Concatenation (mathematics)

In mathematics, concatenation is the joining of two numbers by their numerals. That is, the concatenation of 123 and 456 is 123456. Concatenation of numbers a and b is denoted a||b. Relevant subjects in recreational mathematics include Smarandache-Wellin numbers, home primes, and Champernowne's constant. The convention for sequences at places such as the Online Encyclopedia of Integer Sequences is to have sequences of concatenations include as the first term a number prior to the actual act of concatenation. Therefore, care must be taken to ensure that parties discussing a topic agree either with this convention or with plain language. For example, the first term in the sequence of concatenations of increasing even numbers may be taken to be either 24, as would seem obviously correct, or simply 2, according to convention.

Calculation
The concatenation of numbers depends on the numeric base, which is often understood from context.

Given the numbers p and q in base b, the concatenation p||q is given by


 * $$p||q=p b^{l(q)}+q$$

where


 * $$l(q)=\lfloor log_b(q) \rfloor+1$$

is the number of digits of q in base b, and $$\lfloor x \rfloor$$ is the floor function.

Vector extension
The concatenation of vectors can be understood in two distinct ways; either as a generalization of the above operation for numbers or as a concatenation of lists.

Given two vectors in $$\mathbb{R}^n$$, concatenation can be defined as


 * $$\left( \begin{array}{c} a_1 \\ a_2 \\ \vdots \\ a_n \end{array} \right) || \left( \begin{array}{c} b_1 \\ b_2 \\ \vdots \\ b_n \end{array} \right) = \left( \begin{array}{c} a_1 || b_1 \\ a_2 || b_2 \\ \vdots \\ a_n || b_n \end{array} \right)$$

In the case of vectors in $$\mathbb{R}^1$$, this is equivalent to the above definition for numbers. The further extension to matrices is trivial.

Since vectors can be viewed in a certain way as lists, concatenation may take on another meaning. In this case the concatenation of two lists (a1, a2, ..., an) and (b1, b2, ..., bn) is the list (a1, a2, ..., an, b1, b2, ..., bn). Only the exact context will reveal which meaning is intended.