How to Find the Multiplicative Inverse: Formula and Examples
The multiplicative inverse of a number is the number you multiply it by to get 1. It is also called the reciprocal. This guide shows how to find multiplicative inverse values for every common type of number, from fractions to modular arithmetic.
The multiplicative inverse formula
Inverse of a = 1 ÷ a, because a × (1 ÷ a) = 1
Every number except zero has exactly one multiplicative inverse.
How to find multiplicative inverse values
Whole numbers
Write the number over 1 and flip it. The inverse of 5 is 1/5, because 5 × 1/5 = 1.
Fractions
Swap the numerator and denominator. The inverse of 3/4 is 4/3, since 3/4 × 4/3 = 12/12 = 1.
Mixed numbers
Convert to an improper fraction first, then flip. 2 1/3 = 7/3, so its inverse is 3/7.
Decimals
Divide 1 by the decimal, or turn it into a fraction first. 0.25 = 1/4, so its inverse is 4.
Negative numbers
The inverse keeps the sign. The inverse of −6 is −1/6, since (−6) × (−1/6) = 1.
Zero
Zero has no multiplicative inverse, because nothing times zero equals 1.
Multiplicative inverse examples
| Number | Multiplicative inverse | Check |
|---|---|---|
| 5 | 1/5 | 5 × 1/5 = 1 |
| 3/4 | 4/3 | 3/4 × 4/3 = 1 |
| 2 1/3 | 3/7 | 7/3 × 3/7 = 1 |
| 0.25 | 4 | 0.25 × 4 = 1 |
| −6 | −1/6 | −6 × −1/6 = 1 |
Multiplicative inverse of a complex number
Multiply top and bottom by the conjugate:
1 ÷ (a + bi) = (a − bi) ÷ (a² + b²)
For 3 + 4i, the inverse is (3 − 4i) ÷ 25 = 0.12 − 0.16i.
How to find the modular multiplicative inverse
In modular arithmetic, the inverse of a modulo m is the number x where a × x leaves a remainder of 1 when divided by m. It exists only when a and m share no common factor.
Small numbers: test values
For 3 mod 11, try x = 1, 2, 3, and so on: 3 × 4 = 12 = 11 + 1, so the inverse is 4.
Larger numbers: the extended Euclidean algorithm
For 7 mod 26, run Euclid’s algorithm and work backwards: 26 = 3 × 7 + 5, 7 = 1 × 5 + 2, 5 = 2 × 2 + 1. Back-substituting gives 1 = 3 × 26 − 11 × 7, so 7 × (−11) leaves remainder 1. Adding 26 makes it positive: the inverse is 15, and 7 × 15 = 105 = 4 × 26 + 1. This is the step used in cryptography such as RSA.
The multiplicative inverse calculator returns the reciprocal of any number or fraction, and the modular inverse when you enter a modulus.
Common mistakes
- Confusing inverse with opposite. The additive inverse of 5 is −5; the multiplicative inverse is 1/5.
- Flipping a mixed number directly. Convert to an improper fraction first.
- Changing the sign. A negative number’s inverse is also negative.
Reciprocals show up in logarithms too: logab and logba are reciprocals. See how to calculate logarithms without a calculator.
To work with coordinates, read in how to calculate the distance formula.
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