What you need to know

Things to remember:

  • A rational number is either an integer, a fraction, or a terminating decimal (one that stops), or has a recurring (repeating) digit or digits.

So, what are rational numbers? Well, a rational number is any number that can be written as a fraction with integers on the top and bottom. We have four different types of rational number:

1. Integers, such as -57, – 4, 0, 12, 1000, etc. (Put as a fraction with 1 on the bottom)

\dfrac{-57}{1}

\dfrac{-4}{1}

\dfrac{0}{1}

\dfrac{12}{1}

\dfrac{1000}{1}

2. Terminating decimal numbers (decimals that end), such as 0.2, 0.56, 1.234, etc. (Multiply by a power of 10 to remove the decimal and put it over the power of 10)

\dfrac{2}{10}

\dfrac{56}{100}

\dfrac{1234}{1000}

3. Recurring decimals (decimals that have a repeating digit or digits), like 0.11111… or 0.142857142857… These are more complicated to write as a fraction. But, we usually put a dot or line over the top of numbers to show that is being repeated.

0.111111…=0.\overline{1}

0.142857142857…=\overline{142857}

4. Fractions themselves are rational numbers.

So, why do we have a special name for these numbers? Because there is an opposite type, which we call irrational numbers!

– Irrational numbers are numbers that can’t be written as a fraction, like \pi and \sqrt{2}.

However, not all square roots are irrational. For example, \sqrt{9} can be simply written as “3”, which is a rational number!

Which of these isn’t a rational number?

7                      \dfrac{2}{5}                     8                      -6                     \sqrt{17}

We can check this against our list of 4 types of rational numbers.

7 – Integer

\dfrac{2}{5} – Fraction

8 – Integer

-6 – Integer

\sqrt{17} None

 

\sqrt{17} is none of the four types of rational number, so must be the irrational number!

Example Questions

Hint: See if you can write the square root as an integer.

 

\sqrt{11} – irrational

 

\sqrt{36}=6 – rational

 

\sqrt{81}=9 – rational

 

\sqrt{121}=11 – rational

 

\sqrt{23} – irrational

4 – rational

\pi – irrational

-4 – rational

0.\overline{5} – rational

\sqrt{49}=7 – rational

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