# Parity (math) - Parität (Mathematik)

An integer is called even if it is divisible by two without a remainder ; otherwise it is called odd . The set of integers is broken down into two disjoint subsets of equal power . This parity (from Latin paritas “equality, equal strength”) is a helpful invariant for many questions and is one of the most important tools in elementary number theory .

## Even and odd numbers

### Definition

A natural or whole number is called even if it is divisible by two, otherwise odd . Even numbers are through${\displaystyle \pm 2k}$ characterized by odd numbers ${\displaystyle \pm 2k+1}$ for anything ${\displaystyle k\in \mathbb {N} _{0}}$. Accordingly, the zero is considered even.

That is, odd numbers left in Division by 2 always a remainder of 1, even numbers the rest of 0. They are therefore by their residue class modulo Two characterized. There${\displaystyle (-1)^{0}=1}$ and ${\displaystyle (-1)^{1}=-1}$applies, the parity is sometimes also symbolized with a positive or negative sign, see also: parity bit . However, it is wrong to understand the sign of positive and negative numbers as a division of parity.

### Calculation rules

The rules for calculating parities follow the laws of the remainder class field with the two elements zero and one. These stand for the corresponding remainders modulo 2 and thus for even or odd. In particular, squaring preserves parity.

Addition:

 + 0 1 0 0 1 1 1 0

Multiplication:

 * 0 1 0 0 0 1 0 1

In the decimal , binary and generally in every place value system with an even base, the parity can be recognized by whether the last digit is divisible by 2.

### Remarks

• The house numbers in many European cities run alternately, so that even and odd numbers are each on one side of the street ( orientation numbering ). The idea is based on the simpler continuation of the numbering when the street is later extended.
• In many European countries the north-south highways have an odd number and the west-east highways have an even number.
• The even numbers form an ideal in the ring of whole numbers, the odd ones don't. The even numbers are the sequence A005843 in OEIS , the odd numbers are the sequence A005408 in OEIS .
• In English , the number 2 is sometimes referred to as "the oddest prime". This is a play on words with the meanings strange and odd of the word odd , because the prime number 2 is a special or strange ( odd ) prime because it is the only one not odd ( odd ).
• A natural number ${\displaystyle n\geq 1}$can always be clearly expressed as the product of an (even) power of two and an odd number${\displaystyle u}$ to be written: ${\displaystyle n=2^{s}\cdot u}$, in which ${\displaystyle s\geq 0}$ and ${\displaystyle u\geq 1}$
• Every perfect number known so far is even. It is still unknown whether odd perfect numbers even exist.
• Euclid's proof of the irrationality of the root of 2 is largely based on parity comparisons.
• If you are for ${\displaystyle g}$ one straight and for ${\displaystyle u}$ inserting an odd number applies ${\displaystyle (-1)^{g}=1}$ and ${\displaystyle (-1)^{u}=-1}$.

## Generalizations

The concept of parity is also used more generally in many areas of mathematics :

• Euler's achievement in solving the Königsberg bridge problem lies in the abstract approach: Once you have understood how a district with paths can be understood as a graph , you can easily see that a closed tour of all paths can only exist if at each one Point leaves an even number of lines - because every point that you leave must be reached by a different route. This was not the case with the Königsberg problem; a closed path is not possible there. This is also one of the classic parity arguments.
• The proof of the unsolvability of the original 15-puzzle is carried out with the help of a parity which is ultimately based on the parity of permutations . It can be used to indicate to what extent two stones are swapped or not. The same approach excludes all positions with the Rubik's Cube in which only two curb stones or only two corner stones would be exchanged, or only one curb or one corner stone is rotated.
• The partial subdivision of a set of functions into even and odd functions can only serve as parity with restrictions.
• The parity of the order of zeros and poles provides some information, so there is always a sign change in the case of odd zeros or poles of real-valued functions .
• In order to be able to specify the parity of other mathematical objects, there must be at least one meaningful mapping that assigns an integer to each of these objects. In particular, division by remainder must be possible; for example, no parity can be specified for any real number .