Rational and Irrational Numbers

Explained with examples and non examples

Rational Numbers

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 which numbers the various sets and subsets of rational numbers.

Set of Real Numbers 6enn Diagram

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

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!
pi
$$ \frac{ \sqrt{2}}{3} $$ Although this number can be expressed as a fraction, we need more than to be a rational number. 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 but , like $$ \pi $$ constantly changes and so can not be represented as a quotient of two integers

Practice Problems

Problem 1

Is the number $$ -12 $$ rational or irrational?

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

Problem 2

Is the number $$ \sqrt{ 25} $$ rational or irrational?

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

Is the number $$ 0.09009000900009... $$ rational or irrational?

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

Is the number $$ 0.\overline{201} $$ rational or irrational?

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

Problem 5

Is the number $$ \frac{ \sqrt{3}}{4} $$ rational or irrational?

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

Is the number $$ \frac{ \sqrt{9}}{25} $$ rational or irrational?

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

Is the number $$ \frac{ \pi}{\pi} $$ rational or irrational?

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

Is the number $$ \frac{ \sqrt{2}}{ \sqrt{2} } $$ rational or irrational?

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

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