Math Core

Variables stand for numbers, but which numbers? In Algebra 1 the answer is almost always "any real number." This lesson sorts the real numbers into families, so you can tell what kind of number you're working with and why it matters.

Building up the number system

People invented new kinds of numbers whenever the old ones couldn't answer a question. Each set below contains the one before it.

Definition

Sets of real numbers

Natural numbers (counting numbers): $1, 2, 3, 4, \dots$
Whole numbers: the natural numbers and zero, $0, 1, 2, 3, \dots$
Integers: the whole numbers and their opposites, $\dots, -3, -2, -1, 0, 1, 2, 3, \dots$
Rational numbers: any number that can be written as a fraction $\dfrac{a}{b}$ , where $a$ and $b$ are integers and $b \ne 0$ .
Irrational numbers: real numbers that cannot be written as a fraction of integers.
Real numbers: all rational and irrational numbers together. Every point on the number line is a real number.

Why the growth? Natural numbers can't answer $3 - 5$ , so we need integers. Integers can't answer $3 \div 5$ , so we need rational numbers. And no fraction squares to exactly $2$ , so $\sqrt{2}$ requires irrational numbers.

You can picture the sets as nested boxes: natural numbers sit inside the whole numbers, which sit inside the integers, which sit inside the rational numbers. The irrational numbers are a separate box beside the rationals. Rationals and irrationals together fill the real numbers, and no number is both.

Rational numbers

A number is rational if you can write it as a fraction of integers, even if it isn't shown that way.

Every integer is rational: $-6 = \dfrac{-6}{1}$ .
Every terminating decimal is rational: $0.35 = \dfrac{35}{100} = \dfrac{7}{20}$ .
Every repeating decimal is rational: $0.\overline{3} = 0.333\ldots = \dfrac{1}{3}$ .
Mixed numbers are rational: $2\tfrac{1}{4} = \dfrac{9}{4}$ .

Irrational numbers

An irrational number's decimal goes on forever without repeating a pattern. The most common ones you'll meet are:

Square roots of numbers that aren't perfect squares, like $\sqrt{2} \approx 1.41421\ldots$ , $\sqrt{10}$ and $\sqrt{27}$ .
$\pi$ $\approx 3.14159\ldots$

Common mistake

A square root symbol does not automatically make a number irrational. Check whether the number under the root is a perfect square: $\sqrt{49} = 7$ and $\sqrt{\tfrac{9}{16}} = \tfrac{3}{4}$ are rational. Also, $\dfrac{22}{7}$ is a rational approximation of $\pi$ ; it is not equal to $\pi$ .

Rational or irrational?

Ask: can this number be written as an integer over a nonzero integer? Terminating and repeating decimals: yes, rational. Non-repeating, non-terminating decimals (such as $\sqrt{\text{non-perfect square}}$ and $\pi$ ): no, irrational.

Worked example: Classifying numbers

List every set each number belongs to.

$-8$
$0$
$\sqrt{36}$
$-\dfrac{5}{3}$
$\sqrt{15}$

Solutions.

$-8$ : integer, rational, real. (It's negative, so not whole or natural.)
$0$ : whole number, integer, rational, real. (Zero is not a counting number.)
$\sqrt{36} = 6$ : natural, whole, integer, rational, real.
$-\dfrac{5}{3}$ : rational, real. (It isn't an integer.)
$\sqrt{15}$ : $15$ is not a perfect square, so it is irrational and real.

Estimating square roots

To place an irrational square root on the number line, find the perfect squares on either side of it.

Worked example: Between which integers?

Between which two consecutive integers is $\sqrt{40}$ ?

The perfect squares closest to $40$ are $36 = 6^2$ and $49 = 7^2$ . Since $36 < 40 < 49$ , we get $6 < \sqrt{40} < 7$ .

Because $40$ is much closer to $36$ than to $49$ , $\sqrt{40}$ is a little more than $6$ . (In fact $\sqrt{40} \approx 6.32$ .)

Comparing and ordering real numbers

To compare numbers written in different forms, convert them all to decimals.

Worked example: Ordering mixed forms

Order from least to greatest: $\sqrt{5}$ , $2.2$ , $\dfrac{9}{4}$ , $2.\overline{1}$ .

Convert each to a decimal:

number	decimal
$\sqrt{5}$	$2.236\ldots$
$2.2$	$2.200$
$\dfrac{9}{4}$	$2.250$
$2.\overline{1}$	$2.111\ldots$

From least to greatest: $2.\overline{1}$ , $2.2$ , $\sqrt{5}$ , $\dfrac{9}{4}$ .

Tip

When two decimals are close, write out enough digits to see the first place where they differ. $2.236\ldots$ and $2.250$ agree in the ones and tenths places, but the hundredths digit ( $3$ vs. $5$ ) decides it.

Practice

Practice 1

Which number is irrational?

Practice 2

Which set does $-12$ belong to?

Practice 3

Which statement is true?

Practice 4

Between which two consecutive integers is $\sqrt{70}$ ? Enter the smaller integer first.

Separate answers with commas, e.g. 2, -5

Practice 5

Write $0.375$ as a fraction in lowest terms.

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

Practice 6

Which number is the greatest?

Practice 7

How many of these numbers are rational? $\quad \sqrt{64}, \quad \pi, \quad -0.4, \quad \sqrt{3}, \quad \dfrac{0}{5}$

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