{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

4400irrationals

# 4400irrationals - SOME IRRATIONAL NUMBERS PETE L CLARK...

This preview shows pages 1–2. Sign up to view the full content.

SOME IRRATIONAL NUMBERS PETE L. CLARK Proposition 1. The square root of 2 is irrational. Proof. Suppose not: then there exist integers a and b = 0 such that 2 = a b , meaning that 2 = a 2 b 2 . We may assume that a and b have no common divisor – if they do, divide it out – and in particular that a and b are not both even. Now clear denominators: a 2 = 2 b 2 . So 2 | a 2 . It follows that 2 | a . Notice that this is a direct consequence of Euclid’s Lemma – if p | a 2 , p | a or p | a . On the other hand, we can simply prove the contrapositive: if a is odd, then a 2 is odd. By the Division Theorem, a number is odd iff we can represent it as a = 2 k + 1, and then we just check: (2 k + 1) 2 = 4 k 2 + 4 k + 1 = 2(2 k 2 + 2 k ) + 1 is indeed again odd. So a = 2 A , say. Plugging this into the equation we get (2 A ) 2 = 4 A 2 = 2 b 2 , b 2 = 2 A 2 , so 2 | b 2 and, as above, 2 | b . Thus 2 divides both a and b : contradiction. Comment: This is a truly “classical” proof. In G.H. Hardy’s A Mathematician’s Apology , an extended rumination on the nature and beauty of pure mathematics, he gives just two examples of theorems: this theorem, and Euclid’s proof of the infinitude of primes. As he says, this is inevitably a proof by contradiction (unlike Euclid’s proof, which constructs new primes in a perfectly explicit way). The orig- inal statement is logically more complicated than what we actually prove in that it takes for granted that there is some real number 2 – characterized by being positive and having square equal to 2 – and then shows a “property” of this real number, namely it not being a fraction. But the essence of the matter is that a certain mathematical object does not exist – namely a rational number a b such that ( a b ) 2 = 2. This was the first “impossibility proof” in mathematics.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 3

4400irrationals - SOME IRRATIONAL NUMBERS PETE L CLARK...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online