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Additive
Inverses

Discover how every number has a mirror partner that brings it back to zero: the foundation of working with positive and negative numbers.

7th Grade Math

What Is an Additive Inverse?

The additive inverse of a number is the number you add to it to get zero. Every number has exactly one additive inverse: its opposite.

+7 and -7
\( 7 + (-7) = 0 \)
+3.5 and -3.5
\( 3.5 + (-3.5) = 0 \)
0
\( 0 + 0 = 0 \) (self-inverse)

Visualizing on a Number Line

A number and its additive inverse sit at equal distances from zero, but on opposite sides. When you walk from one to the other through zero, the trips cancel out.

-5 -3 -1 0 +1 +3 +5 +4 -4 4 units 4 units

Opposites in the Real World

Additive inverses appear whenever two actions cancel each other out. If you gain something and then lose the exact same amount, you are back where you started.

Deposit & Withdrawal
Deposit \( \$50 \), then withdraw \( \$50 \): net change is \( \$0 \).
Climb & Descend
Climb 12 floors up, then descend 12 floors: you are back at ground level.
Gain & Loss
Earn 8 points, then lose 8 points: your score is unchanged.

The Additive Inverse Property

For every \( a \):   \( a + (-a) = 0 \)
1
\( a \) can be any rational number: integers, fractions, or decimals.
2
\( -a \) is the opposite sign of \( a \). If \( a = 5 \), then \( -a = -5 \). If \( a = -3 \), then \( -a = 3 \).
3
Zero is its own inverse: \( 0 + 0 = 0 \). It is the only number equal to its additive inverse.

Opposite Arrows Cancel Out

Think of each number as an arrow on the number line. The additive inverse is an arrow of the same length pointing in the opposite direction. Placed tip-to-tail, they return to the origin.

0 +5 -5 Net displacement = 0 -5 +5

Additive Inverse vs. Absolute Value

Students sometimes confuse the additive inverse (opposite) with the absolute value (distance from zero). They are related but different operations.

Additive Inverse Absolute Value
Symbol \( -a \) \( |a| \)
Of \( +6 \) \( -6 \) \( 6 \)
Of \( -6 \) \( +6 \) \( 6 \)
Result sign Flips the sign Always non-negative
Key property \( a + (-a) = 0 \) \( |a| \geq 0 \)

How to Find the Additive Inverse

1
Identify the number. It could be positive, negative, a fraction, or a decimal.
2
Change the sign. If the number is positive, make it negative. If it is negative, make it positive.
3
Verify. Add the original number and its inverse. The result should be \( 0 \).

Example 1: Integer Inverses

Problem: Maria earns \( \$15 \) babysitting, then spends \( \$15 \) on a book. Represent this with additive inverses and find the net change.

$0 $15 +$15 earned -$15 spent Net = $0

Example 2: Fraction Inverses

Problem: A recipe calls for \( \frac{2}{3} \) cup of sugar. You accidentally add \( \frac{2}{3} \) cup too much, so you remove \( \frac{2}{3} \) cup. What is the net change in sugar?

1
Identify the quantities: Added \( +\frac{2}{3} \) cup, removed \( -\frac{2}{3} \) cup.
2
Add them: \( \frac{2}{3} + \left(-\frac{2}{3}\right) = \frac{2}{3} - \frac{2}{3} = 0 \)
3
Interpret: The net change is 0 cups: no extra sugar.

Example 3: Temperature Change

Problem: The temperature rises \( 8.5^\circ \)F during the day, then drops \( 8.5^\circ \)F at night. What is the net temperature change?

+8.5 -8.5 0

Why Zero Is Special

Zero plays a unique role in the world of additive inverses. It is the identity element for addition and the only number that is its own additive inverse.

Identity Element
Adding \( 0 \) to any number leaves it unchanged: \( a + 0 = a \).
Self-Inverse
\( 0 + 0 = 0 \). Zero is the only number equal to its own additive inverse.

The Inverse of a Negative

Students often wonder: what is the additive inverse of a negative number? The rule is the same: flip the sign.

0 -3 +3 opposites -4 -2 +2 +4
1
\( -(-3) = +3 \). The inverse of a negative is its positive counterpart.
2
Verify: \( (-3) + (+3) = 0 \) ✓

Patterns with Inverse Pairs

Every additive inverse pair shares a pattern: equal absolute value, opposite sign. Here is a quick reference.

Number \( a \) Inverse \( -a \) Sum
\( +9 \) \( -9 \) \( 0 \)
\( -2.7 \) \( +2.7 \) \( 0 \)
\( \frac{1}{2} \) \( -\frac{1}{2} \) \( 0 \)
\( -100 \) \( +100 \) \( 0 \)
\( 0 \) \( 0 \) \( 0 \)

Common Mistakes to Avoid

✕ Confusing Inverse with Reciprocal
The additive inverse of \( 5 \) is \( -5 \), not \( \frac{1}{5} \). The reciprocal (multiplicative inverse) is a different concept.
✕ Thinking Negatives Have No Inverse
Every number has an inverse. The additive inverse of \( -4 \) is \( +4 \). Negatives "flip" to positives.
✕ Forgetting the Sum Must Be Zero
If your answer does not produce a sum of \( 0 \), it is not the additive inverse. Always verify: \( a + (-a) = 0 \).

Question 1 of 5

What is the additive inverse of \( -12 \)?

Question 2 of 5

Which situation describes additive inverses combining to make \( 0 \)?

Question 3 of 5

What is the value of \( \frac{3}{5} + \left(-\frac{3}{5}\right) \)?

Question 4 of 5

Which number is its own additive inverse?

Question 5 of 5

A diver descends \( 15 \) meters below sea level, then ascends \( 15 \) meters. Which expression represents the diver's net change in depth?