Since 0, negative numbers and numbers like ÷`2 and p (that canít be expressed as fractions) didnít exist for the Pythagoreans, letís see how they brought them into existence. The Pythagoreans knew (as did the Babylonians by 1700 BC, but they didnít get the credit) that in a right triangle (one with a 90į angle) the blue square plus the green square must equal (in area) the red square. Or in numbers, a2 + b2 = c2: the Pythagorean Theorem. Now imagine that in the diagram below a = b = 1; i.e., the blue and green squares 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 must have asked themselves, is this rational? Is this ÷`2 a rational number? That is, are there whole numbers p and q such that ÷`2 = p/q? What follows is their proof (probably by Hippasus of Metapontum) that ÷`2 is not rational. That understanding changed the Pythagoreans whole view of the universe.
Hippasus of Metapontum's proof of the existence of irrational numbers follows. This discovery so changed the Greek view of the universe that they drowned Hippasus for divulging the existence of irrational numbers to an outsider, Hippasus being the very one who may have actually discovered that irrational numbers exist rather than Pythagoras himself. You may wish to first look at the proof of the Pythagorean Theorem, which implicitly demonstrates the existence of ÷`2 (and therefore irrational numbers) insofar as an isosceles right triangle with sides of length 1 has an hypotenuse of length ÷`2. 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 (are reduced to lowest terms). Now p/q = ÷`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 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 ÷`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 ÷`2 ; or in other words, ÷`2 is not a rational number.
This proof was discovered by the Greeks over 2000 years ago!
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