Math Core

Temperatures below zero, a bank balance that dips into overdraft, a submarine $300$ feet below sea level: all of these need numbers less than zero. Algebra uses negative numbers constantly, so this lesson makes sure you can place them, compare them and measure their size with confidence.

What integers are

Definition

Integers

The integers are the whole numbers and their opposites:

\ldots,\ -3,\ -2,\ -1,\ 0,\ 1,\ 2,\ 3,\ \ldots

Numbers to the right of $0$ are positive. Numbers to the left of $0$ are negative. Zero is neither positive nor negative.

On a horizontal number line, positive numbers sit to the right of $0$ and negative numbers sit to the left. Here are the integers $-4$ , $-1$ and $3$ :

Numbers such as $-2.5$ or $-\dfrac{1}{2}$ also live on the number line, between the integers. They are not integers, but everything in this lesson about order and distance works for them too.

Opposites

Two numbers are opposites if they are the same distance from $0$ but on different sides. The opposite of $5$ is $-5$ , and the opposite of $-5$ is $5$ . The opposite of $0$ is $0$ .

The negative sign can be read as "the opposite of." So $-(-7)$ means "the opposite of $-7$ ," which is $7$ .

5 and −5 are opposites: each is 5 units from 0.

Comparing integers

On a number line, the number farther to the right is greater. That one rule settles every comparison.

$3 > -4$ , because $3$ is to the right of $-4$ . Every positive number is greater than every negative number.
$-1 > -4$ , because $-1$ is to the right of $-4$ .
$-6 < 0$ , because $-6$ is to the left of $0$ .

Common mistake

With negative numbers, a bigger-looking digit means a smaller number. $-8$ is less than $-2$ , because $-8$ is farther to the left. Think of temperature: $-8^\circ$ is colder than $-2^\circ$ .

Absolute value

Definition

Absolute value

The absolute value of a number is its distance from $0$ on the number line. It is written with vertical bars: $|a|$ .

Distance is never negative, so $|a| \ge 0$ for every number $a$ .

For example, $|6| = 6$ and $|-6| = 6$ , because both numbers are $6$ units from $0$ . And $|0| = 0$ .

Absolute value bars are a grouping symbol. Simplify what is inside first, then take the absolute value: $|3 - 10| = |-7| = 7$ .

Tip

A negative sign outside the bars stays: $-|-4|$ means "the opposite of $|-4|$ ," which is $-4$ .

Worked example: Ordering integers

Order from least to greatest: $2,\ -5,\ 0,\ -1,\ -9$ .

Picture them on a number line and read from left to right. The farthest left is $-9$ , then $-5$ , then $-1$ , then $0$ , then $2$ .

-9 < -5 < -1 < 0 < 2

Worked example: Absolute value versus value

Which is greater, $-12$ or $|-3|$ ?

First simplify: $|-3| = 3$ . Now compare $-12$ and $3$ . A positive number is greater than a negative one, so $|-3|$ is greater.

Notice that $-12$ has the larger absolute value ( $12$ ), but the smaller value. Absolute value measures size, not order.

Worked example: Distance between two integers

The temperature at dawn was $-7^\circ\text{F}$ . By noon it was $5^\circ\text{F}$ . How many degrees did it rise?

From $-7$ up to $0$ is $7$ degrees. From $0$ up to $5$ is $5$ more. The total rise is $7 + 5 = 12$ degrees.

Practice

Practice 1

Which statement is true?

Practice 2

What is the opposite of $8$ ?