# Rational Numbers

A **Rational Number** is a real number that can be written as a simple fraction (i.e. as a **ratio**).

### Example:

**1.5 is a rational number** because 1.5 = **3/2** (it can be written as a fraction)

Here are some more examples:

Number | As a Fraction | Rational? |
---|---|---|

5 | 5/1 | Yes |

1.75 | 7/4 | Yes |

.001 | 1/1000 | Yes |

-0.1 | -1/10 | Yes |

0.111... | 1/9 | Yes |

√2 (square root of 2) |
? | NO ! |

Oops! The square root of 2 cannot be written as a simple fraction! And there are many more such numbers, and because they are **not rational** they are called Irrational.

Another famous irrational number is Pi (π):

## Formal Definition of Rational Number

More formally we would say:

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

So, a rational number can be:

p |

q |

Where q is not zero

### Examples:

p | q | p / q | = |
---|---|---|---|

1 | 1 | 1/1 | 1 |

1 | 2 | 1/2 | 0.5 |

55 | 100 | 55/100 | 0.55 |

1 | 1000 | 1/1000 | 0.001 |

253 | 10 | 253/10 | 25.3 |

7 | 0 | 7/0 | No! "q" can't be zero! |

## Using Rational Numbers

If a rational number is still in the form "p/q" it can be a little difficult to use, so I have a special page on how to: |

### Pythagoras' Student

The ancient greek mathematician *Pythagoras* believed that all numbers were rational (could be written as a fraction), but one of his students *Hippasus* proved (using geometry, it is thought) that you could **not** represent the square root of 2 as a fraction, and so it was *irrational*.

However *Pythagoras* could not accept the existence of irrational numbers, because he believed that all numbers had perfect values. But he could not disprove *Hippasus'* "irrational numbers" and so Hippasus was thrown overboard and drowned!