# Rational and Irrational Numbers

Explained with examples and non examples

### 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.

### 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

Let

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

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 $$\frac{1}{9}$$.