Enter a number as an integer/decimal, simple fraction, or mixed number to find its multiplicative inverse (reciprocal). The calculator returns the inverse in both fraction form and decimal form — the value that, when multiplied by your input, equals 1.
Multiplicative Inverse (Decimal)
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Multiplicative Inverse (Fraction)
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Original Number (Decimal)
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Verification (Original × Inverse)
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The multiplicative inverse of a number a is a value b such that a × b = 1. It is also called the reciprocal. For example, the multiplicative inverse of 5 is 1/5, because 5 × (1/5) = 1. Note that 0 does not have a multiplicative inverse, since no number multiplied by 0 can equal 1.
To find the multiplicative inverse of a fraction a/b, simply flip the numerator and denominator to get b/a. For example, the inverse of 3/4 is 4/3. This works because (3/4) × (4/3) = 12/12 = 1.
For any non-zero number x, its multiplicative inverse is simply 1/x. For an integer like 8, the inverse is 1/8 = 0.125. For a decimal like 2.5, the inverse is 1/2.5 = 0.4. You can think of any integer or decimal as a fraction over 1, then flip it.
First, convert the mixed number to an improper fraction. For example, 2 1/3 becomes (2×3+1)/3 = 7/3. Then flip the fraction to get 3/7. So the multiplicative inverse of 2 1/3 is 3/7.
Every non-zero real number has a multiplicative inverse. The number 0 is the only exception — it has no multiplicative inverse, because there is no number you can multiply by 0 to get 1. Negative numbers do have inverses: for example, the inverse of -4 is -1/4.
The multiplicative inverse of a number x is 1/x, because x × (1/x) = 1 (the multiplicative identity). The additive inverse of x is -x, because x + (-x) = 0 (the additive identity). These are two completely different concepts.
The standard multiplicative inverse of x is 1/x in the real numbers. The modular multiplicative inverse of an integer a modulo m is an integer b such that a × b ≡ 1 (mod m). The modular version is used in number theory and cryptography and only applies to integers with a specific modulus.
The multiplicative inverse of 1 is 1 itself, since 1 × 1 = 1. The multiplicative inverse of -1 is also -1, since (-1) × (-1) = 1. These are the only two real numbers that are their own multiplicative inverses.