The concept Zero

The very simple concept of Zero (“null” or “naught” or “nothing”) as in effect a number, contributed greatly to a dramatic development of modern mathematics and since the middle ages with new insights and discoveries, and harder problems being solved every day.

Some people claim that through history within any given century mathematics progressed further than in all previous ones together. There has been a tremendous fundamental shift in methods and philosophy of mathematics, and of science during the twentieth century.

The drastic breakthroughs were inherently possible due to decimal notation coming from Hindu-Arabic traders in the 1200s spreading to the West. In this system “ 0 ” is in practice used as a placeholder, as an empty place or position, this made calculations radically easier faster and more reliable than for instance when using Roman numerals or other counting systems.

The definition

Zero (null) can be defined as the identity for addition, meaning that for any number x we have,

x + 0 = 0 + x = x

It means anything with null added stays the same, or adding nothing leaves a quantity unchanged. It would mean also 0 + 0 = 0. The simple idea is crucial for solving both elementary and advanced algebraic equations.

Examples of “contradictions”

From the definition of “division”, 1 divided by zero or 1/0 is the question “How many times does 0 go into 1?” We must have 0 + 0 + ... + 0 m-times but it is still 0 it has to be zero since 0 + 0 is 0. What is an unending amount of nothings?

It says strictly speaking 1 = 0 too and infinity times zero thus must be zero etc. Zero is the only such problem you can divide by any other. However you will also soon run into these kind of irreconcilable difficulties if you want to believe infinity is number.

Other “Non-numbers”

Necessity is the mother of invention.

The Irrationals

The ancient Greeks believed in all of nature there are only rationals, fractions of integers. It was believed to the point of religion it came as very unsettling a proof that there are in fact others. There are real numbers that are not rational, per definition are irrational. As an example as they discovered, the square root of 2 is irrational.

There has to be such a length since by Pythagoras it is the hypotenuse of a 45o right-
angle triangle with equal sides of one unit.

Assuming root 2 is rational and applying the definition of a rational as strictly the fraction of two integers, and this a number of which the square is 2 will lead to a contradiction. The assumption is wrong and root 2 is irrational.

Twilight Zone

As further extension there are complex numbers, amphibian that live between exists and doesn't exist. There are very basic non-trivial ideas for instance the solution of the equation,

x^2 = x * x = -1

Since the product of two negative numbers is positive, and of two positives is also positive, it follows that no real times itself can be negative. Thus the square of every number must be positive.

So is there a number x that squared equals Negative One? We needed one, we invented one and call it “ i ” for imaginary!