Parity (mathematics)

In mathematics, parity is the property of an integer that tells us whether it is even or odd. An integer is even if it can be divisible by 2, meaning you can split it into pairs without any leftovers. An odd number is one that cannot be divided by 2 in this way, leaving one number unpaired. For example, numbers like −4, 0, and 82 are even, while numbers like −3, 5, 23, and 61 are odd.

Parity is a useful idea in math because it helps us understand patterns and solve problems more easily. For instance, we can quickly see that adding an even number to another even number always gives an even result, while adding an even number to an odd number gives an odd result. This idea applies only to whole numbers, so it does not work with numbers that have decimals or fractions, like 1/2 or 4.6978.

We can also tell if a number is even or odd by looking at its last digit. In the usual decimal system we use every day, if the last digit is 0, 2, 4, 6, or 8, the number is even. If the last digit is 1, 3, 5, 7, or 9, the number is odd. The same rule works in other number systems too. For example, in the binary system, a number ending in 0 is even, and one ending in 1 is odd.

Definition

An even number is a number you can divide perfectly by 2. For example, 4 divided by 2 equals 2, so 4 is even. An odd number is a number that cannot be divided perfectly by 2. For example, 5 divided by 2 leaves a remainder, so 5 is odd.

Even numbers include ... and odd numbers include ... You can think of even numbers as those that can be paired up perfectly, while odd numbers always have one left over.

Properties

In mathematics, we look at whether a number can be divided evenly by 2. This helps us check if calculations are correct. Just like normal adding and multiplying, these operations follow special rules when we only care about whether numbers are even or odd.

When we add or subtract even numbers, we always get an even number. Adding or subtracting an odd number from an even number gives us an odd number. Adding or subtracting two odd numbers always results in an even number.

For multiplication, multiplying two even numbers or an even number with an odd number always gives an even number. Multiplying two odd numbers gives us an odd number.

These rules help us understand how numbers behave in a simple way.
divisibility modular arithmetic field with two elements quotient if and only if dividend factors of two

History

The ancient Greeks thought the number 1, called the monad, was special. They believed it was neither fully odd nor fully even. This idea lasted until the 1800s. In 1826, a writer named Friedrich Wilhelm August Fröbel said teachers should teach students that 1 is neither odd nor even. He believed this showed an important idea: between two different things, there is often a third thing that fits in between.

Higher mathematics

Integer points in spaces with two or more dimensions also have a parity, based on the sum of their coordinates. For example, in chess, the color of a square shows its parity. Bishops move only between squares of the same parity, while knights switch parity with each move. This idea helped solve a famous puzzle: if you remove two opposite corner squares from a chessboard, you cannot cover the rest with dominoes, because each domino covers one square of each parity and the board ends up with an imbalance.

In number theory, even numbers are those divisible by 2, and odd numbers are not. All prime numbers are odd except for 2. Even numbers also have special properties in advanced math, like forming certain types of number structures.

In group theory, the parity of a permutation depends on whether it can be broken down into an even or odd number of swaps. This concept is important in puzzles like the Rubik's Cube, where only even permutations are possible.

In analysis, a function’s parity describes how it behaves when its inputs are negated. Even functions give the same result for a number and its negation, while odd functions give the negation of the result.

In combinatorial game theory, numbers are called "evil" if they have an even number of 1’s in their binary form, and "odious" if they have an odd number of 1’s. These ideas help strategy in certain games.

Additional applications

In information theory, a parity bit added to a binary number helps find simple mistakes. If one bit changes, the number no longer matches the right parity, so we know there was an error.

In wind instruments like the clarinet, only certain notes are played, which are odd multiples of the main note. In some places, house numberings are set so one side of the street has even numbers and the other has odd numbers. In the United States numbered highways, even numbers usually mean roads going east to west, and odd numbers mean roads going north to south. For flight numbers, even numbers often mean flights going east or north, and odd numbers mean flights going west or south.