Numbers |
A History of Numbers |
Propositional Logic | Logical Completeness | The Liar's Paradox
Logical Consistency | Basic Methods of Mathematical Proof | Integers and Natural Numbers
Rational Numbers | Irrational Numbers | Cantor's Diagonal Argument | Imaginary Numbers | The Euler Equation

The Meaning of Number

A couple of centuries BC, the prevelant group of mathematicians-cum-philosophers-cum-cultists, called the Pythagoreans, (after Pythagorus, their leader and guru) postulated the purity of expression granted by the numbers. They stated, but could not prove, that every number was rational or the ratio of two integers. This 'union of ratios' was the symbol of perfection used by the Pythagorean Brotherhood. Their all-encompasing statement, "Everything is Number!" as uttered by Pythagorus, summed up their holistic attitude to mathematics. They were the first to demonstrate the whole number ration of musical notes and so linked maths and music forever.

There was trouble in their rational paradise though. One of the students of the esteemed Pythagorus was investigating the maths of square roots of prime numbers (A prime number is one with no whole number factors apart from one and itself.) He was shocked to discover that the number refused to simplify into a nice fraction. Fiddling around he became even more worried when it seemed that he had proved a flaw in the teachings of the Brotherhood, that fractions were a very small subset of a larger world of numbers. Disconcerted and unable to find a flaw with his reasoning he presented the proof to Pythagorus for guidence and reassurance.

What he got instead was a lynch mob. Pythagorus flew into a rage and declared the poor student an enemy and heretic. Calling the rest of the Brotherhood he ordered the student to be picked up. Leading a procession to the courtyard fountain and pool, the young man was thrown in and held under the surface until he stopped struggling. The fabled proof was lost or (more probably) destroyed. Pythagorus died safe in the knowledge that for the rest of the world, the rationals were all.

Enter: Euclid

Then along came a man called Euclid. This man wrote a book, by which he is famous to school children for 2000 years. This book, called Elements, was considered the second most important book of all time(after the Bible.) In it, the first principles, proofs and axioms of flat-plane (known as Euclidian) geometry were laid down. This book wasn't overturned (or at least shown to be incomplete) until the age of Gauss and Riemann in the 19th century. Euclid was a very clever man and its a pity Elements overshadows his other major accomplishment, the proof of the existance of irrationals. This is laid out in all its glory below.

Irrational Properties

Not only won't irrationals simplify to a whole-number fraction but they have some other odd properties. They possess an infinite number of non-repeating decimal places. Please note that an infinite number of digits doesn't make an irrational (eg a ninth is also represented as 0.1 recurring and has an infinite number of repeating digits) but this is only the slightest restriction. Some famous examples of irrationals are pi (the ratio of the circumference of a perfect circle to its diameter) and e (the natural number) which could be called the fundemental numbers at the foundation of maths (as shown by the Euler Equation) so, as you see, irrationals are very important.

Euclid's Proof

This is a fantastic example of proof by contradiction. Its aim is to try and rearrange SQRT(2) into the form p/q where p and q are two integers and show an absurdity arising as a result of assuming that there is a fraction p/q such that p/q = SQRT(2) where p and q are integers.

Certain knowledge about the properties of fractions and even numbers is required to follow the proof. These are:

1. If you take any integer and double it, the result must be even.
2. If you know the square of a number is even, then the number itself must be even.
3. Fractions can be simplified by dividing both top and bottom by a common factor as long as the result of any such division is integer. Fractions cannot be simplified forever.

Let: SQRT(2) = p/q

If we square both sides then: 2 = p²/q²

This can be rearranged to give: 2q² = p²

Now from point (1) we know that p² must be even. Futhermore, from point (2) we know that p itself must be even. Since p is even it can be re-written in the form p = 2m, where m is some other whole number. This follows from point (1). Plug this back into the equation and we get:

2q² = (2m)² = 4m²

Divide both sides by 2 to get: q² = 2m²

But, by the same arguments we used before, we know that q² must be even, and so q itself must also be even. If this is the case then q can be written as 2n where n is some other integer. If we go back to the beginning, then

SQRT(2) = p/q = 2m/2n

Which can be simplified by dividing top and bottom by 2 to give:

SQRT(2) = m/n which is simpler than p/q

However we find we can repeat the process again to give a still simpler fraction, d/e which can be put through to give another, even simpler, fraction, f/g. In fact, we find that this process can be repeated an infinite number of times. However we know from point (3) that fractions cannot be simplified forever. p/q does not seem to obey this rule (and so strays into absurdity) and therefore SQRT(2) cannot be expressed as a fraction. Therefore SQRT(2) is irrational.

QED