2.6Rational and Irrational Numbers

Integers, fractions, and decimals that stop or repeat all belong to one family: the rational numbers.

Rational number

\displaystyle \text{A number that can be expressed as } \frac{p}{q} \text{ where } p, q \text{ are integers and } q \neq 0.

All integers are rational (e.g. $5 = \frac{5}{1}$ , $-3 = \frac{-3}{1}$ , $0 = \frac{0}{1}$ )
Terminating decimals are rational (e.g. $0.75 = \frac{3}{4}$ )
Recurring decimals are rational (e.g. $0.\overline{3} = \frac{1}{3}$ , $0.\overline{142857} = \frac{1}{7}$ )

Not every number is rational. Some decimals run forever without ever repeating.

$\sqrt{2} = 1.41421356\ldots$ is irrational (never terminates, never repeats)
$\sqrt{3}, \sqrt{5}, \sqrt{7}, \ldots$ are irrational
$\pi = 3.14159265\ldots$ is irrational
$\sqrt{4} = 2$ is rational (it is a perfect square)

The real numbers are the rationals and irrationals together: every point on the number line.

Every real number sits in exactly one of the bottom three boxes.

Classifying numbers

Classify each number as rational or irrational: (a) $\displaystyle 0.125$ (b) $\displaystyle \pi$ (c) $\displaystyle \sqrt{49}$ (d) $\displaystyle 0.\overline{27}$

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(a)

0.125 = \frac{1}{8}

(terminating decimal), so it is rational.

(b)

\pi = 3.14159\ldots

(non-terminating, non-repeating), so it is irrational.

(c)

\sqrt{49} = 7

, so it is rational.

(d)

0.\overline{27} = \frac{3}{11}

(recurring decimal), so it is rational.

Now you try

Is $\sqrt{17}$ a rational number?

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17

is not a perfect square, so

\sqrt{17}

cannot be written as a fraction of two integers and is irrational. Every irrational number is real.

Therefore,

\sqrt{17}

is irrational.

The answer is No.

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Predict

Is $\sqrt{50}$ rational or irrational? Decide before reading on.

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For a whole number

n

\sqrt{n}

is rational exactly when

n

is a perfect square.

The perfect squares are

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, \ldots

49

and

64

are on that list;

50

sits between them, so it is not a perfect square.

\sqrt{50}

is irrational. Compare

\sqrt{49} = 7

, which is rational.

Recall: Perfect squares from Chapter 1, a number whose prime factor indices are all even.→