Enter Values
How to Use
- Enter any integer for
a, including negatives. - Enter a positive integer for
m(the modulus). - Press Compute Additive Inverse to get normalized results.
- Use the buttons to export the results as CSV or PDF.
- Check the steps panel to verify each arithmetic transformation.
Result
| a | m | a mod m | additive inverse b | (a+b) mod m | class [0..m-1] |
|---|---|---|---|---|---|
| No calculation yet. Enter values and compute. | |||||
Example Data Table
| # | a | m | a mod m | additive inverse b | (a+b) mod m |
|---|---|---|---|---|---|
| 1 | 14 | 5 | 4 | 1 | 0 |
| 2 | -7 | 4 | 1 | 3 | 0 |
| 3 | 123456789 | 97 | 39 | 58 | 0 |
| 4 | 0 | 13 | 0 | 0 | 0 |
| 5 | 5 | 5 | 0 | 0 | 0 |
| 6 | -1 | 11 | 10 | 1 | 0 |
| 7 | 37 | 1 | 0 | 0 | 0 |
| 8 | -42 | 9 | 3 | 6 | 0 |
These samples include negatives, zero, equal modulus, and modulus one.
Formula Used
The additive inverse of a modulo m is a number b in the residue class [0, m-1] such that
a + b ≡ 0 (mod m). Therefore, b ≡ -a (mod m). A practical formula is:
b = (-a) mod m = (m - (a mod m)) mod m
We always normalize with a non‑negative remainder, ensuring a canonical representative in [0, m-1].
Properties of Additive Inverses modulo m
- Every residue class has exactly one additive inverse in
[0, m-1]. - If
a ≡ 0 (mod m), then its inverse is0. aand its inverse sum to a multiple ofm.- Inverse is an involution: the inverse of
bisa mod m. - For even
m,m/2is self‑inverse since(m/2 + m/2) ≡ 0.
Common Edge Cases and Worked Examples
- Negative a: normalize first; e.g.,
a=-7, m=4givesa mod m = 1, inverse3. - Zero a: inverse is zero for any positive
m. - m = 1: the only residue is
0; the inverse is0. - a multiple of m: inverse
0sincea mod m = 0.
Residue Class Pairings for m = 12
Each residue r has an additive inverse b such that r + b ≡ 0 (mod m). Canonical representatives are listed in [0, m-1].
| r | additive inverse b = (−r) mod m | (r + b) mod m |
|---|---|---|
| 0 | 0 | 0 |
| 1 | 11 | 0 |
| 2 | 10 | 0 |
| 3 | 9 | 0 |
| 4 | 8 | 0 |
| 5 | 7 | 0 |
| 6 | 6 | 0 |
| 7 | 5 | 0 |
| 8 | 4 | 0 |
| 9 | 3 | 0 |
| 10 | 2 | 0 |
| 11 | 1 | 0 |
Self‑Inverse Residues for m = 12
Residues satisfying 2r ≡ 0 (mod m) are fixed points under inversion. For even m, this set is {0, m/2}. For odd m, only {0}.
| # | Residue r | Verification (2r mod m) |
|---|---|---|
| 1 | 0 | 0 |
| 2 | 6 | 0 |
FAQs
It is a value
b satisfying a + b ≡ 0 (mod m). In other words, b ≡ -a (mod m), normalized to the canonical class [0, m-1].Yes, for any positive modulus
m, every residue class has a unique additive inverse within [0, m-1], including 0 which maps to 0.We normalize using a non‑negative remainder:
a mod m is always in [0, m-1], then compute b = (-a) mod m.No. Additive inverse requires
a + b ≡ 0 (mod m), while multiplicative inverse requires a·x ≡ 1 (mod m) and exists only when gcd(a,m)=1.All integers are congruent to
0 (mod 1), so the additive inverse is always 0.Normalization chooses a canonical representative for each residue class, making comparisons and exports predictable and consistent.
Yes. Use the CSV and PDF buttons to export either the current result or the example dataset for sharing or record keeping.