Operation Properties

Definition

An operation is a function, represented by a symbol like " $+$ " or " $\cdot$ ", that takes one or more inputs and maps them to an output. The most well-known operations, addition and multiplication, are called binary operations because they are functions with two inputs.

To understand the structure of operations, we can take a look at their different properties, like inverses and identities. In general, denote a binary operation that acts on two inputs $a$ and $b$ and produces an output $c$ by $a*b=c$ .

Identities

The identity of an operation is whatever number "does nothing" when you apply it to some number.

Example: the additive identity is $0$ because $0+a=a+0=a$ for any number $a$ .

Example: the multiplicative identity is $1$ because $1 \cdot a = a \cdot 1 = a$ for any number $a$ .

In general, if $e$ is the identity for operation $*$ , then $e * a = a * e = a$ for any $a$ .

Inverses

An inverse for an operation is a number that, when combined with another number using the operation, results in the identity.

Example: for addition, the inverse of a number $a$ is $-a$ because $a + (-a) = 0$ .

Example: for multiplication, the inverse of $a$ is $\frac{1}{a}$ (assuming $a$ is not zero) because $a \cdot \frac{1}{a} = 1$ .

In general, if $a$ has an inverse $a^{-1}$ for operation $*$ , then $a * a^{-1} = a^{-1} * a = e$ , where $e$ is the identity for $*$ .

What about subtraction and division?

We learn subtraction and division almost as soon as we learn about addition and multiplication, right? Well, it turns out that subtraction isn't really that different from addition, and likewise with division and multiplication. Subtracting two numbers like $a-b$ can be thought of as adding $a$ to $-b$ . Similarly, dividing $\frac{a}{b}$ can be thought of as multiplying $a$ by $\frac{1}{b}$ .

These relationships can be summarized as:

a - b = a + (-b)

\frac{a}{b} = a \cdot \frac{1}{b}