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Proof that irrational numbers like √2 exist!

Since 0, negative numbers and numbers like √2 and π (that can’t be expressed as fractions) didn’t exist for the Pythagoreans, let’s see how they brought them into existence. The Pythagoreans knew (as did the Babylonians by 1700 BC, but they didn’t get the credit) that in any right triangle (one with a 90° angle like the yellow one below) that the sum of the squares on the two sides equals the square on the hypotenuse (the long side), or as in the diagram below the area of the blue square plus the area of green square must equal the area of the red square. Expressed in numbers this is a2 + b2 = c2 , the Pythagorean Theorem.

Pythagorean Theorem diagram of three triangles


Now in an isosceles triangle the two "legs" are of equal length, so we can have
a = b = 1; i.e., the blue and green squares above are equal and their sides are 1 unit long. The Pythagorean Theorem tells us that in this case:

12 + 12 = c2, or 1 + 1 = c2, or 2 = c2, or c2 = 2.

So c is a number which when multiplied by itself = 2. Nowadays we have a symbol for c: √2. Since the Pythagoreans knew that such a number has to exist by the above statements, they asked themselves, "Is this number, this √2, a rational number?" (Since a rational number can be expressed as a fraction, then there would have to be whole numbers p and q such that p/q = √2 ). They asked this question because whole numbers and their ratios were the only numbers that they knew existed, and their universe being in harmony depended on these being the only numbers that exist.

What follows is their proof—probably by Hippasus of Metapontum rather than Pythagoras himself—that √2 is not rational. This discovery so changed the Greek view of the universe that when Hippasus divulged the existence of irrational numbers to an outsider, they drowned him in a lake.

You may wish to first look at the proof of the Pythagorean Theorem which implicitly demonstrates the existence of √2 (but not that it is irrational) insofar as an isosceles right triangle with sides of length 1 has an hypotenuse of length √2. Here we will proceed by the method known as "proof by contradiction": we will assume that the √2 is a rational number, and by doing so arrive at a contradiction, telling us that our original assumption must be false.

Assume that the √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 = √2.

If m and n are not reduced to lowest terms (i.e., if they contain common factors), reduce m and n (factor out their common factors) until we are left with new whole numbers p and q that have no common factors. Then

p/q = √2.

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, namely 2q2. If p2 is even it follows that p must be even, since 2 cannot appear in the square of a number unless it is a factor of the number itself (the prime factorization theorem tells us that all numbers are composed of unique prime factors; therefore if 2, a unique prime, appears in p2, its source must be p 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, then they are both divisible by 2. But p and q have no common factors, since starting with m and n we factored out all common factors to arrive at p and q, a fraction reduced to lowest terms! Therefore, by assuming that √2 = a fraction of whole numbers p/q reduced to lowest terms, it logically follows that that fraction p/q is not reduced to lowest terms. And since by 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 √2 . In other words, √2 is not a rational number.

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

This Proof Irrational Numbers Exist page and much of this 600-page website are excerpted from the personalized Fine Art  Book You and the Universe.

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