Pythagoras: proof square root of 2 is irrational number

Pythagoras' proof of the existence of irrational numbers follows. This discovery so changed the Geek view of the universe that they drowned one of their order, Hippasus of Metapontum, for divulging the existence of irrational numbers to an outsider. You may wish to first look at the proof of the Pythagorean Theorem, which implicitly demonstrates the existence of Ö`2 (and therefore irrational numbers) insofar as an isosceles right triangle with sides of length 1 has an hypotenuse of length Ö`2. We will proceed by the method known as "proof by contradiction:" we will assume that the Ö`2 is a rational number, and by so doing arrive at a contradiction, telling us that our original assumption must be false.

Assume that the Ö`2 is a rational number. Therefore, by the definition of a rational number, there must exist whole numbers m and n such that

m/n = Ö`2 .

If m and n are not reduced to lowest terms, i.e., if they contain common factors, factor them out until we are left with new whole numbers p and q that have no common factors (are reduced to lowest terms), and where p/q = Ö`2 .

After squaring both sides of this equation we get: p²/q² = 2 and then multiplying both sides by q² we get

p² = 2q²

But this tells us that p² is even (since it equals a multiple of 2: 2q²), and therefore p itself must be even, since 2 is a prime number and cannot appear in the square of a number unless it is a factor of the number itself. Therefore we can represent p as 2r (where r is a whole number), which by substitution in the above equation gives us (2r)² = 2q² --> (2r)(2r) = 2q² --> 4r² = 2q² -->

2r² = q²

which tells us that q² is even, and therefore by the above reasoning q itself is even. But if both p and q are even, they are both divisible by 2. But p and q have no common factors, since starting with m and n we factored them all out to arrive at p and q. Therefore, by assuming that Ö`2 can be represented by a fraction p/q reduced to lowest terms, it follows that p and q are not reduced to lowest terms. Since following the rules of logical inference our assumption arrives at a contradiction to itself, it follows that our assumption must be false. Therefore there are no whole numbers that as a fraction equal Ö`2 ; or in other words, Ö`2 is not a rational number.

This proof was discovered by the Greeks over 2000 years ago!

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Proof that the √2 is an irrational number.