Concatenation (mathematics)
Encyclopedia
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 number
s, home prime
s, and Champernowne's constant. The convention
for sequence
s 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.
, which is often understood from context.
Given the numbers p and q in base b, the concatenation p||q is given by
where
is the number of digits of q in base b, and
is the floor function.
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
, concatenation can be defined as
In the case of vectors in
, 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, 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
Recreational mathematics
Recreational mathematics is an umbrella term, referring to mathematical puzzles and mathematical games.Not all problems in this field require a knowledge of advanced mathematics, and thus, recreational mathematics often attracts the curiosity of non-mathematicians, and inspires their further study...
include Smarandache-Wellin number
Smarandache-Wellin number
In mathematics, a Smarandache–Wellin number is an integer that in a given base is the concatenation of the first n prime numbers written in that base. Smarandache–Wellin numbers are named after Florentin Smarandache and Paul R...
s, home prime
Home prime
In number theory, the home prime HP of an integer n greater than 1 is the prime obtained by repeatedly factoring the increasing concatenation of prime factors including repetitions. The mth intermediate stage in the process of determining HP is designated HPn...
s, and Champernowne's constant. The convention
Convention (norm)
A convention is a set of agreed, stipulated or generally accepted standards, norms, social norms or criteria, often taking the form of a custom....
for sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
s 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 baseRadix
In mathematical numeral systems, the base or radix for the simplest case is the number of unique digits, including zero, that a positional numeral system uses to represent numbers. For example, for the decimal system the radix is ten, because it uses the ten digits from 0 through 9.In any numeral...
, which is often understood from context.
Given the numbers p and q in base b, the concatenation p||q is given by
where
is the number of digits of q in base b, and
is the floor function.Vector extension
The concatenation of vectorsCoordinate vector
In linear algebra, a coordinate vector is an explicit representation of a vector in an abstract vector space as an ordered list of numbers or, equivalently, as an element of the coordinate space Fn....
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
, concatenation can be defined asIn the case of vectors in
, this is equivalent to the above definition for numbers. The further extension to matricesMatrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
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.