Multiplicative Identity
The multiplicative identity is one: multiplying any number by one leaves that number unchanged.
Formula
a \times 1 = a
Definition
The multiplicative identity is one: multiplying any number by $1$ leaves the number unchanged. Formally, 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 $\mathbb{C}$ this element is $1$, required in a ring to be distinct from $0$ (except in the zero ring), and it is unique. In a ring with unity, the group of units $R^*$ consists of elements with a multiplicative inverse; a field additionally requires every non-zero element to be a unit.
Example
$8 \times 1 = 8$ and $1 \times 254 = 254$: multiplying by $1$ is always the "do-nothing" operation. In $\mathbb{Z}/7\mathbb{Z}$: $5 \times 1 = 5$, and for matrices the identity matrix $I$ satisfies $A \times I = I \times A = A$ for all $A$ of the appropriate size. In $\mathbb{Z}$, the units are $R^* = \{1,-1\}$, while in $\mathbb{Z}/n\mathbb{Z}$, $R^*$ is the group of integers coprime to $n$, with order $\phi(n)$ (Euler's totient function); Fermat's little theorem, $a^{p-1} \equiv 1 \pmod p$ for prime $p$ and $a$ not divisible by $p$, follows from Lagrange's theorem applied to this group of order $p-1$.
Key Insight
Just as zero is the do-nothing number for addition, one is the do-nothing number for multiplication. The multiplicative identity is the "unit" of multiplication, and elements with multiplicative inverses are called units of a ring; in $\mathbb{Z}$, only $1$ and $-1$ qualify. The structure of the group of units, $(\mathbb{Z}/p\mathbb{Z})^*$ in particular, determines the solutions to equations mod $n$ and underlies results like Fermat's little theorem.