are greater than zero: x>0.
• All negative numbers are less than zero: x<0.

We use > or < to show which number has a bigger value. This is useful when using variables or stating a property law. We use ≥ or ≤ to get as close to a number without being it. This is useful in calculus limits. We use absolute value bars “| |” to indicate positive numbers. This is useful when numbers must be positive like distance. Distance is always positive. If –4 is 4 units to the left of 0 and 4 is 4 units to the right of 0 then the distance is |±4|.

When dividing or multiplying by a negative you change the sign of the number: 10÷ –2 = –5 and –10 ÷ –2 = 5
When dividing an inequality flip the inequality sign if divisor is a negative number:

(–6 < 10) ÷ –2 = 3 > –5 3 is greater than –5. signs changed we have to “flip” the inequality sign.

(2x ≥ y)/ –2 = x ≤ ½y Plugging in values for x and y will show that flipping the sign is required.

Fractions

A fraction is a division problem written as a single number. Writing numbers this way is called fractional notation. It has two parts the top number and the bottom number. The top number is to be divided by the bottom number. We end up writing the division problem as a single fraction.

1÷2 = ½ 2÷3 = ⅔ 2÷5= ⅖

The top number is called the numerator, the bottom number is called the denominator. The top number is also called the dividend, the bottom number is also called the divisor. The result of this division is called a fraction, or a quotient.

The numerator is a product of the denominator and the quotient.

1÷2 = 0.5 = ½ 2 ∙ 0.5 = 2 ∙ ½ = 2/2 = 1 2÷3 = 0.6 = ⅔ 3 ∙ 0.6 = 3 ∙ ⅔ = 6/3 = 2 2÷5 = 0.4 = ⅖ 5 ∙ 0.4 = 5 ∙ ⅖ = 10/5 = 2

The numerator divided by the denominator equals a fraction. The dividend divided by a divisor equals a quotient.

We can write any number as itself divided by 1: 1÷1 = 1 10÷1 = 10 5÷1 = 5 Any number divided by 1 is itself: a÷1 = a 1/1 = 1 10/1 = 10 5/1 = 5

1÷1 = 1/1 = 1 10÷10 = 10/10 = 1 5÷5 = 5/5 = 1

In multiplication two numbers called factors multiply to produce a product. A product is the result of multiplication. We say multiplication it the inverse operation of division. Inverse means to reverse the operation and get the original number we started with. If I divide 1 by 2 I will get “one-half”. If I take one-half and multiply it by the number I used to divide I will get 1 back. 1 was my original number. This is called a reciprocal operation.

Any number divided by itself equals 1, (except 0 because we can not divide by 0). When a number is divided by 0 we say the answer is “undefined”. This means the answer is unknowable. Using variables to represent any number we write these laws for fractions:

1. Any non-zero number a divided by itself is 1: a/a = 1. 2. Any reciprocal operation equals 1: a/1 ∙ 1/a = a∙1/1∙a = a/a = 1. 3. Any number a divided 0 is undefinable: a÷0 = a/0 = undefined or “und.” for short.

We also say the answer “does not exist” or dne for short because you can not divide by 0 and get a real number answer. We prove this later.

• To add or subtract fractions you need to have a common denominator.
• To multiply fractions you write the product of the numerators over the product of the denominators.
• To divide fractions you multiply the first fraction with the reciprocal of the second fraction.
• An identity of a number is an operation that does not change the value of the number like multiplying by 1 or adding 0.

Multiplicative identity of 1: a∙1 = a a÷1 = a Additive identity of 0: a+0=a a – 0 = a

We can perform addition, subtraction, multiplication, and division of two fractions. Multiplication is the easiest:

The parentheses are grouping symbols telling what order to perform operations. This is covered during pemdas. Algebra uses the multiplication dot called center dot or interpunct instead of the multiplication cross arithmetic uses. We combine the numerators on top and denominators on the bottom and multiply. To multiply is to find a product. We say “we take the product of numerators and divide by the product of the denominators”. Since a fraction is a division problem you should know the different ways of writing them and names of their symbols. 1÷2 = 1/2 = ½ . Even though 1/2 and ½ both have forward slash consider ½ to have a fraction bar since the numbers are smaller and the numerator is raised above the slash.

1. The fraction in the illustration use a fraction bar “ – ” to indicate division.
2. On a computer we normally use forward slash “ / ” to indicate division. 1/2 = ½ .
3. The division symbol “ ÷ ” is called obelus. It is a Latin word.

Multiplication of fractions

For any two fraction a/b and c/d, a/b ∙ c/d = ac/bd. The numerator is the product of the two numerators and the denominator is the product of the two denominators. Multiply both numerators to make a new numerator, then multiply both denominators to make a new denominator.

Multiply by 1 gives that same number: ⅘ ∙ 5/5 = 4∙5/5∙5 = 20/25 . Since 5/5 = 1 all we did was multiply ⅘ by 1.

Notice that when multiplying we combine the two fractions using 1 fraction bar. Doing this means writing two multiplication dots instead of 1. This is a big deal. We changed the way our equation looked. Changing the way it looks is called “algebraic manipulation”. Changing the way it looks makes it easier to solve, it gets us started in the right direction.

Two factors produce a product. a÷b = ab, here we have 4∙5 = 20 and 5∙5 = 25. We say the factors of 20 are 4 and 5 and that the factors of 25 are 5 and 5.

Since we multiplied â…˜ by 1 and got 20/25 then â…˜ = 20/25. â…˜ and 20/25 are the same fraction therefore the same number.

We can simply a fraction by reducing it to it's simplest form. It works like this:

20÷5 /25÷5 = 4∙(5÷5)/5∙(5÷5) 4∙(5÷5)/5∙(5÷5) = 4∙(1)/5∙(1) = 4∙1/5∙1= 4/5

We can manipulate fractions this way to cancel factors that appear in the number and the denominator. If a factor is in both the top and bottom then the are a fraction equal to 1 and are said to cancel. Here are some different ways of writing this:

When we cancel factors in a fraction we are reducing the fraction to its simplest form. Simplest form for a fraction removes fractions equal to 1.

½ ∙1 = ½ ½ ∙2 = 2/4 ½ ∙3 = 3/6 ½ ∙10 = 10/20

2/4 , 3/6 , and 10/20 all equal ½ . They are multiples of ½ made by multiplying ½ by a fraction equal to 1. The fractions equal to 1 used were 1/1, 2/2, 3/3, and 10/10.

½ is the simplest form of these multiples of ½. To reduce a fraction to simplest terms we divide the top and bottom by the same factor. This factor is called the greatest common factor or GCF.

10/20 The greatest factor that is common to both 10 and 20 is 10.

10/20 = 1∙10/2∙10 and then we separate into two fractions ½∙ 10/10 and then we write 10/10 as 1. Observe: 10/20 = 1∙10/2∙10 = ½ ∙10/10 = ½∙1 = ½. We are left with ½ as the simplest form of 10/20. Canceling the other fractions using their gcf's of 2 for 2/4 and 3 for 3/6 yields similar results.

The original picture had a factor of 2 in the top and bottom. We can reduce the fraction by canceling 2/2 to 1.

1/2∙ 2/3 = 1∙2/2∙3 = 1∙2/2∙3 = 1/3 ∙ 2/2 = 1/3 ∙1 = 1/3. There are trickier ways to write this where you can swap denominators of both fraction and get same result. This is advanced: ½ ∙ ⅔ = ⅓ ∙2/2.

Because multiplication is commutative we can switch the order of numbers to be multiplied: 2∙3 = 3∙2. ½ ∙⅔ = ⅓∙2/2 the denominators have switched places. We really did this:

1/2∙2/3 = 1∙2/2∙3 = 1∙2/3∙2. Writing this using fraction notation we get: ½ ∙⅔ = 1∙2/2∙3 = 1∙2/3∙2. We changed the order of the factors in the second fractions denominator. We have 1∙2/2∙3 = 1∙2/3∙2. When we separate 1∙2/2∙3 it becomes ½ ∙ ⅔. When we separate 1∙2/3∙2 it becomes ⅓ ∙ 2/2.

1. When multiplying fractions combine all fractions by fraction using a single fraction bar. 2. Write all terms as being multiplied by each other

Canceling

There are two types of canceling: Additive and multiplicative. Additive cancellation is when you start at a number then add or subtract a number and to cancel you add it's additive inverse. Multiplicative cancellation is when you reduce a fraction by removing factors that as a fraction equal 1/1 = 1.

3+7 =10 I start with 3 and add a number “7” which results in a sum of “10”. 3+7 – 7 = 3 I cancel “+7” with “ – 7”. The operation is canceled and I have my original number

3–7 = –4 I start with 3 and subtract a number “7” which results in a difference of “ –4”. 3–7+7 = 3 I cancel “–7” with “ +7”. The operation is canceled and I have my original number 3

An additive inverse of a number is the number with the opposite value sign. 3 and –3 are additive inverses. The definition of an additive inverse is two numbers that sum to 0. By definition two additive inverses cancel each other under addition. “Under addition” means addition the operation we are using.

3 + (–7) = –4 We are adding –7. 3 + (–7) = 3 – 7 Adding a negative is the same as subtracting a positive. 3 + (–7) +7 = 3 We cancel the addition of –7 by adding 7 (or by subtracting –7).

3 – (–7) = 10 We are subtracting –7. 3 – (–7) = 3 + 7 Subtracting a negative is the same as adding a positive. 3 – (–7) –7 = 3 we cancel the subtraction of –7 by subtracting 7 (or by adding –7).

15/40 I have the fraction 15/40. 3∙5/2∙2∙2∙5 I find the prime factors of 15 and 40. This is known as the prime factorization. 3∙5/2∙2∙2∙5 I cross out or “strike through” factors when they matching the top and bottom. 3/2∙2∙2 I write my factors with the canceled factors removed. 3/8 = ⅜ I write my final result in fraction notation.

15/40 A fraction not in simplest form. 3∙5/2∙2∙2∙5 Writing numerator and denominator as the products of their prime factors. 3∙5/2∙2∙2∙5 Removing pairs of matching factors. 3/2∙2∙2 Writing the factors without the canceled factors. ⅜ Simplifying by performing all operations until none are left.

The fraction 3/2∙2∙2 still has operations left. We still have to multiply the factors in the denominator. We need to find the product of these factors so we can write it as a single number. Once we have found the product and written ⅜ there are no more operations and the fraction is as reduced as I can make it. Something in simplest form is when there are no more operations to perform and all fractions are fully reduced.

A prime factorization is the unique set of factors for an individual number. Every number has a set of factors that is unique to that number. Every number has a prime set of factors, no two numbers have the same unique set of factors. Prime factorization is used when you have trouble find the greatest common factor. If you know the gcf then using it to cancel is faster. By the way even numbers or whole numbers are called integers.

15/40 What is the largest factor that 15 and 40 both have? 15/40 What is the largest number that divides 15 and 40 and gives an even number? 15÷5/40÷5 The largest number is 5. 3/8 = ⅜ Dividing by 5 will give even numbers. We call this “ integer solutions”.

Fractions should have integers for the top and bottom. If they don't you have a complex fraction like 2 = 1/½ or 1 = ½/½. To avoid complex fraction find the gcf or use prime factorization. Complex fractions are used for rational expressions or ratios. This is covered after polynomials where rational expressions look like polynomials but are not. If you have a complex fraction clear the fractions or decimals to get an integer for the top and bottom. ½÷½ = 1 because a number divided by itself is 1. This applies to fractions, decimals, or variables.

½÷½ = 2(½÷½) = 2∙½ ÷ 2∙½ = 2/2 ÷ 2/2 = 1÷1 = 1 ½÷½ = 2(½) ÷ 2(½) = 1

0.5÷0.5 = 2(0.5÷0.5) = 2∙0.5 ÷ 2∙0.5 = = 1÷1 = 1 0.5÷0.5 = 2(0.5) ÷ 2(0.5) = 1

1.5/3 The numerator contains a decimal 0.5 which as a fraction is ½. 1.5/3 Multiplying what to ½ will equal an integer? 2∙ ½ = 1. 2/2∙ (1.5/3) Multiply the top and bottom by 2 using 2/2. 2/2 ∙ (1.5/3) 2∙1.5 = 3 and 2∙3 = 6 so we have 3/6 which is 3÷6 which is ½.

2/2 ∙ (1.5/3) 2∙1.5/2∙3 Combining fractions as one fraction. 3/6 Finding products. 3÷3/6÷3 Factoring a 3 out of 3 and 6 because 3 and 6 have a gcf of 3. ½ 3÷3 = 1 and 6÷3 = 2 so we have 1∙2 which is ½.

There are shortcuts we instinctively learn to use later. We have many options for writing equations because some forms are easier than others depending on the circumstances. 1.5/3 had a decimal in the numerator. This makes it a complex fraction where we have a fraction in a fraction. Fractions are not whole numbers so they are not integers. Normally we want integers. Having a fraction implies division and so this is not the simplest form because we have the operation of division to do.

Addition of fractions

Because 1.5 = 1+½, the fraction 1.5/3 can be written messy as (1+½)÷(3) or 1+½/3. We can separate this into two fractions.

1.5/3 We have a complex fraction. 1+½/3 Splitting 1.5 into its whole and fractional parts of 1 and 0.5 written as 1 and ½. 1/3 + ½/3 or 1/3 + 0.5/3 We can write ½ ÷3 as ½/3 or 0.5/3. 1/3 + 0.5/3 For this “ 1/3 + 0.5/3 ” looks better than “ ½/3 ” because it is easier to read. 1/3 + 2/2( 0.5/3) We have to multiply the top and bottom by 2 so we multiply by 2/2. 1/3 + 2∙0.5/2∙3 When multiplying write as one fraction to combine the operations. 1/3 + 1/6 We are adding two fractions. To add we need a common denominator. 1/3+1/6 = 2/2 ∙ 1/3 + 1/6 This is hard to read because we are not using fraction notation. 1/3+1/6 = 2/2 ∙ 1/3 + 1/6 This is easier to read. 2/2 ∙ 1/3 + 1/6 To add we scale ⅓ by 2/2 to get 2/6. One third equals two sixths so value is the same. 2/6 + 1/6 To add fraction with the same denominator find the sum of the numerators. 3/6 2+1 = 3 so we have 3/6. ½ 3/6 reduces to ½ when you factor out the gcf of 3. 3/3 = 1 and 3/6 =2 we have 1/2.

This was a easy example made more complicated by bad notation. The next examples with be from first basics. Fractions are parts of a whole number of something. When we take something and divide it into equal parts each part is an equal fraction of each other. If I have a piece of candy and divide it into two equal parts I have two halves. That is I have 2 parts that are each ½. If I add the parts I should get the original number of whole pieces of candy I had: ½ + ½ = 1 whole piece of candy.

This is the difference between a fraction and a piece. A piece is 1 whole piece and a fraction is a fraction of that whole piece. Pieces are not fractions but rather 100% of something. When we have “a piece of a piece” we call that a fraction.

½ + ½ =1 This says that ½ of 1 piece plus ½ of 1 piece equals 1 whole piece.

½ of a piece plus ½ of a piece = 1 whole piece. We could say ½ a candy + ½ a candy = 1 candy. I have a chocolate candy bar that is 4 squares each the same size. Each square would then be ¼ of a whole bar. I have an orange and it has 8 equal slices. Each slice is ⅛ of a whole orange. If I eat 2 slices I ate 2/8 of a whole orange.

â…›+â…› = 1+1/8 = 2/8 Add the numerators. The denominator stays the same.

If I ate 3 squares of the candy bar then:

¼ + ¼ + ¼ = ¾ I ate ¾ of the whole bar. 1 – ¾ = ¼ I have ¼ of the bar left. 4/4 – ¾ = ¼ I started with 4 out of 4 squares, then ate 3 out of 4, and now have 1 out of 4 left.

The numerator is how many we have and the denominator is how many it takes to have a whole piece.

If I eat 1 square then I ate ¼ of the bar. If I eat 2 squares then I ate 2/4 of the bar. 2/4 reduces to ½ or the same as saying ½ of the bar. If I eat 3 squares then I ate ¾ of the bar. If I eat 4 squares then I ate 4/4 of the bar. I ate the entire bar. 4/4 = 1 so I ate 1 whole bar.

If I eat 2 out of 8 orange slice then I ate 2/8 of the orange. This reduces to ¼. Really I ate ¼ of the entire orange. If I eat 7 orange slice then ⅛+⅛+⅛+⅛+⅛+⅛+⅛=⅞. I ate 7 out of 8 which as a fraction is ⅞. To add these fractions we check to make sure we have a common denominator. A common denominator means that all fractions have the same denominator. All fractions have the same denominator so we can add these fractions by simply adding the top numbers: ⅛+⅛+⅛+⅛+⅛+⅛+⅛= (1+1+1+1+1+1+1)÷8 = (1+1+1+1+1+1+1)/8 = ⅛.

Repeated addition is called multiplication. We can multiply instead of adding here since we are adding the same number over and over. ⅞ = (1+1+1+1+1+1+1)÷8 = 7(1)÷8 = 7÷8 = ⅞.

Writing (1+1+1+1+1+1+1) as 7(1) is called factoring. We factored “1” out seven times so we have 7 groups of “1”. We write this as 7∙1 or 7(1). Parentheses implies we need to multiply if there is no other operation sign outside the ().

Factoring is a form of division. Here are the gory details:

1+1+1+1+1+1+1 (1+1+1+1+1+1+1) (1÷1+1÷1+1÷1+1÷1+1÷1+1÷1+1÷1) or (1/1+1/1+1/1+1/1+1/1+1/1+1/1) or (1/1+1/1+1/1+1/1+1/1+1/1+1/1) = 7(1)

We count how many “1's” are in the () and then write that number outside the () and write 1 inside the () only once.

More Factoring

2+4+6+8+10 All of these numbers are multiples of 2. 2 is the least common multiple or LCM. (2+4+6+8+10) Please group this series of numbers with (). 2(2/2+4/2+6/2+8/2+10/2) Please write the LCM outside () and divide every number by the LCM 2(2/2+4/2+6/2+8/2+10/2) This writes the same thing above but in fraction notation. 2(1+2+3+4+5) Simply fractions. 2(15) Sum terms inside (). 30 Find product of 2∙15 to be 30.

If we sum the series we still get 30:

2+4+6+8+10 6+6+8+10 12+8+10 20+10 30

Sometimes factoring is faster for harder problems or allows us to cancel or do something else. Notice that we divided every number, called a term, by the factor we were using? To factor find a common factor of all the terms in the (). If they have a common factor other than 1 you can factor that out by dividing every term by that factor.

⅛+⅛+⅛+⅛+⅛+⅛+⅛ Write our sum. (⅛+⅛+⅛+⅛+⅛+⅛+⅛) Write it in (). (⅛÷⅛ + ⅛÷⅛ + ⅛÷⅛ + ⅛÷⅛ + ⅛÷⅛ + ⅛÷⅛ + ⅛÷⅛) Divide every term by ⅛. ⅛(1+1+1+1+1+1+1) This means we factored out ⅛ out of every term. ⅛(7) Whats in the () equals 7 (⅛+⅛+⅛+⅛+⅛+⅛+⅛) Our sum. 7(⅛) This says we have 7 factors of ⅛.

More Adding Fractions

Find a common denominator to add fractions:

⅖ + ¾ These have different denominators. (4/4 ∙ ⅖ ) + (5/5 ∙ ¾) This creates a common denominator by cross multiplying. (8/20) + (15/20) We now have a common denominator of 20. 23/20 We add 8+15 = 23.

Cross multiplying multiples the the top and bottom of the first fraction with the second fractions denominator. Cross multiplying multiples the the top and bottom of the second fraction with the first fractions denominator.

a/b + c/d = ad/bd + bc/bd = ad+bc/bd a/b + c/d = ad/bd + bc/bd = ad+bc/bd

We cross multiply when we struggle to find a lcm. Other fractions are easy to find a lcm so we do not need to cross multiply. If we denominators are multiples of each other then we only multiply the smaller denominator to math the bigger one.

Examples of adding fractions

4/8 + 5/8 = 9/8 Add numerators since denominators are same.

2/5 + 5/10 5 and 10 are multiples of 5 so use 10 as the lcm. Multiply the first fraction by 2/2. 4/10 + 5/10 = 9/10 2/2 ∙ 2/5 = 2∙2/2∙5 = 4/10. 2/5 + 5/8 The LCM of 45 and 8 is 40. 16/40 + 25/40 Cross multiply to find common denominator since 5 and 8 are not multiples. 31/40 31/40 is simplest form because 31 and 40 do not share any factors to factor out.

24/100 + 3/50 Find lcm. 12/50 + 3/50 Use 50 as lcm by factoring 2 out of 24 and 100. 24÷2/100÷2 = 12/50. 15/50 Check for gcf of 15 and 50. The gcf is 5. Factor 5 out of 15 and and 50. 3/10 This is the simplest form.

Subtracting fractions

Same as adding except subtract numerators instead of adding them. Use same methods to combine fractions.

1. Check to see if denominators are the same.
2. If they are same subtract numerators and write difference as the new numerator over given denominator.
3. LCM stands for least common multiple and GCF stands for greatest common factor.
4. If denominators are not the same find LCM and write one of the fractions so that you can subtract.
5. If denominators are not multiples of each other then the LCM will be found by cross multiplying.
6. Once denominators are the same and you have subtracted reduce the fraction by factoring out the GFC.

7/8 – 3/8 = 4/8 = ½ 4/8 reduces to ½. 4 and 8 share a factor of 2 that cancels.

4/8 We have 4/8. 2∙2/2∙4 Writing as its factors shows that the greatest common factor is 2. 2/2 ∙2/4 We separate the fraction into two fractions to treat 2/2 separately. 1∙2/4 2/2 is said to “ cancel “ to 1. We rewrite 2/2 as 1. 2/4 Multiplying by 1 cancels since multiplying by 1 does not change the number. 2∙1/2∙2 Writing as its factors shows that 2 and 4 have a gcf of 2. 2/2 ∙1/2 Separate the two fractions. Rewrite the cancellation as the gcf over itself. 1∙ ½ Write the cancellation as 1. ½ Multiplying by 1 cancels the factor 1. ½ is the simplest form of 4/8.

Obviously there are steps that can be combined or skipped. This shows every pro45cess to completion so there are no questions about a particular step. For simple problems basic steps are combined. For harder problems it helps to be more cautious and write steps out more fully as needed. We could have simply said 4/8 = ½ and been done.

3/8 – 7/8 = –4/8 = –4/8 = –½ 3 – 7 = – 4.

When you have a negative fraction write the negative sign in front of the fraction and not in the numerator nor denominator. Some formulas have subtraction performed in either the numerator or denominator. When a formula does many things to create a fraction we tend to call it a quotient. Famously the formula for slope of a linear equation can have a negative denominator if written backwards. The numerator would be positive but the denominator is negative. We move the negative sign to the front of the fraction to quote the entire fraction as negative not just the top or bottom. The word quotient means divide and comes from the word “ quote”.

If we have a quotient of –3/7 then: –3/7 = 3/–7 = – 3/7 Move the sign to the front. Here is what is going on: Dividing a positive by a negative or a negative by a positive gives a negative result.

–10/2 = –5 because: –10 ∙ 2 = –5 10/–2 = –5 because: 10 ÷ –2 = –5

Either way we get –5 or – 5/1 .

Remember that any whole number (or fraction) can be written as itself over 1 because any number divided by 1 is itself. This is the called the division identity fo 1 where n ÷ 1 = n.

2 = 2/1 3 = 3/1 2+1 = 2+1/1 5∙7 = 5∙7/1 x = x/1 2x+y –7 = 2x+y –7/1

Indeed even equations can be written as themselves divided by 1. For any two fractions a/d and b/d: a/d + b/d = a+b/d and a/d – b/d = a – b/d. If denominators are unlike multiply one fraction by the lcm/lcm so that you have two common denominators. If denominators are not multiples of each other the lcm can be found by cross multiplying: a/b + c/d = ad+bc/bd.

a/b+c/d = d/d∙a/b + b/b∙c/d = d∙a/d∙b +b∙c/b∙d = da/db +bc/bd = ad/bd +bc/bd = ad+bc/bd

a/b+c/d We have two fractions to be added with different denominators. d/d∙a/b + b/b∙c/d We cross multiply to find the LCM to use as a denominator for both fractions. d∙a/d∙b +b∙c/b∙d We combine fractions like this if we want to show our steps more fully. da/db +bc/bd Writing variables without operation signs between them means they are being multiplied. ad/bd +bc/bd Reorder the variables in alphabetical order. ad+bc/bd The final answer. If we had numbers instead of variables we would combine numerators.

Division of fractions

To divide two fractions multiply by the reciprocal of the divisor: a/b ÷ c/d = a/b ∙ d/c = ad/bc.

½ ÷⅔ = ½ ∙ 3/2 = ¾ ⅔ ÷ ⅗ = ⅔ ∙ 5/3 = 10/9 ⅔ ÷ ¼ = ⅔ ∙4/1 = ⅔ ∙4 = 8/3 10÷⅕ = 10∙5 = 50

When dividing by a fraction the rule used says to “flip and multiply”. We say “flip” to mean flip the numerator and denominator of the dividing fraction. Flipping means to swap the top number with the bottom number. Flipping a fraction gives you the reciprocal of that fraction. The reciprocal of a whole number is a fraction with that number on the bottom and 1 as the numerator as this is the opposite of a whole number which can be written as that number over 1 as the denominator.

6÷7/3 = 6∙3/7 = 6/1 ∙ 3/7 = 6∙3/1∙7 = 18/7 This show some steps which can be eliminated once you know the rules. 6÷7/3 = (6∙3)÷7 = 18÷7 = 18/7 This shows a sloppy way of doing it but is valid. 6÷7/3 = 6∙3/7 = 18/3 This is the fastest and preferred method. Too many steps confuse beginners.

10÷2 = 10∙½ = 5 Dividing by a whole number is the same as multiplying by its reciprocal. 10÷½ = 10∙2 = 20 Dividing by a fraction is the same as multiplying by its reciprocal.

⅔÷¼ = ⅔∙4/1 = 8/3 Dividing by a fraction is the same as multiplying by its reciprocal. ⅔÷4 = ⅔∙¼ = 2/12 = 1/6 Dividing by a whole number is the same as multiplying by its reciprocal. ⅔÷4 = ⅔∙¼ = 2/3∙4 = 1/6 Note that dividing by 4 is the same as multiplying the denominator by 4.

Dividing a fraction by a number is the same as multiplying the fraction's denominator by that number.

⅔÷ 4 = 2÷(3∙4) = 2÷12 = 1/6 = 1/6 ⅔÷4 We are dividing by 4... ⅔÷4 ... ⅔∙¼ ... 2/3∙4 … so we are multiplying the denominator of ⅔ by 4. 2/12 1/6

⅔÷¼ = 2÷(3∙¼) = 2÷¾ = 8/3 ⅔÷¼ We are diving by ¼. ⅔∙4/1 Flip and multiply... 8/3 Also we could...

⅔÷¼ We are diving by ¼... 2/3∙¼ ...so we are multiplying the denominator by ¼. 2/¾ The denominator is a fraction so the entire term is a complex fraction. 2÷¾ We are dividing by a fraction so we need to flip and multiply. 2∙4/3 Multiply 2∙4 to get a numerator of 8. 8/3

We do not want complex fractions. Flip and multiply avoids them. If you have a complex fraction you need to know how to solve it. These complicated forms are to be avoided if possible. If you have a complex fraction use these methods as shown to eliminate them.

Fractions plus, minus, times, or divided with a whole number

7+½ = 7.5 We can convert ½ to 0.5 and just add as decimals or... 7+½ 14/2+ ½ ...we can convert 7 to a fraction with a like denominator as ½. 15/2 15÷2 = 7.5

Since ½ has a denominator of 2 we multiply 7/1 ∙2/2 = 14/2 which equals 7 because 14∙2 = 7. Multiplying something by a fraction that is equal to 1 does not change the value of the number but changes the form. When we have a whole number and need to perform operations on it with a fraction we convert the number into a fraction by writing the number over 1. All whole numbers are fractions that have a denominator of 1.

4+5/23 = 4∙23/1∙23 + 5/23 = 96/23 + 5/23 = 101/23

101/23 is in simplest form but is an improper fraction. We can convert it to a mixed fraction if we like. 101/23 = 23+23+23+23+9/23 = 4+9/23 we can pull 23 out of 101 4 times and be left with a remainder of 9.

When multiplying a whole number to a fraction we can usually do this mentally. The denominator stays the same and the whole number and the fractions numerator multiply to produce a new number. If the fraction has 1 for a numerator then the whole number becomes the numerator. Just move whole number up top.

4∙½ = 4/2 = 2 4∙⅓ = 4/3 4∙12/5 = 48/5 x∙3/14 = 3x/14 3∙x/4 = 3x/4 7∙x/y = 7x/y

Let x be any whole number and p/q be any fraction. If you do something with a fraction then x is written like x/1.

Addition: x + p/q = x/1 + p/q = q/q ∙ x/1 + p/q = qx/q + p/q = qx+p/q Subtraction: x – p/q = x/1 – p/q = q/q ∙ x/1 – p/q = qx/q – p/q = qx – p/q Multiplication: x ∙ p/q = x/1 ∙ p/q = px/q Division: x ÷ p/q = x/1 ÷ p/q = x/1 ∙ q/p = qx/q

Make x, p, q and numbers and follow the formulas listed to solve any arithmetic operation. Multiply by the LCM to clear decimals to change complex fractions to regular fractions. Any number divided by itself is 1: x÷x = x/x = 1.

Multiplying by 1 as a fraction of lcm/lcm finds equivalent fractions that are scaled multiples of the original. Scaled multiples of a fraction equal the original and thus do not change the value, we say “ multiplying by 1 does not change the value”. Use lcm/lcm to scale a fraction to find a common denominator so that fractions can be added. If x/y + x/5y then to add multiply x/y with 5/5 to get 5x/5y + x/5y so that denominators are the same. lcm/lcm can be used here because y and 2y are multiples of y. If x/y + x/z then lcm/lcm will not work because y and z are not multiples of each other. To find the lowest common multiple you need to cross-multiply. This will find a common denominator as yz. x/y + x/z = xy/yz + xz/yz = 2xyz/yz .

Canceling only works by reducing fractions: 48x/120xy = 2∙2∙3∙4∙x/2∙3∙4∙5∙x∙y = 2/5y Terms do not cancel under addition: x+2/y+2 ≠ x/y. Because x+2 is (x+2) “the entire sum is the numerator”... you can not cancel addition with division since they are not reciprocal operations. x+3/3 ≠ x. The constants do not cancel because they are not factors. For division only factors cancel not added or subtracted parts.

Fraction notation Division notation using obelus Division notation using slash Result ½/½ ½ ÷ ½ ½ / ½ 1 0.5/0.5 0.5÷0.5 0.5/0.5 1 x/x x÷x x/x 1 ½ 1÷2 1/2 0.5 2/7 2÷7 2/7 0.285714 2x/4y 2x÷4y 2x/4y x/2y

Standard math does not use this symbols but are included for reference. The x and y are inverted: Fraction notation Division notation using backslash Division notation using vertical bar Result ½/½ ½ \ ½ ½ | ½ 1 0.5/0.5 0.5\0.5 0.5|0.5 1 x/x x\x x|x 1 ½ 2\1 2|1 0.5 2/7 7\2 7|2 0.285714 2x/4y 4y\2x 4y|2x x/2y

Percentages

Balancing an Equation

Properties

The Property Laws use letters like a or b or c to mean any three numbers that are not the same. Using letters in place of numbers is called using variables. Variables are very important in used in every algebra problem. In fact variables is what separates algebra from arithmetic. These laws are taken directly from the glossary at the beginning of the book.

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. A number times 1 is itself.

Multiplicative property of zero: For any number a, a·0 = 0. This states number times zero is zero.

Division of zero: Division by zero is undefined. It is undefined since dividing by zero does not give us a singular value but potentially infinite values. Whereas “definition of a number” means that a number is unique, “undefined” means such a unique number either does not exist or a domain error has occurred.

Properties of equality: The following properties are true for any a, b, or c.

• Reflexive: a = a. This is similar to the identity of a number.
• Symmetric: if a = b then b = a. This just restates substitutive property twice.
• Transitive: if a = b and b = c then a = c. Two numbers are equal if they both equal a third number
• Substitutive: if a = b then a may be replaced by b. This allows known values to replace given variables.

Distributive property: For any numbers a, b, c:

a (b + c) = ab + ac and (b + c) a = ba + ca
a (b – c) = ab – ac 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 in front 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.

Addition principle: If a = b then a+c = b+c. This says if we add something to one side of an equation we must must add same amount on the other. This is main concept of algebra. What follows is any operation done on one side must be done on the other. The addition principle is shorthand for all these other instances of balancing an equation.

Exponent Laws

• 30 = 1 Any numbers raised to zero equals 1.
• 31 = 3 Any number raised by 1 equals itself.
• 32 = 9 A number squared is itself times itself. 3·3 = 9. There are two base factors.
• 33 = 27 A number cubed is three factors of itself multiplied.

• 31 = 3 The exponent tells how many bases to multiply.
• 32 = 3 · 3
• 33 = 3 · 3 ·3
• 34 = 3 · 3 · 3 · 3

Sometimes the exponent is blank and has a empty box to fill in or nth for the exponent. 2nth means number like “1st 2nd …nth” . We are saying a number in a series. If 34 = bnth and plug in b=3 and nth= 4.

Formula for Powers:

“bnth = b raised this number of times” where b is the base and nth is the power. Write b nth times and multiply to find the power of b. Power can mean either the exponent or value of b raised to the power of nth.

Example: If 34 = 81 then 4 is “the power 3 is raised to” or “the exponent of 3 is 4” . 81 is “the power of 34”.

Using power to describe both the result and exponent is confusing; nonetheless:

(23 · 3) = 3·2·2·2= 24 because (3,6,12,24) are multiples of 3. A multiple is repeated addition of a base number.
33 = 3·3·3·3= 81 because (3,9,27,81) are powers of 3.
Powers are repeated multiplication

Power can mean the base, exponent, entire term, or the total answer. This is ambiguous but how it is. When multiples of a number are the result of a power operation they are called powers of the base. Sometimes we say things like “100 is the square of 10” but we also say “100 is the second power of 10”. We say 9 is a power of 3, 27 is a power of 3, 81 is a power of 3. My preference for this text:

34 = 81 where the base is 3, the exponent is 4, the exponential term is 34, and the power or product is 81.

Reciprocal Law

For any non zero number a, a/1 · 1/a = 1 and -a, -a/1 · -1/a = 1. A reciprocal the number that multiplies to 1. Very generally if a number is whole then it can be written as number/1. Five can be written as 5, 5/1, or 5/1. Whole number are called integers. Integers can be written as fractions by writing them with a denominator of 1. Since this means divide by 1 we do not change the value because any number divided by 1 equals itself. The reciprocal of an integer always has a numerator of 1 since an integer always has a denominator of 1. We can say ½ is the reciprocal of 2 and 2 is the reciprocal of ½. We are flipping the top and bottom numbers.

Fractions are called rationals. They are not integers so not whole numbers. The have numbers different than 1 in them. Fractions have the form numerator/denominator. The reciprocal of a fraction is denominator/numerator.

The reciprocal of 2/3 is 3/2. This can be generalized to a/b becomes b/a. To generalize means to write a formula using variables to represent all numbers that can be used. Lastly an exponent of –1 will create a reciprocal and invert your number such as (3/2)-1 = 2/3.

Negative Exponent Law

An exponential term with positive exponent has the form base^exponent = p/1. If we have a negative exponent the form is base^–exponent = 1/p. A positive exponent creates an answer. A negative exponent creates the reciprocal of this answer. A negative exponent will create a reciprocal of positive version of exponential term and be written as 1/product.

You write the that the answer is written negative sign in front of the power negates the multiplication by inducing division of 1 using the the power as the divisor. To find a negative power, write the power as a denominator with one as the numerator. 34 = â…“-4 and 3-4 = â…“4 as they are reciprocal.

Writing exponents as the denominator with 1 as the numerator:

2-2 = 1/22 = ¼
3-1 = â…“
3-2 = â…‘
3-3 = â…Ÿ27

Negative powers are the negation of positive powers. Negation here means reversing an operation. We are not multiplying by a negative number. Powering a number means multiplying by groups, then it's negation must mean dividing by groups. This is why no power can be a negative number unless it is imaginary. ( i2 = -1).

Negation Laws

Multiplication negates division and visa versa. A negation is an “undoing”; adding is the negation of subtraction.

2· 3 = 6
6 ÷ 3 = 2 or Division undoes multiplication
6 · ⅓ = 2 Multiplying by the reciprocal divides (undoing multiplication)

27+3= 30
30- 3 = 27 or 30 + (-3) = 27 Subtraction undoes addition or negating by adding a negative sum

10+1= 11 A positive sum
-(10+1) = -11 Negating a positive sum by multiplying by -1

Usually we simplify math by negating negative sums so we can add the difference. This gives the absolute value. Absolute value is always positive. Use as a positive or negative as situation warrants.

0–10–1 = ? A negative series
-1(0+10+1) Writing in () to isolate negation sign on the outside
0+10+1(-) Flipping signs by multiplying each term by -1
11(-) Summation of terms
–11 Negating the positive back to a negative so sign is not lost

It is important when negating problems using absolute value for ease calculation you put the minus sign in parentheses so that you do not forget the original sign of the equation. If you want to use a positive value you stop at 11(-) but you must have the sign there. If you need to convert the answer back it's original form having the sign properly isolated so you can keep track of your negations is critical.

93 – 100 a negative sum
–(93 –100) = (– 93 +100)(–1) distributing the sum out flips the signs
(100 – 93)(–1) = 7(–1) writing as (100 – 7)(–1)

Negation is useful for removing negative operators so that equations can be simplified. When all like terms are combined you change the sign of the answer to show the actual value.

Negating Negative Exponents

Negating a power means to turn a power into it's reciprocal. A reciprocal is what ever multiplied to it makes it equal one. The reciprocal of any whole number is it's fractional opposite. 5 shares reciprocity with ⅕ because 5 · 0.2 = 1. Written fractionally it is 5/1 · 1/5 = 5/5. Reducing by dividing by 5 makes it 1/1. both 5/5 and 1/1 equal 1. any number divided by itself equals 1.

If we are undoing the negative of a power we have to use its reciprocal:

3-4 a negative power
1 · ⅓ · ⅓ · ⅓ · ⅓ expanded out using reciprocal notation
1 · 1/81 multiplying
1/81 the reciprocal of 81/1

81 · 1/81 = 1 so they are reciprocal. Also note that (81/1 · 1/81) = 1. This is true for any number.

Negative Factor Law

Negative sums having an even number of terms multiplied equals a positive sum always. For any non zero numbers -a · -b = ab. Negative sums having an odd number of terms multiplied equals a negative sum always. For any non zero numbers -a · b = -ab.

If you have three terms all with negative signs the two negative signs cancel out making the sum positive but the third sign negates it and your sum is negative.

If you have four terms with all negative signs they all cancel out making the sum positive. When two numbers have a negative sign it cancels out leaving the product positive. Each pair of negative terms cancel out their (-) so if you have an even number of negative terms the sum will always be positive. If you have an odd number of terms the sum will be negative because the last term is negating it.

Two Negatives Rule

You can only multiply two numbers at a time. With 5 numbers to be multiplied You start at the left of the expression and multiply the first two terms creating a new product with four left over. You multiply that with the third term creating a newer product leaving fourth and fifth term. You multiply your product with the fourth term and the product of that with the last term. This is the full meaning of the above.

-3 · -3 · -3 · -3 · -3 – 35
9 · -3 · -3 · -3 multiplying...
-27 · -3 · -3 multiplying...
81 · -3 multiplying...
-243 final product

We performed four multiplication operations using five terms which created four products. Our products were (– 3, 9, -27, 81, – 243). The first two terms canceled out the signs. Each additional operation changed the sign from positive to negative. Since we had an odd number of terms the first four canceled out the signs and the last term negated it making it a negative. The pattern alternates. Even negatives are positive, odd are negative.

-30 = –3/– 3
-31 = -3 (-3) -32 = 9 (-3 · -3) -33 = -27 (-3 · -3 · -3) -34 = 81 (-3 · -3 · -3 · -3) -35 = -243 (-3 · -3 · -3 · -3 · -3)

Negative Bases

-3-1 = -⅓ ( –3/ –3 ∙ –3) because (–31)/(–32) is –31-2 then –3-1 = –⅓
-3-2 = ⅑ (-⅓)2 because –3-1 = –⅓ then –3-2 = –⅓∙– ⅓
-3-3 = -⅟29 (-⅓ · -⅓ · -⅓)

Write the power under a fraction bar and use 1 as the numerator. Numerator is the top number and denominator is the bottom number. The answer of a division operation is called the quotient. A fraction is a division problem. Also (dividend ÷ divisor =quotient) if strictly speaking division and not fractions.

In “ -3-3 = -⅟29 ” it's factors are (-⅓, -⅓, -⅓). Taking a fraction of a fraction will make it smaller. When we are raising a number to a negative power that is what we are doing. We are dividing by groups of the base.

-3-1 = -⅓ (-0.33333) – a third -3-2 = ⅑ (0.111111) + a third of a third -3-3 = -⅟29 (-0.0370370) – a third of a third of a third

Adding Powers

32 + 33  32+3 Adding the exponent in the superscript area multiplies instead of adds
32 + 33 = 9 + 27 The terms are separated by the add sign making them discrete quantities

Raising to a power, expanding, the expansion of are all ways to say the final value of the exponential term. Expansion can mean writing out in expansion notation like (3 · 3 · 3 · 3 · 3) or solving like 243.

Subtracting Powers

321 - 33 The bases are the same so we can combine exponents
321-3 Write one copy of the base and subtract exponents in the index
318 Combine the exponents
10,460,353,203 Expand to solve

Exponential means that the bigger the exponent you larger the interval between powers become. A version of exponential notation is used to write very large numbers called scientific notation. This bakes the number and reduces it to the ones unit with a limited number of decimals and then writes ten multiply sign in front of it with powers of ten being used to represent the length of the sum.

Multiplying Powers

32· 34 standard form
32 +4 grouping
36 multiplying the exponents as factors to get a productive
729 simplifying

36
(3 ·3 · 3 · 3 · 3· 3) write out all your bases so you can easily times them.
(9 · 9 · 9) or (27·27) multiply by grouping pairs and squaring
9· 81) or 729

Do Not Multiply Different Bases By Adding Exponents

4a² · 4b² can not be simplified any further adding exponents since a  b. Since the base is different you would be grouping together different groups of groups. 4a²Â·4b² = 4²a²b² = 4(32) · 4(22) not 16(ab⁴) because a²b² ≠ ab⁴. The mistake is adding exponents of base a to base b. 4(32) · 4(22) = 4(3∙3) · 4(2∙2) = 4(9) · 4(4)

• a2b2 ≠ ab4
• 3222 ≠ 3∙24 is right but coefficients need to be multiplied
• 9∙4 ≠ 3∙8 is right (does equal 4a24b2 )

3n2 · 7n2 can simplified further since they have the same variable. If bases are different you would be grouping together different groups of groups. If the variables same they are called like terms.

4x+4 can not be simplified further. We can write them to have the same variable but its power is 0 and it cancels making it unusable. Hence 4x1 +4x0 ≠ 4x1 because 4x+4≠ 4x. Any constant has a variable base raised to a power of 0 which makes the variable equal 1, which makes the term any constant times 1, which cancels and you are left with the original constant.

• 4x+4 These can not be added together because 4 doesn't have x..
• 4x+4x0 Writing 4 as 4x0 doesn't help because x will cancel itself.
• 4x+4∙x0 Raise the power first then multiply by the coefficient (follow PEMDAS).
• 4x+4∙1 x0= 1 following the zero power law.
• 4x+4 We can not write these as one term, only as two terms.

3n2 · 7n2 We will use n = 5. 3(52) · 7(52) Substitute. 3(25) · 7(25) Expansion. 75 · 175 Distribution. 13,125 Simplification.

English has four different tenses for math words. These will come up. You are not limited in how you express math. I am going to prefer the simplest versions highlighted in blue but may use other versions to show process. It is important to learn math and English at the same time so you can read and count. I want literacy in both.

• 3n2 · 7n2 stating a product as two factors
• 21(n4) combining like terms
• 21(54) substitute
• 21(625) simplify
• 13125 distribute
• 3n2 · 7n2 = 21n4 equate

Finding like terms

3n3 · 7n2 (3·7) · (n3 · n2) Multiply your constants ( the coefficient 3 and 7) (21) · (n3 · n2) Multiply your variables (the bases n2 and n2) (21) · (n3+ 2 ) Add to multiply exponents (21) · (n5) This works 21n5 This is as simple as we can make it because there are no more operations left

3n3 and 7n2 are like because they share the same base of n. We only care about the variable not the coefficients as coefficients are constants and constants always add (excerpt for imaginary numbers of the form a+bi). We you write in full 3n3 ∙ 7n2 can easily be seen as 3∙n3 ∙ 7∙n2 = 3∙7∙n3 ∙n2 = 21∙n3+2=5 = 21∙n5.

Commutative Property

You can rewrite equations by changing the order in which the number is listed without changing result.The commutative property for addition and subtraction:

1+2+3 = 1+3+2 = 2+1+3 = 2+3+1 = 3+1+2 = 3+2+1
1–2–3 = 1–3–2 = 2–1–3 = 2–3–1 = 3–1–2 = 3–2–1
1∙2∙3 = 1∙3∙2 = 2∙1∙3 = 2∙3∙1 = 3∙1∙2 = 3∙2∙1

Division is not commutative as changing order effects result:

Of the six combinations we get three different answers. Because 1 cancels out when used as a denominator using a number other than 1 will give six different answers. Using 0 or 1 in proofs often is a bad way to check validity of an answer. Use numbers 2 and higher. You can commute exponential terms and radicals because they are simplified as constants, unless division operators come into conflict. If the division operator is present nothing can be moved to the left or right of it but anything on the left can be moved anywhere on the left and anything on the right can be moved on the right:

• 2∙ √3 + 82 ÷ 4 – 23 ∙ 5 We have all operators present
• 2∙ √3 + 82 ÷ 4 – 23 ∙ 5 The blue is left and the red is right of the obelus
• (2∙ √3 + 82 )÷ (4 – 23 ∙ 5) Group using parentheses
• 2∙ √3 + 8^2 / 4 – 2^3 ∙ 5 Or write as a fraction

As a fraction everything left is the numerator and everything right is the denominator. It is easier to perform this calculation using parentheses but writing as a fraction shows you how division keeps things separate. The numerator (or LH) can be rewritten in different orders up to 3 ways :

Because PEMA still applies we must do multiplication before the addition, as such there is no other way to re-order the terms. If we only had multiplication or addition only then we could have 6 re-orderings.

Find 82, multiply by 2, write as coefficient of √3. Changing order does nothing to change the order of operations.

The Associative Property

The associative property says you can perform the operations in different orders and still get same answer.

1. The associative property of addition says you can add in any order
2. The associative property of multiplication says you can multiply in any order

Whether we group the first two or last two numbers the answer is still the same.
This will be true for any addition problem in that how you group terms does not change the result. Because of this addition is said be to associative.

Multiplication is also associative: 2∙3∙4 can be (2∙3)∙4 or 2∙(3∙4)

In general the associative property says that how you group addition or multiplication does not change result. The reason being deals with PEMDAS and the order of operations. For any operation and it's reciprocal operation math is associative. Math is not associative when you have operations that are not reciprocal such + or – with ∙ or

• Addition is reciprocal with subtraction
• Multiplication is reciprocal with division
• Exponentiation is reciprocal with radicalization

PEMDAS says order matters, associativity appears to say order does not matter. More correctly associativity should use the word “group or grouping” rather than the word “order”. PEMDAS out ranks any property law, its order must be strictly adhered. The only time PEMDAS appears to be broken is when performing a reciprocal operation. We can't really talk about associative property without talking about PEMDAS and order of operations.

PEMDAS lists the order of operations, the discreet separate operations. Parentheses are always the first operation to do. What associativity does is not say order does not matter but that you can change how things are grouped. In other words parentheses always come first (according to PEMDAS) but you can change where the parentheses are ( according to associativity).Some people politely point out that sometimes it is okay to break the order of PEMDAS like 2∙3÷4 where they state you can divide first instead of multiplying. Well done. They are reciprocal operations and therefore both multiplication.

[2] Using PEMA over PEMDAS is preferred by some teachers because illustrates the reciprocity of same level operations. PEMA is superior but PEMDAS is what is easier to learn. PEMA is necessary because it teaches reciprocal operations and does so in a concise manner.

• Parentheses
• Exponents and radicals
• Multiply and divide
• Add and subtract

PEMDAS uses six letters to state five operations and does not show reciprocity while PEMA uses four letters to state six operations and does show reciprocity. PEMA uses to less letters, includes radicals[4], and shows reciprocity. This makes PEMA better than PEMDAS for three reasons.

Always do parentheses first (or any other grouping symbol).Consider a operation and its reciprocal “on the same level” and essentially to be the same operation therefore you can do them in alternate orders (not “any” order). Saying “any order” tends to confuse people and lose track of the correct order of operations. Without order of operations complex equations would have more than one answer, which is incorrect, and people would differ on the result leading to math being nonfunctional[5]. How should we do this problem: 2∙3÷4

There are two operation “∙” and “÷”, whichever one we perform first we get the same result. A dumb way to say this is “because ∙ and ÷ are the same operation. Close but inaccurate. This is the same inaccuracy as saying taking a square root is the same as exponents[6] in that is shows a lack of fundamental understanding. A more accurate response is that ∙ and ÷ are reciprocal and alternate orders of performing them are possible because reciprocity allows it. A closer look:

There are different ways to perform this but all follow the same order of operations whether you call it PEMDAS BODMAS, BIDMAS, PEMA or whatever. The point is the result is the same. Calling multiplication the same as division is an inaccuracy because it isn't the “same” it is the reciprocal and reciprocal means the “opposite”. Same does not mean opposite. Reciprocal operations are antonyms NOT synonyms. Likewise exponents are NOT the same as radicals. Defining them as such is a gross inaccuracy. PEMDAS and BODMAS/BIDMAS does not explicitly state radicals but only exponents/indices. They only list five operations and group exponents and radicals under a singular title. This lacks consistency. Using the same logic they should also group multiply with divide and add with subtract but don't. PEMA does and therefore is more logical and concise and more instructive. Calling five or more different things merely “power” or “powers” is confusing and lacks specificy. You might as well just say “number” as you are not differentiating one part of an equation or another. This leads to much pointing to an object to illustrate it rather than simply referring to it by a well defined name and being completely understood.

Ideally all equations should have well defined labels for each part of the equation. Consider that in a binary operation there are three numbers, two numbers are combined using an operator and a third number left as the result. Even in long equations with many terms only two numbers can be operated at a time. Of the six operations the four basic operations are well understood and defined, it is only when reaching higher math such as powering and rooting there is less definition and a greater lack of clarity as to what is what. At any time up to five things can be referred to as a “power” under PEMDAS. This is sloppy and does not provide enough clarity. Math should be well defined by clarifying terms with unique designations as to eliminate confusion when referring to a specific term. The following diagram lists the six operations with an unambiguous naming convention.

The picture does not completely give details how to pronounce but is drawn from that perspective, further explanation is required namely with radicals. Also a fraction is considered division here but special attention is given to renaming the dividend and divisor as numerator and denominator respectively. There may be more ways to write these operations but this is as far as I got. Not everyone is going to talk like an egghead all the time but if they do this is the proposed method to do it. Anything else is considered slang or inaccuracies for casual discussion (which is okay).

The number left of the operator (or top in the case of fractions) is considered the first term and is in blue.
The number right of the operator (or bottom in the case of fractions) is considered the second term and is in red.
The number right of the equals sign is considered the result and is in green. All math consists of a binary operation in which three numbers are involved; the first number to the left of the operator, the second number to the right of the operator, the third number right of the equals/equality sign as the answer. Most only refer to the left hand or right hand of the equation this takes things farther.

This is all for now. This shows how the associative property fits with Pemdas and reciprocity to form a more complete picture rather than teaching individually and leaving students to connect the dots themselves. The hope is that students connect the dots in a triangle and understand these concepts and use them together as a working unit rather than individual concepts that are scattered across math.

Inverses, Reciprocals, Identities, and Canceling

Inverse operations: These are called the reciprocal operations. They are the reverse operation that cancels another. An inverse operation works backwards from the answer to find the starting number. This can be used to check your answers. When working backwards if you arrive at the starting number then you have the right answer.

Inverse of a function: Functions are equations with many operations. The inverse of a function cancels another function. When two functions cancel what happens is they are multiplied and the answer is 1. 1 itself cancels when it is a factor in a multiplication operation, as we will see.

Inverse of a number: The inverse of a number is the multiplicative reciprocal of a number. The reciprocal of a number is when you write a number as a fraction and then invert the numerator with the denominator. In multiplication when two number multiplied result to 1 as the answer they are reciprocals. A reciprocal is the number you multiply to equal one. The purpose of this is to cancel the number or term in the equation.

When you cancel a number it is erased from the equation. There are three ways to cancel something:

Additive inverses: Additive inverses are zero sums. If two numbers are added together and the total is 0 then the numbers used are additive inverses. Adding opposite numbers cancels them both. Although canceling is not the proper term this is the easiest way to explain it. If 8+10–8 then we can change the order to 8–8+10 = 0+10. We see that 8 and –8 are opposite of each other. These are called additive inverses. Additive inverses are used in inverse operations. Additive means adding but can also mean adding a negative number like 4 + (–1) = 4 – 1. Likewise an additive inverse we can have 4 + (–4) = 4 – 4, or simply –4+4. In either case both –4+4 and 4–4 both equal zero. This cancels them out, or more properly sums to 0.

Reciprocals: This is the proper usage of canceling. When you multiply a number, be it an integer or a rational, and the product is 1 and the number is said to be canceled. Informally canceling is the act of reducing a fraction so that its quotient is 1. This is better explained under the fractions section. Reciprocals deal heavily with multiplication. Under property laws there exists the law of reciprocal multiplication.

Division: You can divide a number, term or entire equation by itself and the answer is 1. This is obvious for simple problems like 3÷3 = 1 but not more complicated problems like √2/√2 = 1 or (3x2+4x+7)/(3x2+4x+7) = 1.

Identities: Identities are formulas that make canceling easier. They are used in long equations with many parts to simplify the problem before or as you work.