### Rational Numbers

**Definition** : Can be expressed as the quotient of two integers (ie a fraction) with a denominator that is not zero.

Many people are surprised to know that a repeating decimal is a rational number. The venn diagram below shows examples of all the different types of rational, irrational numbers including integers, whole numbers, repeating decimals and more.

#### Set of Real Numbers Venn Diagram

### Examples of Rational Numbers

5 | You can express 5 as $$ \frac{5}{1} $$ which is the quotient of the integer 5 and 1. | |

2 | You can express 2 as $$ \frac{2}{1} $$ which is the quotient of the integer 2 and 1. | |

$$ \sqrt{9} $$ | Is rational because you can simplify the square root to 3 which is the quotient of the integer 3 and 1. | |

$$ .\overline{11} $$ | All repeating decimals are rational. It's a little bit tricker to show why so I will do that elsewhere. | |

$$ .9 $$ | Is rational because it can be expressed as $$ \frac{9}{10} $$ (All terminating decimals are also rational numbers). | |

$$ .73 $$ | is rational because it can be expressed as $$ \frac{73}{100} $$. | |

$$ 1.5 $$ | is rational because it can be expressed as $$ \frac{3}{2} $$. |

# Irrational Numbers

**Definition: **Can not be expressed as the quotient of two integers (ie a fraction) such that the denominator is not zero.

#### Examples of Irrational Numbers

$$ \sqrt{7} $$ | Unlike $$ \sqrt{9} $$, you cannot simplify $$ \sqrt{7} $$ . | |

$$ \frac{5}{0} $$ | If a fraction, has a dominator of zero, then it's irrational | |

$$ \sqrt{5} $$ | Unlike $$ \sqrt{9} $$, you cannot simplify $$ \sqrt{5} $$ . | |

$$ \pi $$ | $$ \pi $$ is probably the most famous irrational number out there! | |

$$ \frac{ \sqrt{2}}{3} $$ | Although this number can be expressed as a fraction, we need more than that, for the number to be rational . The fraction's numerator and denominator must both be integers, and $$\sqrt{2} $$ cannot be expressed as an integer. | |

$$.2020020002 ...$$ | This non terminating decimal does not repeat . So, just like $$ \pi $$, it constantly changes and can not be represented as a quotient of two integers. |

**Practice** Problems

##### Problem 1

Rational because it can be written as $$ -\frac{12}{1}$$, a quotient of two integers.

##### Problem 2

Rational, because you can simplify $$ \sqrt{25} $$ to the integer $$ 5 $$ which of course can be written as $$ \frac{5}{1} $$, a quotient of two integers.

##### Problem 3

This is irrational, the ellipses mark $$ \color{red}{...} $$ at the end of the number $$ \boxed{ 0.09009000900009 \color{red}{...}} $$, means that the pattern of increasing the number of zeroes continues to increase and that this number never terminates and never repeats.

##### Problem 4

This is rational. All repeating decimals are rational (see bottom of page for a proof.)

##### Problem 5

This is irrational. You cannot simplify $$ \sqrt{3} $$ which means that we can **not** express this number as a quotient of two integers.

##### Problem 6

Unlike the last problem , this ** is ** rational. You can simplify $$ \sqrt{9} \text{ and also } \sqrt{25} $$. If you simplify these square roots, then you end up with $$ \frac{3}{5} $$, which satisfies our definition of a rational number (ie it can be expressed as a quotient of two integers).

##### Problem 7

This is rational because you can simplify the fraction to be the quotient of two integers (both being the number 1).

$ \frac{ \pi}{\pi } = \frac{ \cancel {\pi} } { \cancel {\pi} } = \frac{1}{1}=1 $

##### Problem 8

This is rational because you can simplify the fraction to be the quotient of two inters (both being the number 1).

$ \frac{ \sqrt{2}}{\sqrt{2} } = \frac{ \cancel {\sqrt{2}} } { \cancel {\sqrt{2}}} = \frac{1}{1}=1 $

### Proof that repeating decimals are rational numbers

$$ x = .\overline{1} $$

Multiply both sides by 10$$ 10 \cdot x = 10 \cdot .\overline{1} \\ 10x = 1.\overline{1} $$

Subtract equation 1 from 2$$ 10x - 1x = 1.\overline{1} - .\overline{1} \\ 9x = 1 \\ x = \frac{1}{9} $$

Yes, the repeating decimal $$ .\overline{1} $$ is equivalent to the fraction $$ \frac{1}{9} $$.