| 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 | Imaginary Numbers | The Euler Equation
A couple of centuries BC, the prevalent group of mathematicians-cum-philosophers-cum-cultists, called the Pythagoreans, (after Pythagoras, their leader and guru) postulated the purity of expression granted by numbers. They stated, but could not prove, that every number was rational, or the ratio of two integers, an integer being a whole number or a thing complete in itself. This 'union of ratios' was the symbol of perfection used by the Pythagorean brotherhood. Their all-encompassing statement, 'Everything is Number!' as uttered by Pythagoras, summed up their holistic attitude to mathematics. They were the first to demonstrate the whole number ratio 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 Pythagoras 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 Pythagoras for guidance and reassurance.
What he got instead was a lynch mob. Pythagoras 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. Pythagoras died safe in the knowledge that for the rest of the world, the rationals were all.
Then along came a man called Euclid. This man wrote a book, which made him famous to school children for 2,000 years. This book, called Elements, was considered by some to be 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 it's a pity Elements overshadows his other major accomplishment, the proof of the existence of irrationals. This is laid out in all its glory below.
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 base of natural logarithms) which could be called the fundamental numbers at the foundation of maths (as shown by the Euler Equation). So, as you see, irrationals are very important.
This is a fantastic example of proof by contradiction. Its aim is to try and rearrange SQRT(2) into the form a/b where a and b are two integers, and then to show a contradiction arising as a result of this assumption.
Certain knowledge about the properties of fractions and even numbers is required to follow the proof. These are:
If you take any number and double it, the result must be even.
If you know the square of a number is even, then the number itself must be even.
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) = a/b where a and b are both integers.
Now we repeatedly apply rule (3) to a/b until the fraction can no longer be simplified. This gives us:
SQRT(2) = p/q where p and q do not share any common factors.
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
However this contradicts the original condition that p and q do not share any common factors, and therefore there cannot exist two integers a and b where SQRT(2) = a/b
Therefore SQRT(2) is irrational. QED.