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# Rational Numbers

A rational number is not so called because it is 'rational' in the way the word is usually used - logical or reasonable as opposed to irrational - an argument that makes no logical sense!

It is named thus because it is a number that can be experssed as a ratio - a fraction - of a ratio of integers - RATIOnal... For example:

1.25 is a rational number, because we could write it as a fraction of 5 and 4. We could write it as 5/4

### Formal Definition of Rational Number

A rational number is a number that can be expressed in the form p/q where p and q are integers and q is not equal to zero.

So, a rational number is any number that can be expressed as:

p/q (where q is not zero).

Any number that cannot be expressed as a ratio is an irrational number.

Pythagoras (the ancient greek mathematician) and his followers believed that all numbers were rational, but one of his students Hippasus proved (using geometry, it is thought) that you could not write the square root of 2 as a fraction, and so it was irrational. Pi (π) is also an irrational number.

Followers of Pythagoras could not accept the existence of irrational numbers, and rumour has it that Hippasus was drowned at sea as a punishment from the gods for proving his wrong!