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Additive Inverse: The Number's Opposite (a + (−a) = 0)

Additive Inverse of a Number

The opposite of a number. For example, the additive inverse of 12 is –12. The additive inverse of –3 is 3. Formally, the additive inverse of x is –x. Note: The sum of a number and its additive inverse is 0.

See also

Inverse, negative number

Key Formula

x+(−x)=0x + (-x) = 0
Where:
  • xx = Any real number
  • −x-x = The additive inverse (opposite) of x

Worked Example

Problem: Find the additive inverse of −7 and verify your answer.
Step 1: Identify the number. The given number is −7.
x=−7x = -7
Step 2: Apply the definition: the additive inverse of x is −x. Negate the number.
−x=−(−7)=7-x = -(-7) = 7
Step 3: Verify by adding the number and its additive inverse. The sum should be 0.
−7+7=0✓-7 + 7 = 0 \checkmark
Answer: The additive inverse of −7 is 7.

Another Example

Problem: Find the additive inverse of 3/4.
Step 1: The given number is 3/4.
x=34x = \frac{3}{4}
Step 2: The additive inverse is the negation of the number.
−x=−34-x = -\frac{3}{4}
Step 3: Check: the sum of the number and its additive inverse must equal zero.
34+(−34)=0✓\frac{3}{4} + \left(-\frac{3}{4}\right) = 0 \checkmark
Answer: The additive inverse of 3/4 is −3/4.

Frequently Asked Questions

Is the additive inverse the same as the opposite of a number?
Yes. The terms 'additive inverse' and 'opposite' mean the same thing. The additive inverse of xx is −x-x, which is exactly the opposite of xx on the number line. Both terms describe the number that, when added to xx, gives 0.
What is the additive inverse of 0?
The additive inverse of 0 is 0 itself, because 0+0=00 + 0 = 0. Zero is the only number that is its own additive inverse.

Additive Inverse vs. Multiplicative Inverse (Reciprocal)

The additive inverse of xx is −x-x, because x+(−x)=0x + (-x) = 0. The multiplicative inverse of xx (where x≠0x \neq 0) is 1x\frac{1}{x}, because x⋅1x=1x \cdot \frac{1}{x} = 1. One undoes addition (returning to the additive identity 0), while the other undoes multiplication (returning to the multiplicative identity 1). For example, the additive inverse of 5 is −5, but the multiplicative inverse of 5 is 1/5.

Why It Matters

The additive inverse is essential for solving equations. When you see x+9=14x + 9 = 14, you add the additive inverse of 9 (which is −9) to both sides to isolate xx. More broadly, the concept appears throughout algebra as the foundation of subtraction, since subtracting a number is defined as adding its additive inverse: a−b=a+(−b)a - b = a + (-b).

Common Mistakes

Mistake: Thinking the additive inverse of a negative number is still negative.
Correction: The additive inverse of a negative number is positive. For instance, the additive inverse of −8 is +8, not −8. Always negate the sign: −(−8)=8-(-8) = 8.
Mistake: Confusing additive inverse with multiplicative inverse (reciprocal).
Correction: The additive inverse of 4 is −4 (since 4+(−4)=04 + (-4) = 0), not 1/4. The reciprocal 1/4 is the multiplicative inverse. These are different operations with different identity elements (0 vs. 1).

Related Terms

  • Sum — A number plus its inverse sums to zero
  • Inverse — General concept that includes additive inverse
  • Negative Number — Additive inverses of positive numbers
  • Multiplicative Inverse — Analogous concept for multiplication
  • Identity — Zero is the additive identity involved
  • Absolute Value — A number and its inverse share the same absolute value
  • Opposite — Synonym for additive inverse