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Opposites on the
Number Line

Discover how opposite numbers mirror each other across zero and why the opposite of an opposite brings you right back.

6th Grade Math

What Are Opposite Numbers?

Two numbers are opposites if they are the same distance from zero on a number line but sit on opposite sides. For example, \(3\) and \(-3\) are opposites.

-4 -3 0 3 4 3 units 3 units

Zero Is Its Own Opposite

Every number has exactly one opposite, and zero is unique: the opposite of \(0\) is \(0\) itself. Zero sits right in the center of the number line, so it has no "other side" to jump to.

-3 -1 0 1 3
\(-0 = 0\)

Reading the Negative Sign

The symbol \(-\) in front of a number means "the opposite of." So \(-5\) reads as "the opposite of \(5\)," and \(-(-5)\) reads as "the opposite of the opposite of \(5\)."

\(-5\)
"The opposite of \(5\)": start at \(5\), jump across zero to land at \(-5\).
\(-(-5)\)
"The opposite of \(-5\)": jump back across zero to land at \(5\) again.

The Opposite Rule

1
Opposites are equidistant from zero. If a number \(a\) is \(d\) units from zero, then \(-a\) is also \(d\) units from zero, on the other side.
2
Opposites sum to zero. For any number \(a\): \(a + (-a) = 0\).
3
The opposite of the opposite is the original. \(-(-a) = a\) for every number \(a\).

\(-(-a) = a\) on the Number Line

-4 -2 0 2 4 3 -3
Step 1: \( -(3) = -3 \)
Step 2: \( -(-3) = 3 \)

Negative vs. Opposite

Negative Number Opposite
Meaning A number less than zero The number on the other side of zero
Example \(-7\) is negative The opposite of \(-7\) is \(7\)
Sign Always has a \(-\) sign Can be positive, negative, or zero
Key idea Describes a location Describes a relationship

How to Find the Opposite

1
Locate the number on the number line.
2
Count the distance from that number to zero.
3
Move the same distance to the other side of zero.
4
Write the answer with the opposite sign. If you started positive, write negative, and vice versa.

Example: Opposite of \(6\)

-8 -6 -4 0 4 6 8 flip across 0

Example: Opposite of \(-2\)

-4 -3 -2 0 2 3 4 flip across 0

Example: What Is \(-(-7)\)?

Read it from the outside in: "the opposite of \(-7\)." Since \(-7\) is 7 units to the left of zero, its opposite is 7 units to the right.

-7 0 7
\( -(-7) = 7 \)

Opposites for Fractions and Decimals

The same rules apply to all rational numbers. Just change the sign:

Fractions
The opposite of \(\frac{3}{4}\) is \(-\frac{3}{4}\).
The opposite of \(-\frac{1}{2}\) is \(\frac{1}{2}\).
Decimals
The opposite of \(2.5\) is \(-2.5\).
The opposite of \(-0.75\) is \(0.75\).

Opposites Add to Zero

When you add a number and its opposite, you always get zero. This is sometimes called the additive inverse property.

-5 0 5 add -5 land on 0

Opposites in the Real World

Elevation
\(200\) feet above sea level and \(-200\) feet below sea level are opposites.
Temperature
\(10^\circ\text{C}\) above zero and \(-10^\circ\text{C}\) below zero are opposites.
Money
Earning \(\$50\) and spending \(\$50\) cancel out: \(50 + (-50) = 0\).

Common Mistakes to Avoid

✕ "Opposite always means negative"
The opposite of \(-4\) is \(4\), which is positive. Opposite just means the other side of zero.
✕ Forgetting \(-(-a) = a\)
Students sometimes write \(-(-3) = -3\). Two negatives cancel: \(-(-3) = 3\).
✕ Thinking \(0\) has no opposite
Zero does have an opposite: itself. \(-0 = 0\).

Summary at a Glance

Number Opposite Sum Rule Used
\(5\) \(-5\) \(0\) Change sign
\(-8\) \(8\) \(0\) Change sign
\(0\) \(0\) \(0\) \(0\) is its own opposite
\(-(-3)\) \(-3\) \(0\) \(-(-a) = a\), then change sign
\(\frac{1}{2}\) \(-\frac{1}{2}\) \(0\) Works for fractions too

Question 1 of 5

What is the opposite of \(9\)?

Question 2 of 5

What does \(-(-4)\) equal?

Question 3 of 5

Which statement about zero is true?

Question 4 of 5

A number \(a\) is located 5 units to the left of zero on the number line. Which expression gives \(a\)?

Question 5 of 5

What is the opposite of \(-\frac{3}{5}\)?