Integer If the domain is restricted to then each and every member of has one and only one corresponding member of N and by the definition of cardinal equality the two sets have equal cardinality. Integer An integer (from the Latin "integer" meaning "whole") is a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not. The set of integers consists of zero (), the positive natural numbers (, , , …), also called "whole numbers" or "counting numbers", and their additive inverses (the negative integers, i.e.,
Transposable integer at all the steps: and also note that: By observing the remainders at each step, we can thus perform a desired cyclic permutation by multiplication. E.g., In this manner, cyclical left or right shift of any number of positions can be performed. Less importantly, this technique can be applied to any integer to shift cyclically right or left by any given number of places for the following reason: An integer "X" shift cyclically right by "k" positions when it is multiplied by an integer "n". Prove its formula. Proof First recognize that "X" is the repeating digits of a repeating