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- 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*,*b*∈**Z**. We may as well assume that*a*,*b*have no common factors else we could cancel them out.

Then 2*b*^{2}=*a*^{2}and so*a*is even. But then*a*^{2}is divisible by 4 and so*b*^{2}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.

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

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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