Integertranslate

Integertranslate is a mathematical operation that shifts integers by a fixed integer amount t, mapping n to n + t for all integers n.

Formally, for any subset S of integers, Integertranslate(S, t) = {n + t | n in S}. If t = 0, the set is unchanged. It is a bijection from the integers to themselves; its inverse is Integertranslate(·, -t).

Properties include that composing translations adds offsets: T_t composed with T_s equals T_{t+s}. Distances are preserved, since |(n+t) - (m+t)| = |n - m|. Parity can change depending on t: parity is preserved if t is even and flipped if t is odd. On modular arithmetic modulo m, translation induces a permutation of residue classes.

Examples help illustrate the concept. S = {0, 1, 2, 3} translated by t = 4 yields {4, 5, 6, 7}. On Z5, translating residues by t = 1 maps {0, 2, 4} to {1, 3, 0} modulo 5.

Applications of integer translation appear in data processing for offsetting values, in discrete geometry for shifting lattice points, and as a simple analogue of the Caesar cipher in cryptography. In programming, it is commonly implemented by adding a constant to each element of a collection.

See also translation, affine transformation, modular arithmetic, and Caesar cipher.