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Multiplicative Inverse — Definition, Formula & Examples

The multiplicative inverse of a number is what you multiply that number by to get 1. For example, the multiplicative inverse of 5 is 15\frac{1}{5} because 5×15=15 \times \frac{1}{5} = 1.

For any nonzero number aa, its multiplicative inverse is the unique number a−1a^{-1} such that a×a−1=1a \times a^{-1} = 1. The multiplicative inverse is also called the reciprocal. Zero has no multiplicative inverse because no number multiplied by 0 equals 1.

Key Formula

a×1a=1a \times \frac{1}{a} = 1
Where:
  • aa = Any nonzero number
  • 1a\frac{1}{a} = The multiplicative inverse (reciprocal) of a

How It Works

To find the multiplicative inverse, flip the number. For a whole number like 7, write it as 71\frac{7}{1} and flip it to 17\frac{1}{7}. For a fraction like 34\frac{3}{4}, swap the numerator and denominator to get 43\frac{4}{3}. For a negative number like −2-2, the inverse is −12-\frac{1}{2} — the result is still negative because a negative times a negative would give a positive product, but you need the signs to match so the product is exactly 1.

Worked Example

Problem: Find the multiplicative inverse of 38\frac{3}{8} and verify your answer.
Flip the fraction: Swap the numerator and denominator.
38→83\frac{3}{8} \rightarrow \frac{8}{3}
Verify: Multiply the original number by its inverse to confirm the product is 1.
38×83=2424=1\frac{3}{8} \times \frac{8}{3} = \frac{24}{24} = 1
Answer: The multiplicative inverse of 38\frac{3}{8} is 83\frac{8}{3}.

Why It Matters

Dividing by a number is the same as multiplying by its inverse, which is why understanding reciprocals is essential for fraction division. This concept also appears when solving equations — multiplying both sides by a multiplicative inverse isolates the variable.

Common Mistakes

Mistake: Confusing the multiplicative inverse with the additive inverse (opposite sign).
Correction: The additive inverse of 5 is −5-5 (they add to 0), while the multiplicative inverse of 5 is 15\frac{1}{5} (they multiply to 1). These are different operations with different purposes.