Rational and irrational numbers

You already know how to work with fractions, decimals and negative numbers. In 8th grade you meet a surprising fact: some numbers on the number line can never be written as a fraction. This lesson shows you how to tell the two kinds of numbers apart.

Rational numbers

The word rational comes from ratio. A rational number is any number you can write as a ratio (a fraction) of two integers.

The key word is can. A number doesn't have to look like a fraction to be rational. You just need to be able to rewrite it as one.

number	written as a fraction
$7$	$\dfrac{7}{1}$
$-4$	$\dfrac{-4}{1}$
$0$	$\dfrac{0}{1}$
$0.9$	$\dfrac{9}{10}$
$-1.25$	$-\dfrac{125}{100} = -\dfrac{5}{4}$
$3\tfrac{1}{2}$	$\dfrac{7}{2}$

So every integer, every terminating decimal and every mixed number is rational.

Decimals of rational numbers

To turn a fraction into a decimal, divide the numerator by the denominator. Only two things can happen:

• The division ends (terminates): $\dfrac{3}{8} = 3 \div 8 = 0.375$.
• The division repeats forever: $\dfrac{5}{11} = 0.454545\ldots$

We write a repeating decimal with a bar over the digits that repeat: $0.454545\ldots = 0.\overline{45}$.

Why must it end or repeat? When you divide by $11$, each remainder is one of $0, 1, 2, \dots, 10$. If you ever get a remainder of $0$, the decimal ends. If not, you only have $10$ possible remainders, so sooner or later a remainder comes back. From that point on, the same steps (and the same digits) happen again and again.

Irrational numbers

Some decimals go on forever and never fall into a repeating pattern. Since every fraction's decimal ends or repeats, these numbers cannot be fractions.

The most famous examples are:

• $\pi = 3.14159265\ldots$, the ratio of a circle's circumference to its diameter.
• $\sqrt{2} = 1.41421356\ldots$, the number that gives $2$ when you multiply it by itself.
• The square root of any whole number that is not a perfect square, such as $\sqrt{3}$, $\sqrt{10}$ or $\sqrt{50}$.

A decimal can also be built to never repeat, like $0.101001000100001\ldots$ (one more $0$ each time). There is a pattern, but no block of digits repeats, so it is irrational.

Together, the rational and irrational numbers make up the real numbers, which fill the whole number line. Every real number is exactly one of the two: rational or irrational, never both.

Practice

Practice 1

Which number is irrational?

A$\sqrt{49}$B$0.\overline{3}$C$\sqrt{12}$D$-\dfrac{8}{3}$

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Practice 2

Write $\dfrac{9}{20}$ as a decimal.

Your answer

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

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Practice 3

Which is the correct way to write $\dfrac{7}{12}$ as a decimal?

A$0.58\overline{3}$B$0.\overline{583}$C$0.\overline{58}$D$0.583$

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Practice 4

Which statement is true?

AEvery decimal that goes on forever is irrational.BEvery integer is a rational number.C$\sqrt{36}$ is irrational because it has a square root sign.DSome numbers are both rational and irrational.

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Practice 5

How many of these numbers are irrational?

$\sqrt{81}, \quad \sqrt{8}, \quad 5.\overline{21}, \quad 2\pi, \quad \sqrt{30}, \quad -\dfrac{4}{9}$

Your answer

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

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Practice 6

Maya's calculator shows $\dfrac{3}{17} = 0.1764705882$. She says, "I see no pattern, so $\dfrac{3}{17}$ is irrational." What is wrong with her reasoning?

ANothing. The digits don't repeat, so it is irrational.BThe decimal terminates after 10 digits, so it is rational.C$\dfrac{3}{17}$ is irrational only because $17$ is prime.DIt is a fraction of integers, so its decimal must repeat; the repeating block is just longer than the screen.

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Practice 7

The decimal $0.12112111211112\ldots$ keeps adding one more $1$ before each $2$. What kind of number is it?

ARational, because it has a patternBRational, because the block $12$ repeatsCIrrational, because no block of digits repeats foreverDNot a real number

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