Math Core

Walk 3 steps forward, then 3 steps backward, and you're right back where you started. Numbers work the same way: every number has a partner on the other side of zero, the same distance away. These partners are called opposites.

Mirror images across zero

Look at $4$ and $-4$ on the number line. Each one is exactly 4 units from $0$ , but they are on opposite sides.

4 and -4 are both 4 units from 0, on opposite sides.

If you folded the number line at $0$ , the point $4$ would land right on top of $-4$ . Zero acts like a mirror.

Definition

Opposites

Two numbers are opposites if they are the same distance from $0$ on the number line but on opposite sides of $0$ . The opposite of $a$ is written $-a$ .

A few pairs of opposites:

number	opposite
$7$	$-7$
$-2$	$2$
$1.5$	$-1.5$
$-\dfrac{3}{4}$	$\dfrac{3}{4}$

Opposites work for fractions and decimals too, not just integers.

What is the opposite of zero? Zero is 0 units from itself, and it doesn't have a side. So the opposite of $0$ is $0$ . It is the only number that is its own opposite.

Two meanings of the minus sign

The sign " $-$ " in front of a number can be read as "the opposite of." So $-5$ means "the opposite of 5," which is negative five. This reading lets you make sense of a number with more than one sign.

Worked example: The opposite of an opposite

What is $-(-6)$ ?

Read it as "the opposite of the opposite of 6."

The opposite of $6$ is $-6$ . That flips from the right side of zero to the left side.
The opposite of $-6$ is $6$ . That flips back to the right side.

So $-(-6) = 6$ .

Flipping sides

Taking the opposite flips a number to the other side of $0$ . Flip twice and you're back where you started:

$-(-a) = a$

Worked example: Stacked signs

Simplify $-(-(-3))$ .

Work from the inside out.

$-3$ is negative three.
$-(-3)$ is the opposite of $-3$ , which is $3$ .
$-(-(-3))$ is the opposite of $3$ , which is $-3$ .

So $-(-(-3)) = -3$ .

Tip

Count the minus signs in front of a number. An even number of signs gives a positive result. An odd number of signs gives a negative result. Check: $-(-(-3))$ has 3 signs (odd), and the answer is $-3$ .

Common mistake

"Opposite" does not mean "negative." The opposite of a negative number is positive. For example, the opposite of $-9$ is $9$ , not $-9$ .

Opposites in real life

Opposite situations cancel each other out. Gaining 10 yards and losing 10 yards puts a team back where it started. Depositing $50 and then withdrawing $50 leaves the account unchanged.

Worked example: A real-world opposite

A hiker climbs to 120 feet above a lake's surface, written $120$ . What integer describes a diver who is the same distance below the surface?

The diver is the same distance from $0$ but on the other side, so the diver's position is the opposite of $120$ : $-120$ .