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.
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.
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.
“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.
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.
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 = ⅓-4 and 3-4 = ⅓4 as they are reciprocal.
2-2 = 1/22 = ¼
3-1 = ⅓
3-2 = ⅑
3-3 = ⅟27
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).
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 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 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.
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)
-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
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.
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.
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
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. 4a2·4b2 = 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)
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.
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.
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.
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
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:
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 :
2 ∙ √3 + 82 = √3 ∙ 2 + 82 = 82 + √3∙ 2Because 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.
(2√3) ∙ 82 = 2 ∙ 82 ∙ √3 = 64 ∙ 2 ∙ √3 = 128√3Find 82, multiply by 2, write as coefficient of √3. Changing order does nothing to change the order of operations.
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.
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
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.
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
2∙3÷4 2∙3÷4 6÷4 2∙¾ 1.5 1.5There 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:
2∙3÷4 2∙3÷4 2∙3÷4 2∙3∙¼ 2∙¾ 6/4 2∙¾ 2∙3/2∙2 3/2 6/4 3/2 1.5 1.5 1.5There 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.