Multiplicative Identity
ArithmeticThe multiplicative identity is one: multiplying any number by one leaves that number unchanged.
Formula
a \times 1 = a
Definition
The multiplicative identity is 1. When you multiply any number by 1, the number stays the same.
Example
$8 \times 1 = 8$. $1 \times 254 = 254$. Multiplying by $1$ is always the "identity" operation.
Key Insight
Just as zero is the do-nothing number for addition, one is the do-nothing number for multiplication.
Definition
The multiplicative identity of a number system is the element $1$ such that $a \times 1 = 1 \times a = a$ for all $a$. In $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$, the multiplicative identity is $1$. In a ring, $1$ is required to be distinct from $0$ (except in the zero ring). The multiplicative identity is unique.
Example
In $\mathbb{Z}/7\mathbb{Z}$: $5 \times 1 = 5$. In matrices: the identity matrix $I$ satisfies $A \times I = I \times A = A$ for all $A$ of the appropriate size.
Key Insight
The multiplicative identity $1$ is the "unit" of multiplication. Elements with multiplicative inverses are called units of a ring. In $\mathbb{Z}$, only $1$ and $-1$ are units.
Definition
In a ring $(R, +, *)$, the multiplicative identity $1$ satisfies $1 * a = a * 1 = a$ for all $a$. A ring with a multiplicative identity is called a "ring with unity" or a "unital ring." A field additionally requires every non-zero element to have a multiplicative inverse. The group of units of a ring $R$ is denoted $R^*$.
Example
In $\mathbb{Z}$, $R^* = \{1, -1\}$ (units). In $\mathbb{Z}/n\mathbb{Z}$, $R^*$ is the group of integers coprime to $n$, with order $\phi(n)$ (Euler's totient function). This group is abelian and its structure determines the solutions to equations mod $n$.
Key Insight
Fermat's little theorem: if $p$ is prime and $a$ is not divisible by $p$, then $a^{p-1} \equiv 1 \pmod p$. This is just Lagrange's theorem applied to the multiplicative group $(\mathbb{Z}/p\mathbb{Z})^*$ of order $p-1$, where every element satisfies $x^{p-1} = 1$ (the multiplicative identity).