Calculator Input
Enter one value and modulus for a direct answer. Add a list or range to generate a larger table and graph.
Example Data Table
These examples show how different integers map to their additive inverses under the same modulus.
| Input a | Modulus n | a mod n | Additive inverse | Check |
|---|---|---|---|---|
| 7 | 11 | 7 | 4 | (7 + 4) mod 11 = 0 |
| -3 | 11 | 8 | 3 | (8 + 3) mod 11 = 0 |
| 25 | 12 | 1 | 11 | (1 + 11) mod 12 = 0 |
| 18 | 9 | 0 | 0 | (0 + 0) mod 9 = 0 |
Formula Used
Definition: The additive inverse of a modulo n is a number b such that:
a + b ≡ 0 (mod n)
b ≡ -a (mod n)
b = (n - (a mod n)) mod n
If the normalized residue is 0, the additive inverse is also 0. This calculator always normalizes the residue first.
How to Use This Calculator
- Enter the integer value a.
- Enter a positive modulus n.
- Optionally provide a list of integers for batch results.
- Or enter a start, end, and step for a range.
- Click Calculate Additive Inverse.
- Read the result card above the form for the direct answer.
- Review the table and graph for patterns across many values.
- Use the CSV or PDF buttons to save your output.
FAQs
1) What is an additive inverse modulo n?
It is the number that adds to a given residue and produces zero under a chosen modulus. In modular arithmetic, it is written as (-a) mod n.
2) Why does the calculator normalize the input first?
Normalization converts any integer, including negative values, into its standard residue class between 0 and n-1. That makes the inverse easy to compute and compare.
3) Can negative integers have additive inverses modulo n?
Yes. Every integer maps to a residue class. Once the calculator finds that residue, it computes the modular additive inverse from the residue, not from the original sign.
4) Why is zero its own additive inverse?
Zero already satisfies the identity condition because 0 + 0 remains 0 under every positive modulus. So its additive inverse is always zero.
5) Does the modulus need to be prime?
No. Additive inverses exist for every residue class under any positive modulus. Primality matters for multiplicative inverses, not for additive inverses.
6) What happens when the modulus is 1?
All integers are congruent to zero modulo 1. That means the normalized residue is 0, and the additive inverse is also 0.
7) Why do some graph points look symmetric?
Additive inverses often mirror residue values around the modulus because each inverse pairs with a residue to complete a full modular cycle back to zero.
8) Can this calculator help with congruence problems?
Yes. It helps you rearrange modular equations, check residue classes, and study patterns across many integers when solving number theory exercises.