What Does Unity Mean in Mathematics?

The Mathematical Definition of Unity

Number One
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The word unity carries many meanings in the English language, but it is perhaps best known for its most simple and straightforward definition, which is "the state of being one; oneness." While the word carries its own unique meaning in the field of mathematics, the unique use does not stray too far, at least symbolically, from this definition. In fact, in mathematics, unity is simply a synonym for the number "one" (1), the integer between the integers zero (0) and two (2).

The number one (1) represents a single entity and it is our unit of counting. It is the first non-zero number of our natural numbers, which are those numbers used for counting and ordering, and the first of our positive integers or whole numbers. The number 1 is also the first odd number of the natural numbers.

The number one (1) actually goes by several names, unity being just one of them. The number 1 is also known as unit, identity, and multiplicative identity.

Unity as an Identity Element

Unity, or the number one, also represents an identity element, which is to say that when combined with another number in a certain mathematic operation, the number combined with the identity remains unchanged. For example, in the addition of real numbers, zero (0) is an identity element as any number added to zero remains unchanged (e.g, a + 0 = a and 0 + a = a). Unity, or one, is also an identity element when applied to numerical multiplication equations as any real number multiplied by unity remains unchanged (e.g., a x 1 = a and 1 x a = a). It is because of this unique characteristic of unity that is called the multiplicative identity.

Identity elements are always their own factorial, which is to say that the product of all positive integers less than or equal to unity (1) is unity (1). Identity elements like unity are also always their own square, cube, and so on. That is to the say that unity squared (1^2) or cubed (1^3) is equal to unity (1).

The Meaning of "Root of Unity" 

The root of unity refers to the state in which for any integer n, the nth root of a number k is a number that, when multiplied by itself n times, yields the number k. A root of unity in, most simply put, any number which when multiplied by itself any number of times always equals 1. Therefore, an nth root of unity is any number k that satisfies the following equation:

k^n  = 1 (k to the nth power equals 1), where n is a positive integer.

Roots of unity are also sometimes called de Moivre numbers, after the French mathematician Abraham de Moivre. Roots of unity are traditionally used in branches of mathematics like number theory.

When considering real numbers, the only two that fit this definition of roots of unity are the numbers one (1) and negative one (-1). But the concept of the root of unity does not generally appear within such a simple context. Instead, the root of unity becomes a topic for mathematical discussion when dealing with complex numbers, which are those numbers that can be expressed in the form bi, where and b are real numbers and i is the square root of negative one (-1) or an imaginary number. In fact, the number i is itself also a root of unity.