

Rational and Irrational NumbersExplained with examples and non examplesRational NumbersCan 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.
Irrational NumbersCan not be expressed as the quotient of two integers (ie a fraction) such that the denominator is not zeroExamples of Irrational Numbers
problem 1)
Is the number $$ \fbox{ 12 } $$ rational or irrational ?
Rational because it an be written as $$ \frac{12}{1}$$, a quotient of two integers.
problem 2) Is the number $$ \boxed{ \sqrt{ 25} } $$ 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 $$ \boxed{ 0.09009000900009...} $$ 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 $$ \boxed{ 0.\overline{201}} $$ rational or irrational ?
This is rational. All repeating decimals are rational (see bottom of page for a proof.)
problem 5) Is the number $$ \boxed { \frac{ \sqrt{3}}{4} } $$ 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 $$ \boxed { \frac{ \sqrt{9}}{25} } $$ 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 $$ \boxed { \frac{ \pi}{\pi} } $$ 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 $$ \boxed { \frac{ \sqrt{2}}{ \sqrt{2} } }$$ 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
Next
Rational Number Game
Further Reading
