MT2002 Analysis

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## Rationals and irrationals

Definition

The rational numbers Q are all fractions a/b with a, b integers.

Theorem
If r, s are rationals, so are r + s, r - s, r × s and (if s ≠ 0) r /s .

Proof
An exercise in arithmetic with fractions.

The Greeks believed a long time ago that fractions were sufficient to describe real phenomena. However, the Pythagoreans, a religious and philosophical school founded by Pythagoras of Samos (569BC to 475BC) in Southern Italy, discovered the following result.

Theorem
The real number √2 is not a rational number.

Proof
(By contradiction) Suppose that √2 = a/b for a, bZ. We may as well assume that a, b have no common factors else we could cancel them out.
Then 2b2 = a2 and so a is even. But then a2 is divisible by 4 and so b2 is even. But then b is even and so a and b do have a common factor. Thus we have a contradiction.

The Pythagoreans realised that they could produce a line of length √2 from a right-angled triangle and so they were forced to the conclusion that rational numbers were not sufficient to describe their geometric system.

### An aside on decimals

Decimals were first introduced into Europe by the Flemish/Belgian mathematician Simon Stevin (1548 to 1620) though they had been used earlier by some Indian, Arabic and Chinese mathematicians.

Some facts about decimals

Every real number has a unique decimal expansion -- except that terminating decimals (which end in ...00000...) can also be expanded to finish in ... 99999 ...
For example, 1 = 1.0000 ... = 0.9999... , 1/4 = 0.250000 ... = 0.249999 ... , etc.
Numbers with such terminating decimal expansions are of the form m/n with the denominator n having 2 or 5 as its only prime factors.

Rational numbers have decimal expansions which repeat periodically.
For example 1/6 = 0.166666 ... which we write
while 1/7 = 0.14285714285 ... =
The recurring decimal
For example 0.123123 ... = 123/999 = 41/333 .
An irrational number like √2 has a decimal expansion which does not repeat:
√2 = 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070 ...

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JOC September 2001