Multiplicative Inverse Calculator
Enter any integer, decimal, simple fraction, or mixed number and this calculator returns its multiplicative inverse (reciprocal) as a simplified fraction, with full step-by-step working. Switch to modular inverse mode to find the inverse of an integer modulo m using the extended Euclidean algorithm.
What is a multiplicative inverse?
The multiplicative inverse of a number x is the value that, when multiplied by x, gives exactly 1. It is also called the reciprocal. For any non-zero number x, the multiplicative inverse is written as 1/x or x to the power of -1. For example, the multiplicative inverse of 4 is 1/4, because 4 × 1/4 = 1. For a fraction a/b, flipping the numerator and denominator gives b/a, the reciprocal. Zero is the one exception: it has no multiplicative inverse because dividing by zero is undefined.
How to find the multiplicative inverse
For an integer n, write it as n/1 and flip to get 1/n. For a decimal, convert it to a fraction first by multiplying numerator and denominator by the appropriate power of 10 (e.g., 0.25 = 25/100 = 1/4), then flip. For a simple fraction a/b, the inverse is b/a. For a mixed number such as 1 and 2/3, first convert to an improper fraction (5/3), then flip to get 3/5. Always simplify the result to its lowest terms by dividing numerator and denominator by their greatest common divisor.
Modular multiplicative inverse
In modular arithmetic, the modular inverse of an integer a modulo m is the integer x such that a multiplied by x leaves a remainder of 1 when divided by m (written a * x ≡ 1 mod m). This only exists when a and m are coprime, meaning they share no common factors other than 1. The most efficient method to compute it is the extended Euclidean algorithm, which finds x directly. Modular inverses are central to RSA public-key cryptography, Diffie-Hellman key exchange, and many problems in competitive programming and number theory.
Practical uses of the reciprocal
Division by a fraction is identical to multiplication by its inverse: dividing by 3/4 gives the same result as multiplying by 4/3. This rule appears constantly in algebra, physics formulas, and spreadsheet calculations. In linear algebra, the inverse of a matrix M is the matrix M-inverse such that M multiplied by M-inverse equals the identity matrix, a direct generalization of the scalar reciprocal. In signal processing, the inverse of a frequency (1/f) gives the period. Anytime you want to undo a multiplication, the multiplicative inverse is the tool.
Common multiplicative inverses
| Number | Inverse (fraction) | Inverse (decimal) |
|---|---|---|
| 1 | 1/1 = 1 | 1.0 |
| 2 | 1/2 | 0.5 |
| 3 | 1/3 | 0.3333... |
| 4 | 1/4 | 0.25 |
| 5 | 1/5 | 0.2 |
| 6 | 1/6 | 0.1667... |
| 8 | 1/8 | 0.125 |
| 10 | 1/10 | 0.1 |
| 1/2 | 2/1 = 2 | 2.0 |
| 1/3 | 3/1 = 3 | 3.0 |
| 2/3 | 3/2 | 1.5 |
| 3/4 | 4/3 | 1.3333... |
| 1/4 | 4/1 = 4 | 4.0 |
| -1 | -1/1 = -1 | -1.0 |
| -2 | -1/2 | -0.5 |
Quick reference for frequently used numbers and their reciprocals.
Frequently asked questions
What is the multiplicative inverse of 0?
Zero has no multiplicative inverse. For any number x to be the inverse of 0, you would need 0 × x = 1, but 0 multiplied by anything is always 0, never 1. Division by zero is undefined, so zero stands alone as the only real number without a reciprocal.
Is the multiplicative inverse the same as the reciprocal?
Yes, the terms are interchangeable. Both refer to the number you multiply by to get 1. The word "reciprocal" is more common in everyday arithmetic, while "multiplicative inverse" is the formal algebraic term used in group theory and abstract algebra.
What is the multiplicative inverse of a negative number?
A negative number has a negative inverse. The multiplicative inverse of -5 is -1/5, because -5 × -1/5 = 1. The sign of the inverse always matches the sign of the original number, since multiplying two negatives gives a positive.
How do I find the multiplicative inverse of a mixed number?
First convert the mixed number to an improper fraction: for w a/b, the improper fraction is (w * b + a) / b. Then flip it: the inverse is b / (w * b + a). For example, 2 and 1/3 becomes 7/3, and its inverse is 3/7.
When does a modular inverse not exist?
The modular inverse of a modulo m does not exist if a and m share a common factor greater than 1 (they are not coprime). For example, 4 has no inverse modulo 6, because gcd(4, 6) = 2 (not 1). When gcd(a, m) = 1, a unique inverse in the range [1, m-1] is guaranteed to exist.