7.4 Properties of identity, inverses, and zero

Recognize the identity properties of addition and multiplication

What happens when we add zero to any number? Adding zero doesn’t change the value. For this reason, we call $0$ the additive identity .

For example,

What happens when you multiply any number by one? Multiplying by one doesn’t change the value. So we call $1$ the multiplicative identity .

For example,

Use the inverse properties of addition and multiplication

What number added to 5 gives the additive identity, 0?
$5+\_\_\_\_\_=0$
What number added to −6 gives the additive identity, 0?
$\mathrm{-6}+\_\_\_\_\_=0$

Notice that in each case, the missing number was the opposite of the number.

We call $-a$ the additive inverse of $a.$ The opposite of a number is its additive inverse. A number and its opposite add to $0,$ which is the additive identity.

What number multiplied by $\frac{2}{3}$ gives the multiplicative identity, $1?$ In other words, two-thirds times what results in $1?$

$\frac{2}{3}·\_\_\_=1$

What number multiplied by $2$ gives the multiplicative identity, $1?$ In other words two times what results in $1?$

$2·\_\_\_=1$

Notice that in each case, the missing number was the reciprocal of the number.

We call $\frac{1}{a}$ the multiplicative inverse of $a(a\ne 0)\text{.}$ The reciprocal of a number is its multiplicative inverse. A number and its reciprocal multiply to $1,$ which is the multiplicative identity.

We’ll formally state the Inverse Properties here: