Multiplicative Inverse Calculator
This multiplicative inverse calculator finds the reciprocal of any number or fraction, the value that multiplies with it to give 1. Enter a number or fraction and it returns the inverse as a fraction and a decimal. Add a modulus and the multiplicative inverse calculator also finds the modular multiplicative inverse.
Results
The Multiplicative Inverse Calculator is part of our Algebra collection. Browse more algebra to size, convert, and check your work alongside this tool.
What the multiplicative inverse calculator shows
The multiplicative inverse of a number, also called its reciprocal, is 1 divided by that number. A number times its inverse always equals 1. The multiplicative inverse calculator flips fractions, handles negatives and decimals, and, with a modulus, finds the inverse used in modular arithmetic and cryptography.
Multiplicative inverse solver rules
Zero has no inverse. A modular inverse exists only when the number and the modulus share no common factor; the calculator finds it with the extended Euclidean algorithm.
Find multiplicative inverse: examples
The inverse of 3/4:
The modular inverse of 3 mod 11 is 4, because 3 × 4 = 12 = 11 + 1. The modular inverse of 7 mod 26 is 15.
How to use the multiplicative inverse calculator
Enter the number as a numerator, or a fraction as a numerator and denominator. Leave the modulus at 0 for an ordinary reciprocal. The multiplicative inverse calculator returns the inverse as a simplified fraction and as a decimal. Enter a modulus to also find the modular inverse of a whole number.
Reading the multiplicative inverse calculator result
The fraction form is exact; the decimal is rounded. A negative number has a negative inverse. If no modular inverse exists, the calculator says the number is not coprime with the modulus, as with 4 mod 6.
When to find multiplicative inverse values
- Dividing fractions by multiplying by the reciprocal
- Solving equations such as 3x = 7
- Modular arithmetic and RSA cryptography exercises
- Checking homework with the multiplicative inverse calculator
Common multiplicative inverse calculator mistakes
- Confusing inverse with opposite.
- Flipping a mixed number directly. Convert to an improper fraction first.
- Expecting every number to have a modular inverse.
Tips for accurate results
- Check by multiplying the number by its inverse.
- Keep the sign of negative numbers.
- Make sure the number and modulus are coprime.
Multiplicative inverse calculator with steps for modular arithmetic
For a modular inverse, the extended Euclidean algorithm runs Euclid's steps forwards and then works backwards. For 7 mod 26: 26 = 3 × 7 + 5, 7 = 1 × 5 + 2, 5 = 2 × 2 + 1. Back-substituting gives 1 = 3 × 26 − 11 × 7, so −11 works, and adding 26 gives 15. This is the step RSA encryption uses to build its private key. Reciprocals also turn up in percentage work: undoing a 20 percent fall means dividing by 0.8, which the percentage change calculator handles.
Find multiplicative inverse vs opposite
The multiplicative inverse of 5 is 1/5; the additive inverse, or opposite, is −5. They answer different questions: which number multiplies to 1, and which adds to 0.
Multiplicative inverse calculator reference
| Number | Inverse | Check |
|---|---|---|
| 5 | 1/5 | 5 × 1/5 = 1 |
| 3/4 | 4/3 | 3/4 × 4/3 = 1 |
| 0.25 | 4 | 0.25 × 4 = 1 |
| −6 | −1/6 | −6 × −1/6 = 1 |
| 3 (mod 11) | 4 | 3 × 4 = 12 ≡ 1 |
Multiplicative Inverse Calculator FAQ
What is a multiplicative inverse?
The number that multiplies with a given number to give 1, also called the reciprocal.
How do I find the inverse of a fraction?
Swap the numerator and denominator: the inverse of 3/4 is 4/3.
Does zero have a multiplicative inverse?
No, because nothing times zero equals 1.
What is a modular inverse?
A whole number x with a × x leaving remainder 1 when divided by the modulus.