Recognizing 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,

\[\begin{array}{ccccc}\hfill 13+0\hfill & \phantom{\rule{2em}{0ex}}& \hfill −14+0\hfill & \phantom{\rule{2em}{0ex}}& \hfill 0+(−3x)\hfill \\ \hfill 13\hfill & \phantom{\rule{2em}{0ex}}& \hfill -14\hfill & \phantom{\rule{2em}{0ex}}& \hfill -3x\hfill \end{array}\]

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,

\[\begin{array}{ccccc}\hfill 43·1\hfill & \phantom{\rule{2em}{0ex}}& \hfill -27·1\hfill & \phantom{\rule{2em}{0ex}}& \hfill 1·\frac{6y}{5}\hfill \\ \hfill 43\hfill & \phantom{\rule{2em}{0ex}}& \hfill -27\hfill & \phantom{\rule{2em}{0ex}}& \hfill \frac{6y}{5}\hfill \end{array}\]

Definition: Identity Properties

The identity property of addition: for any real number \(a,\)

\(\begin{array}{}\hfill a+0=a\phantom{\rule{2em}{0ex}}0+a=a\hfill \\ \hfill \text{0 is called the}\phantom{\rule{2em}{0ex}}\mathbf{\text{additive identity}}\hfill \end{array}\)

The identity property of multiplication: for any real number \(a\)

\(\begin{array}{c}\hfill a·1=a\phantom{\rule{2em}{0ex}}1·a=a\hfill \\ \hfill \text{1 is called the}\phantom{\rule{2em}{0ex}}\mathbf{\text{multiplicative identity}}\hfill \end{array}\)

Example

Identify whether each equation demonstrates the identity property of addition or multiplication.

\(\phantom{\rule{0.2em}{0ex}}7+0=7\)
\(\phantom{\rule{0.2em}{0ex}}-16\left(1\right)=-16\)

Solution

(a)
	\(7+0=7\)
We are adding 0.	We are using the identity property of addition.

(b)
	\(-16\left(1\right)=-16\)
We are multiplying by 1.	We are using the identity property of multiplication.

This lesson is part of:

Properties of Real Numbers

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