Parity (mathematics)

An integer is called even if it is divisible by two without a remainder; otherwise it is called odd. The set of integers is thus decomposed into two equally powerful disjoint subsets. This parity (from the Latin: paritas "equality, equal strength " ) is a useful invariant for many issues and one of the most important tools in elementary number theory.

Even and odd numbers

Definition

A natural or integer is called even if it is divisible by two, otherwise odd. Even numbers are characterized by odd numbers by for any. Accordingly, the zero is considered to be straight.

That is, odd numbers leave when divided by 2 is always a remainder of 1, even numbers the rest 0 You will therefore characterized by its residue class modulo two. As is true and that parity is sometimes symbolized with positive or negative sign, see also: parity bit. However, it is wrong to understand the sign of positive and negative numbers as parity classification.

Calculation rules

The rules for computing parities follow the laws of the residue class field with two elements. Here are zero and one for the corresponding residues modulo 2, and thus for even or odd. In particular, squaring receives the parity.

Addition:

Multiplication:

In decimal, binary, and generally in any place value system with straight base recognize the parity as to whether the last digit is divisible by 2.

Comments

• The house numbers in many European cities run alternately, so that even and odd numbers for each lie on a street. The idea is based on the simple continuation of the numbering at a later extension of the road.

• The even numbers form an ideal in the ring of integers, the odd do not. The even numbers are the result A005843 in OEIS, the odd are the result A005408 in OEIS.

• In English, the number 2 is called sometimes called " the oddest prime". This is a play on words with the meanings of strange and odd the word odd, because the prime number 2 is a special or odd ( odd ) prime number because it is not the only odd ( odd ).

• A natural number can always be written uniquely as a product of (even) power of two and an odd number :, where and

• Each previously known perfect number is even. Whether odd perfect numbers exist at all, is still unknown.

• Euclid's proof of the irrationality of the square root of 2 is due in large part to parity comparisons, with the gold Bach's guess but the parity plays only a minor role.

Generalizations

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

• Euler's performance in solving the Königsberg bridge problem lies in the abstract approach: have you first understand how a neighborhood with paths can be considered as a graph, one can easily see that a closed tour can only exist on all the way, if at any point an even number of lines going - because every point that you leave, you have to have reached a different route. When Königsberg problem, this was not the case; 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 performed using a parity based ultimately on the parity of permutations. With it, you can specify how far two stones are reversed or not. The same approach closes at the Rubik's Cube from all positions at which only two edges of stones or only two cornerstones were reversed, or only a curb or a cornerstone is rotated.

• The partial division of a lot of features into even and odd functions can only be used with restrictions as parity.

• The parity of the order of zeros and poles gives some information, so it comes at odd zeros or poles of real-valued functions always a sign change.

• In order to specify the parity of other mathematical objects must have at least a reasonable figure exist, each of these objects assigns an integer. In particular Division shall be possible with the rest, for any real number can be, for example, specify no parity.