Inverse Behavior

Everything in math has an "opposite."  It is sort of like a religion, with its light and dark, yin and yang, or whatever.  In math, opposites "cancel" each other out.  Sort of like matter and antimatter.  But in doing so, these reactions make math a lot easier.

There are four things one can do with numbers: Add, subtract, multiply and divide.  The latter two are "short cuts" of the former two. That is to say, multiplication is just adding; and division is just subtracting until you come out even or with something left over.

Addition and Subtraction

Opposite Numbers

So, just how do the "inverse" reactions help in math?  First, numbers live in "parallel universes" on either side of the "neutral zone," aka zero.

    -10 -9 -8 -7 -6  -5 -4 -3 -2 -1|    |+1 +2 +3 +4 +5 +6  +7 +8 +9 +10

<-------------------------------------- 0--------------------------------------------->

Addition

The trouble with numbers, is that they only get "stronger" the farther they get from the "neutral zone." This is a great truth, for a number can increase indefinitely by adding just one unit at a time.  The end is not in site, for it is always beyond the biggest number.

But many times, that power is diminished as necessary reversals happen.  If too much momentum is lost, the number changes sides and wears the sign of the opposing "universe"

Inverse reaction (subtraction)

 But for this illustration, numbers are stubborn, they "teleport" into the other universe!  But alas, they are inexplicitly drawn to their counterpart, resulting in annihilation of both of them.

-6+6

=0

Fractions

Trying to get to the other side of the zero by just turning around is not a good path for an signed number either.  As they try to approach zero, they get weaker and weaker. Finally, they reach the "event horizon," (+/-1) and they begin to break into pieces, but never quite die.  Just as they could always advance away from zero, they will be stuck between +/-1 and 0 unless they again turn around and progress away from "absolute" zero.

  < 1/20  2/19  3/18  4/17  5/16  6/15  7/14  8/13  9/12 10/11 

0=======================1

Scotty, now would be a good time.  Beam me up, now!

Back to reality

Alright, enough fun.  Back to reality.  Hopefully, it might help someone "see" how negative numbers fit in to the scheme of things.  You really can add one more forever in each direction.  That is known as infinity.  It is also true that there are an infinite number of divisions between each of the whole numbers and their opposing negative selves.

Going in a negative direction along the line is called Subtraction, the inverse of Addition.  If one were to go over into the opposing "universe" the inverse would be Addition.  When you just lay a negative number next to a positive number, you have a "subtraction" problem using the bigger of the two.  The sign of the bigger number "wins" as the "difference" is determined,'

-8+5 = -(8-5) = -(3) = -3    Notice the subtraction was done inside a negative parenthesis.

+17-13 = +(17-13) = +4     Just for consistency, the same format is used. Positive wins!


I've read that the first functioning "computer" did not add, but subtracted.  It was a mechanical "difference" machine.  That brings us to Division.  Not like a battalion in the war of numbers, but in the process of dividing whole numbers by "natural" numbers.  Only these numbers can be "trusted" for they all follow a fast rule:  To be "rational" a number must be able to exist in the following form

a

b

(where a is an integer and b is a natural number)

What is a natural number?  It is any positive integer.  This excludes 0. which is neutral.  In other words, you cannot divide by 0.  In fact, you cannot divide by anything but a positive integer.

Integers:  {. . . -3, -2, -1, 0, +1,+2, +3 . . .}

Called "signed" numbers, these are all numerals, be they negative or positive.  It includes the "supernatural" number 0  -- just kidding.  As opposed to "natural,"  zero stands apart, but has powers beyond ordinary integers.

Whole Numbers:  {0, 1, 2, 3 ...}

Zero holds it's own as a "whole" number.  It is healthy, and stands in places all other numbers do, but without any voice.  It is barely noticed.  But when missing, the other numbers are greatly diminished.  The zero is second only to the ruling integer, being his "right hand" man.  I know, weak analogy, but alas, it fits.

Natural Numbers:  {1, 2, 3 ...}

One thing zero is forbidden to do is divide.  It is against all the "natural" laws of numbers. It is beyond logic to say that a whole number can be divided into pieces that have NO value.  If  zero were used as a divisor, then math would be impossible.  It is easy to see this.  Consider this:

For x = 0, verify  2x/x = 1.

2x/x = 1

2x*(1/2)= 1*(1/2)

x = 1/2

0  =/= 1/2

2*(x/x) = 1

2*1 = 1

2 =/= 1

Since by definition x/x  is always 1, and 0/x is by definition 0, then 0/0 would become 1!  Something out of nothing?  Nope, not happening.

Division cannot be by a fraction either. In the case of "division by a fraction" the inverse of division is used.  That is, the "fraction" in the denominator is turned over to become its own "reciprocal" and then the inverse of division, that is, multiplication, takes over.

So, what do I know?

+A-A = -A+A = 0

A-A = 0

A/A = 1

A = A/1 (a rational number)

1/A * A/1 = 1

A/B * B/A = 1 

Next: In Their Prime: Factors that matter