Operations

Inverse Operators

Every operation has an inverse operation that takes the answer and works backwards to find 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.

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.

Terminology and Intuition

There are six basic operators, 3 are positive and make number bigger, the other 3 are negative and make the number smaller.
The parent operators are + and – . All operations come from these as you will see.

Please note there are other symbols for division such as long division box “ ⟌ ” , horizontal fraction bar, and sometimes ratio colon “ : ” although a ratio is slightly different meaning and values at times.

The plus sign + is the parent of multiplication and powers.
The minus sign is the parent of division and radicals

Adding is simply adding. Different numbers may be added.
Multiplying is adding groups of the same number. Count the number of terms and multiply this number (groups) by the value of the term (amount per group).
Powering is multiplying groups of the same number. Count the number of factors and write it as an exponent.

34 = (3∙3∙3∙3) = 3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3 = 81

Subtracting is the reciprocal operation of addition.
Dividing is subtracting groups of the same number. Count number of terms and divide by this number (groups).
Rooting is dividing in groups of the same number. Count number of factors and divide by n – 1 factors.

∜81 = 81÷(3∙3∙3) = 81–3–3–3–3–3–3–3–3–3–3–3–3–3–3–3–3–3–3–3–3–3–3–3–3–3–3 = 3

If we add 27 terms of 3 we have 81. We can group this as 4 groups, means 4 terms, of 3. Groups is the number of terms, amount is the numerical of the term. Numerals are the symbol for a number as opposed to an English word. “Two” is an English word while “2” is an Arabic numeral. When describing a value we use numerals, anything else we use words. And so four groups of 3 is written as 3∙3∙3∙3 or 3+3+3+3 not 4∙4∙4 or 4+4+4. If we have four groups of three we can write it as a power 34. This grouped our multiplication, which was a grouping of our addition. We have groups of groups written in a more compact way.

27 terms of 3 gave 81. If we subtract all these terms we have zero. If we instead subtract n–1 terms we will be left with a single term and have 3. Division and rooting removes all but 1 term of a summation. The exponent is the number of factors we multiply by or the number of roots we are taking. If multiplying we multiply all factors, if rooting we divide by n–1 factors. We needed the 4th root of 81 so “n–1 roots” is 4–1 is 3, we divide by 3 factors.

All values in math can be called numbers. To be more specific certain numbers have names to distinguish them from others in an equation. This creates classifications of numbers. Also a number may be referred to by many names depending on the context and associated action or operation to be performed.

Addition: augend + addend = Total

Subtraction: subtrahend – minuend = Difference

Multiplication: factor ∙ factor = Product

Division: dividend ÷ divisor = Quotient

All of these words call also be called numbers but this creates more specify. “A number plus a number equals a number” isn't really informative. Sum simply means number so can be used anywhere. It can be used in subtraction like “ a sum minus a sum equals a difference”. Addition and subtraction are very basic so I will not spend much time on them. You can sum positive or negative numbers, summing negatives is like subtraction, you can call two numbers in a subtraction problem sums but subtrahend and minuend are more specific as it distinguishes them from sums in an addition problem. For basic math not important for hard math actually useful. The goal is not to be rigorous or pedantic but provide more clarity when needed, not at the expense of it.

The plus sign births 3 ways to multiply:

Early math: 2 × 3 = 6
Algebra: 2 ∙ 3 = 6
Reciprocal: 2 ÷ \( {\frac{1}{3}} \) = 6

Dividing by a fraction invokes multiplication. It doesn't need to be proven here but will show anyways:

2 ÷ 1/3 We are asking how many times 1/3 goes into 2
1/3 + 1/3 + 1/3 = 1 We know this (3/3 =1 so that 1/3 goes into 1 three times)
1/3 + 1/3 + 1/3 + 1/3 + 1/3 + 1/3 = 2 So we know this

This is not the cleanest approach but natural. It shows that 1/3 goes into 2 six times.
Said differently 6 times 1/3 equals 2: \( 6∙1/3 = 2\)

\(6 ∙ 1/3 = 2\)       This appears to be a multiplication
\(6/1 ∙ 1/3 = 2\)
\(6∙1/1∙3 = 2\)
\({\frac{6}{3}} = 2\)
\(6÷3 = 2\)       This appears to be a division

Understand that a fraction is a division problem and that the fraction bar is a division operator. We went from a multiplication operator to a division operator. This is because the fraction is really a division problem we are scaling or multiplying by the first number.

We use the reciprocal operator and the reciprocal number to create this equivalence:
\(2 ÷ {\frac{1}{3}}\)    =    \(2 ∙ {\frac{3}{1}}\)    =    \(6\)

We could find the quotient directly:

2 ÷ \( {\frac{1}{3}}\)
\(2 ÷ 0.\overline{3}\)

\(0.\overline{3} ⟌2\) This is long division with 2 in a division box

3.3 | 20 For clarity move the numbers by a power of 10

How many times does 3.3 go into 20? The answer is 6. If we divide 20 by 6 we get 3.3 and if we divide 20 by 3 we get 6.6. This is just something known about repeating decimals but we find it through long division for ourselves. This bring up a separate topic in math about repeating numbers. After exhaustive research we summarize the rule to be that when dividing by a fraction we multiply by the reciprocal, affectionately known as “flip and multiply”.

For a clearer demonstration to prove that “ \( {\frac{a}{b}}\) ÷ \( {\frac{c}{d}}\) = \( {\frac{a}{b}}\) ⋅ \( {\frac{d}{c}}\) ” we chose a non repeating decimal:

Reciprocal division: \(10 ÷ {\frac{1}{2}}\)
Obelus: 10÷0.5
Box: 0.5⟌10

Long divison using the division box:

0.5 goes into 1 two times:


      2       
0.5⟌10


2 times 0.5 is 1, we write the 1 and subtract:


      2       
0.5⟌10
  – 1      
     0


We bring down the next number which is 0:


      2      
0.5⟌10          
  – 1      
     00


0.5 goes into 0 to times so we write 0 after 2 and we are finshed:


      20       
0.5⟌10          
  – 1      
     00
  

There are other ways to do long division such as clearing the decimal. We would then ask 100÷5 and get 20 but this is not as direct as I showed since it changes the numbers slightly.

10 ÷ ½ = 10 ∙ 2 = 20
10 ÷ ½ equals 10 ∙ 2 since ½ goes into “10” twenty times.

It can be written different ways but all equal the correct answer 20.
Using flip and multiply is fastest and cleanest. You can cross multiply:

10 ÷ ½        10 ÷ ½        10 ÷ ½ 
10/1 ÷ ½      10/1 ÷ ½      10∙2 
10∙2 ÷1∙1     10∙2 /1∙1     20
20 ÷1         20 /1     
20            20      

This gives 3 ways to multiply. The parent operation minus births 3 ways to divide:

Early math: 10 ÷ 2
Algebra: 10/2
Reciprocal: 10 ∙ ½

Conclusion

All math is a binary process meaning you can only operate with two numbers and one operator at a time. All rooting problems break down to division and division breaks down to subtraction. Subtraction is the parent operator of higher negative operations. Likewise powering breaks down to multiplication which breaks down to addition with addition being the parent operation of higher positive operations. This is the meaning of addition gives birth to multiplication which gives birth to powering and subtraction gives birth to division which gives birth to rooting.