Calculator Inputs
Use integers, decimals, or fractions like 7/9. Modular modes require integer entries.
Example Data Table
| Mode | Element | Modulus | Inverse | Verification |
|---|---|---|---|---|
| Additive inverse | 12 | — | -12 | 12 + (-12) = 0 |
| Multiplicative inverse | 5 | — | 0.2 | 5 × 0.2 = 1 |
| Additive inverse modulo n | 8 | 11 | 3 | (8 + 3) mod 11 = 0 |
| Multiplicative inverse modulo n | 7 | 26 | 15 | (7 × 15) mod 26 = 1 |
Formula Used
1) Additive inverse
For an element a, the additive inverse is -a.
a + (-a) = 0
2) Multiplicative inverse
For a nonzero element a, the multiplicative inverse is 1/a.
a × (1/a) = 1, provided a ≠ 0.
3) Additive inverse modulo n
The additive inverse of a modulo n is the residue that makes the sum congruent to zero.
a + b ≡ 0 (mod n), so b ≡ -a (mod n).
4) Multiplicative inverse modulo n
The modular multiplicative inverse exists only when gcd(a, n) = 1.
a × b ≡ 1 (mod n)
This calculator uses the extended Euclidean algorithm to solve ax + ny = 1, then converts x into the least nonnegative residue.
How to Use This Calculator
- Choose the inverse type from the mode list.
- Enter the element value as an integer, decimal, or fraction.
- Supply a modulus only for modular calculations.
- Set the display precision for decimal results.
- Keep the steps option enabled if you want full working.
- Press the calculate button.
- Read the result card that appears above the form.
- Use the CSV or PDF buttons to export the summary.
FAQs
1) What is an inverse element?
An inverse element combines with a chosen element to produce the identity. Under addition, the identity is 0. Under multiplication, the identity is 1. In modular systems, the same idea applies within a fixed modulus.
2) Does every element have an inverse?
No. Every number has an additive inverse, but zero lacks a multiplicative inverse. In modular multiplication, an inverse exists only when the element and modulus are coprime.
3) Why is gcd important in modular inverses?
The condition gcd(a, n) = 1 guarantees that the congruence a×x ≡ 1 (mod n) has a solution. If the gcd is larger than 1, no modular multiplicative inverse exists.
4) Can I enter fractions?
Yes. The calculator accepts entries like 7/9 for additive and ordinary multiplicative inverses. Modular modes require integer input because modular arithmetic works with integer residues.
5) What does normalization mean in modular arithmetic?
Normalization converts a value into its standard residue class, usually from 0 to n−1. For example, −3 mod 11 becomes 8.
6) Why might the calculator show no inverse?
It shows no inverse when the operation cannot reach the identity. Common examples are zero in ordinary multiplication and non-coprime values in modular multiplication.
7) What is the identity element here?
The identity is the neutral element for the chosen operation. It is 0 for addition, 1 for multiplication, 0 mod n for modular addition, and 1 mod n for modular multiplication.
8) When should I use this calculator?
Use it for algebra homework, number theory checks, modular arithmetic practice, proof verification, and quick classroom demonstrations where inverse existence and validation matter.