Definition:Modulo Addition
From ProofWiki
Contents |
Definition
Let $z \in \R$.
Let $\R_z$ be the set of all residue classes modulo $z$ of $\R$.
We define the addition operation on $\R_z$, defined as follows:
- $\left[\!\left[{a}\right]\!\right]_z +_z \left[\!\left[{b}\right]\!\right]_z = \left[\!\left[{a + b}\right]\!\right]_z$
This can be shown to be a well-defined operation.
This operation is called addition modulo $z$.
Also denoted as
Although the operation of addition modulo $z$ is denoted by the symbol $+_z$, if there is no danger of confusion, the symbol $+$ is often used instead.
The notation for addition of two residue classes modulo $z$ is not usually $\left[\!\left[{a}\right]\!\right]_z +_z \left[\!\left[{b}\right]\!\right]_z$.
What is more normally seen is $a + b \pmod z$.
Also see
- Modulo Multiplication, which is defined only on an integer modulus.
Sources
- J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 2.6$
- Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 2$: Example $2.3$
- George McCarty: Topology: An Introduction with Application to Topological Groups (1967)... (previous)... (next): $\text{II}$: A Little Number Theory
- Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (1968): $\S 1.2.4$: Law $\mathbf A$, Exercise $24$
- C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 1.6$
- B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra (1970)... (previous)... (next): $\S 1.2$: Some examples of rings: Ring Example $2$
- Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 18 \alpha$
- Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 14.3 \ \text {(i), (ii)}$
- John F. Humphreys: A Course in Group Theory (1996)... (previous)... (next): $\S 2$: Example $2.30$
