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Properties of Identity, Inverses, and Zero

Recognize the identity properties of addition and multiplication

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,

\[\begin{array}{lllll}13+0 & \ & -14+0 & \ & 0+(-3x) \\ 13 & \ & -14 & \ & -3x\end{array}\]

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,

\[\begin{array}{lllll}43\cdot 1 & \ & -27\cdot 1 & \ & 1\cdot \frac{6y}{5} \\ 43 & \ & -27 & \ & \frac{6y}{5}\end{array}\]

Example

Try it.

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

ⓐ \(\ 7+0=7\)
ⓑ \(\ -16(1)=-16\)

Solution

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

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

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?
\(-6+_____=0\)

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 \(1?\)

\(\frac{2}{3}\cdot ___=1\)

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

\(2\cdot ___=1\)

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:

Example

Try it.

Find the additive inverse of each expression: ⓐ \(13\) ⓑ \(-\frac{5}{8}\) ⓒ \(\ 0.6\).

Solution

To find the additive inverse, we find the opposite.

ⓐ The additive inverse of \(13\) is its opposite, \(-13.\)
ⓑ The additive inverse of \(-\frac{5}{8}\) is its opposite, \(\frac{5}{8}.\)
ⓒ The additive inverse of \(0.6\) is its opposite, \(-0.6.\)

Example

Try it.

Find the multiplicative inverse: ⓐ \(9\\) ⓑ \(-\frac{1}{9}\\) ⓒ \(0.9\).

Solution

To find the multiplicative inverse, we find the reciprocal.

ⓐ The multiplicative inverse of \(9\) is its reciprocal, \(\frac{1}{9}.\)
ⓑ The multiplicative inverse of \(-\frac{1}{9}\) is its reciprocal, \(-9.\)
ⓒ To find the multiplicative inverse of \(0.9,\) we first convert \(0.9\) to a fraction, \(\frac{9}{10}.\) Then we find the reciprocal, \(\frac{10}{9}.\)

Use the Properties of Zero

We have already learned that zero is the additive identity, since it can be added to any number without changing the number’s identity. But zero also has some special properties when it comes to multiplication and division.

What happens when you multiply a number by \(0?\) Multiplying by \(0\) makes the product equal zero. The product of any real number and \(0\) is \(0.\)

Example

Try it.

Simplify: ⓐ \(\ -8\cdot 0\) ⓑ \(\ \frac{5}{12}\cdot 0\) ⓒ \(\ 0(2.94)\).

Solution

ⓐ
	\(-8⋅0\)
The product of any real number and 0 is 0.	\(0\)

ⓑ
	\(\frac{5}{12}\cdot 0\)
The product of any real number and 0 is 0.	\(0\)

ⓒ
	\(0(2.94)\)
The product of any real number and 0 is 0.	\(0\)

Condensed — the full section is in OpenStax Prealgebra 2e.

Simplify Expressions using the Properties of Identities, Inverses, and Zero

We will now practice using the properties of identities, inverses, and zero to simplify expressions.

Example

Try it.

Simplify: \(3x+15-3x.\)

Solution

	\(3x+15-3x\)
Notice the additive inverses, \(3x\) and \(-3x\).	\(0+15\)
Add.	\(15\)

Example

Try it.

Simplify: \(4(0.25q).\)

Solution

	\(4(0.25q)\)
Regroup, using the associative property.	\([4(0.25)]q\)
Multiply.	\(1.00q\)
Simplify; 1 is the multiplicative identity.	\(q\)

Example

Try it.

Simplify: \(\frac{0}{n+5}\), where \(n\ne -5\).

Solution

	\(\frac{0}{n+5}\)
Zero divided by any real number except itself is zero.	\(0\)

Example

Try it.

Simplify: \(\frac{10-3p}{0}.\)

Solution

	\(\frac{10-3p}{0}\)
Division by zero is undefined.	undefined

Example

Try it.

Simplify: \(\frac{3}{4}\cdot \frac{4}{3}(6x+12).\)

Solution

We cannot combine the terms in parentheses, so we multiply the two fractions first.

	\(\frac{3}{4}\cdot \frac{4}{3}(6x+12)\)
Multiply; the product of reciprocals is 1.	\(1(6x+12)\)
Simplify by recognizing the multiplicative identity.	\(6x+12\)

All the properties of real numbers we have used in this chapter are summarized in .

Property	Of Addition	Of Multiplication
Commutative Property
If a and b are real numbers then…	\(a+b=b+a\)	\(a\cdot b=b\cdot a\)
Associative Property
If a, b, and c are real numbers then…	\((a+b)+c=a+(b+c)\)	\((a\cdot b)\cdot c=a\cdot (b\cdot c)\)
Identity Property	\(0\) is the additive identity	\(1\) is the multiplicative identity
For any real number a,	\(\begin{array}{l}a+0=a \\ 0+a=a\end{array}\)	\(\begin{array}{l}a\cdot 1=a \\ 1\cdot a=a\end{array}\)
Inverse Property	\(-\text{a}\)is the additive inverse of \(a\)	\(a,a\ne 0\) \(1/\text{a}\) is the multiplicative inverse of \(a\)
For any real number a,	\(a+\text{(}\text{-}\text{a}\text{)}\ =0\)	\(a\cdot \frac{1}{a}=1\)
Distributive Property \(\\)If \(a,b,c\) are real numbers, then \(a(b+c)=ab+ac\)
Properties of Zero
For any real number a,	\(\begin{array}{l}a⋅0=0 \\ 0⋅a=0\end{array}\)
For any real number \(a,a\ne 0\)	\(\frac{0}{a}=0\) \(\)\(\frac{a}{0}\) is undefined

Key Concepts

Identity Properties
- Identity Property of Addition: For any real number a: \(a+0=a\ 0+a=a\\) 0 is the additive identity
- Identity Property of Multiplication: For any real number a: \(a⋅1=a\ 1⋅a=a\\) 1 is the multiplicative identity
Inverse Properties
- Inverse Property of Addition: For any real number a: \(a+(-a)=0\ -a\) is the additive inverse of a
- Inverse Property of Multiplication: For any real number a: \((a\ne 0)\ a⋅\frac{1}{a}=1\ \frac{1}{a}\) is the multiplicative inverse of a
Properties of Zero
- Multiplication by Zero: For any real number a, \(\begin{array}{lllllll}a⋅0=0 & & & 0⋅a=0 & & & \text{The product of any number and 0 is 0.}\end{array}\)
- Division of Zero: For any real number a, \(\begin{array}{lllllll}\frac{0}{a}=0 & & & & & & \text{Zero divided by any real number, except itself, is zero.}\end{array}\)
- Division by Zero: For any real number a, \(\frac{a}{0}\) is undefined and \(a\div 0\) is undefined. Division by zero is undefined.

Properties of Identity, Inverses, and Zero

Recognize the Identity Properties of Addition and Multiplication

In the following exercises, identify whether each example is using the identity property of addition or multiplication.

Try it.

\(101+0=101\)

Try it.

\(\frac{3}{5}(1)=\frac{3}{5}\)

Solution

identity property of multiplication

Try it.

\(-9\cdot 1=-9\)

Try it.

\(0+64=64\)

Solution

identity property of addition

Use the Inverse Properties of Addition and Multiplication

In the following exercises, find the multiplicative inverse.

Try it.

\(8\)

Try it.

\(14\)

Solution

\(\frac{1}{14}\)

Try it.

\(-17\)

Try it.

\(-19\)

Solution

\(-\frac{1}{19}\)

Try it.

\(\frac{7}{12}\)

Try it.

\(\frac{8}{13}\)

Solution

\(\frac{13}{8}\)

Try it.

\(-\frac{3}{10}\)

Try it.

\(-\frac{5}{12}\)

Solution

\(-\frac{12}{5}\)

Try it.

\(0.8\)

Try it.

\(0.4\)

Solution

\(\frac{5}{2}\)

Try it.

\(-0.2\)

Try it.

\(-0.5\)

Solution

−2

Use the Properties of Zero

In the following exercises, simplify using the properties of zero.

Try it.

\(48\cdot 0\)

Try it.

\(\frac{0}{6}\)

Solution

Try it.

\(\frac{3}{0}\)

Try it.

\(22\cdot 0\)

Solution

Try it.

\(0\div \frac{11}{12}\)

Try it.

\(\frac{6}{0}\)

Solution

undefined

Try it.

\(\frac{0}{3}\)

Try it.

\(0\div \frac{7}{15}\)

Solution

Try it.

\(0\cdot \frac{8}{15}\)

Try it.

\((-3.14)(0)\)

Solution

Try it.

\(5.72\div 0\)

Try it.

\(\frac{\frac{1}{10}}{0}\)

Solution

undefined

Simplify Expressions using the Properties of Identities, Inverses, and Zero

In the following exercises, simplify using the properties of identities, inverses, and zero.

Try it.

\(19a+44-19a\)

Try it.

\(27c+16-27c\)

Solution

Try it.

\(38+11r-38\)

Try it.

\(92+31s-92\)

Solution

31s

Try it.

\(10(0.1d)\)

Try it.

\(100(0.01p)\)

Solution

Try it.

\(5(0.6q)\)

Try it.

\(40(0.05n)\)

Solution

Try it.

\(\frac{0}{r+20}\), where \(r\ne -20\)

Try it.

\(\frac{0}{s+13}\), where \(s\ne -13\)

Solution

Try it.

\(\frac{0}{u-4.99}\), where \(u\ne 4.99\)

Try it.

\(\frac{0}{v-65.1}\), where \(v\ne 65.1\)

Solution

Try it.

\(0\div (x-\frac{1}{2})\), where \(x\ne \frac{1}{2}\)

Try it.

\(0\div (y-\frac{1}{6})\), where \(y\ne \frac{1}{6}\)

Solution

Try it.

\(\frac{32-5a}{0}\), where \(32-5a\ne 0\)

Try it.

\(\frac{28-9b}{0}\), where \(28-9b\ne 0\)

Solution

undefined

Try it.

\(\frac{2.1+0.4c}{0}\), where \(2.1+0.4c\ne 0\)

Try it.

\(\frac{1.75+9f}{0}\), where \(1.75+9f\ne 0\)

Solution

undefined

Try it.

\((\frac{3}{4}+\frac{9}{10}m)\div 0\), where \(\frac{3}{4}+\frac{9}{10}m\ne 0\)

Try it.

\((\frac{5}{16}n-\frac{3}{7})\div 0\), where \(\frac{5}{16}n-\frac{3}{7}\ne 0\)

Solution

undefined

Try it.

\(\frac{9}{10}\cdot \frac{10}{9}(18p-21)\)

Try it.

\(\frac{5}{7}\cdot \frac{7}{5}(20q-35)\)

Solution

20q − 35

Try it.

\(15\cdot \frac{3}{5}(4d+10)\)

Try it.

\(18\cdot \frac{5}{6}(15h+24)\)

Solution

225h + 360

Try it.

In your own words, describe the difference between the additive inverse and the multiplicative inverse of a number.

Try it.

How can the use of the properties of real numbers make it easier to simplify expressions?

Solution

Answers will vary.

Condensed — the full section is in OpenStax Prealgebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Find the opposite of \(-4.\)
If you missed this problem, review .

Αποκάλυψέ την.

\(4\)
Find the reciprocal of \(\frac{5}{2}.\)
If you missed this problem, review .

Αποκάλυψέ την.

\(\frac{2}{5}\)
Multiply: \(\frac{3a}{5}\cdot \frac{9}{2a}.\)
If you missed this problem, review .

Αποκάλυψέ την.

\(\frac{27}{10}\)
Identify whether each equation demonstrates the identity property of addition or multiplication.
1. ⓐ \(\ 7+0=7\)
2. ⓑ \(\ -16(1)=-16\)
Αποκάλυψέ την.

ⓐ
\(7+0=7\)
We are adding 0. We are using the identity property of addition.
ⓑ
\(-16(1)=-16\)
We are multiplying by 1. We are using the identity property of multiplication.
Identify whether each equation demonstrates the identity property of addition or multiplication:
ⓐ \(\ 23+0=23\)ⓑ \(\ -37(1)=-37.\)
Αποκάλυψέ την.
1. ⓐ identity property of addition
2. ⓑ identity property of multiplication
Identify whether each equation demonstrates the identity property of addition or multiplication:
ⓐ \(\ 1\cdot 29=29\)ⓑ \(\ 14+0=14.\)
Αποκάλυψέ την.
1. ⓐ identity property of multiplication
2. ⓑ identity property of addition
Find the additive inverse of each expression: ⓐ \(13\) ⓑ \(-\frac{5}{8}\) ⓒ \(\ 0.6\).
Αποκάλυψέ την.
To find the additive inverse, we find the opposite.
1. ⓐ The additive inverse of \(13\) is its opposite, \(-13.\)
2. ⓑ The additive inverse of \(-\frac{5}{8}\) is its opposite, \(\frac{5}{8}.\)
3. ⓒ The additive inverse of \(0.6\) is its opposite, \(-0.6.\)
Find the additive inverse: ⓐ \(18\) ⓑ \(\frac{7}{9}\) ⓒ \(1.2\).
Αποκάλυψέ την.
1. ⓐ \(\ -18\\)
2. ⓑ \(\ -\frac{7}{9}\)
3. ⓒ \(\ -1.2\)
Find the additive inverse: ⓐ \(47\) ⓑ \(\frac{7}{13}\) ⓒ \(\ 8.4\).
Αποκάλυψέ την.
1. ⓐ \(\ -47\)
2. ⓑ \(\ -\frac{7}{13}\)
3. ⓒ \(\ -8.4\)
Find the multiplicative inverse: ⓐ \(9\\) ⓑ \(-\frac{1}{9}\\) ⓒ \(0.9\).
Αποκάλυψέ την.
To find the multiplicative inverse, we find the reciprocal.
1. ⓐ The multiplicative inverse of \(9\) is its reciprocal, \(\frac{1}{9}.\)
2. ⓑ The multiplicative inverse of \(-\frac{1}{9}\) is its reciprocal, \(-9.\)
3. ⓒ To find the multiplicative inverse of \(0.9,\) we first convert \(0.9\) to a fraction, \(\frac{9}{10}.\) Then we find the reciprocal, \(\frac{10}{9}.\)
Find the multiplicative inverse: ⓐ \(\ 5\\) ⓑ \(\ -\frac{1}{7}\\) ⓒ \(\ 0.3\).
Αποκάλυψέ την.
1. ⓐ \(\ \frac{1}{5}\)
2. ⓑ \(\ -7\)
3. ⓒ \(\ \frac{10}{3}\)
Find the multiplicative inverse: ⓐ \(\ 18\) ⓑ \(\ -\frac{4}{5}\) ⓒ \(\ 0.6\).
Αποκάλυψέ την.
1. ⓐ \(\ \frac{1}{18}\)
2. ⓑ \(\ -\frac{5}{4}\)
3. ⓒ \(\ \frac{5}{3}\)
Simplify: ⓐ \(\ -8\cdot 0\) ⓑ \(\ \frac{5}{12}\cdot 0\) ⓒ \(\ 0(2.94)\).

Αποκάλυψέ την.

ⓐ
\(-8⋅0\)
The product of any real number and 0 is 0. \(0\)
ⓑ
\(\frac{5}{12}\cdot 0\)
The product of any real number and 0 is 0. \(0\)
ⓒ
\(0(2.94)\)
The product of any real number and 0 is 0. \(0\)
Simplify: ⓐ \(\ 0\div 5\\) ⓑ \(\frac{0}{-2}\) ⓒ \(\ 0\div \frac{7}{8}\).

Αποκάλυψέ την.

ⓐ
\(0\div 5\)
Zero divided by any real number, except 0, is zero. \(0\)
ⓑ
\(\frac{0}{-2}\)
Zero divided by any real number, except 0, is zero. \(0\)
ⓒ
\(0\div \frac{7}{8}\)
Zero divided by any real number, except 0, is zero. \(0\)
Simplify: ⓐ \(\ 7.5\div 0\) ⓑ \(\ \frac{-32}{0}\) ⓒ \(\ \frac{4}{9}\div 0\).

Αποκάλυψέ την.

ⓐ
\(7.5\div 0\)
Division by zero is undefined. undefined
ⓑ
\(\frac{-32}{0}\)
Division by zero is undefined. undefined
ⓒ
\(\frac{4}{9}\div 0\)
Division by zero is undefined. undefined
Simplify: ⓐ \(\ 16.4\div 0\) ⓑ \(\ \frac{-2}{0}\) ⓒ \(\ \frac{1}{5}\div 0\).
Αποκάλυψέ την.
1. ⓐ undefined
2. ⓑ undefined
3. ⓒ undefined
Simplify: ⓐ \(\ \frac{-5}{0}\) ⓑ \(\ 96.9\div 0\) ⓒ \(\ \frac{4}{15}\div 0\)
Αποκάλυψέ την.
1. ⓐ undefined
2. ⓑ undefined
3. ⓒ undefined
Simplify: \(3x+15-3x.\)

Αποκάλυψέ την.

\(3x+15-3x\)
Notice the additive inverses, \(3x\) and \(-3x\). \(0+15\)
Add. \(15\)
Simplify: \(-12z+9+12z.\)

Αποκάλυψέ την.

9
Simplify: \(-25u-18+25u.\)

Αποκάλυψέ την.

−18
Simplify: \(4(0.25q).\)

Αποκάλυψέ την.

\(4(0.25q)\)
Regroup, using the associative property. \([4(0.25)]q\)
Multiply. \(1.00q\)
Simplify; 1 is the multiplicative identity. \(q\)
Simplify: \(2(0.5p).\)

Αποκάλυψέ την.

p
Simplify: \(25(0.04r).\)

Αποκάλυψέ την.

r
Simplify: \(\frac{0}{n+5}\), where \(n\ne -5\).

Αποκάλυψέ την.

\(\frac{0}{n+5}\)
Zero divided by any real number except itself is zero. \(0\)
Simplify: \(\frac{0}{m+7}\), where \(m\ne -7\).

Αποκάλυψέ την.

0
Simplify: \(\frac{0}{d-4}\), where \(d\ne 4\).

Αποκάλυψέ την.

0
Simplify: \(\frac{10-3p}{0}.\)

Αποκάλυψέ την.

\(\frac{10-3p}{0}\)
Division by zero is undefined. undefined
Simplify: \(\frac{18-6c}{0}.\)

Αποκάλυψέ την.

undefined
Simplify: \(\frac{15-4q}{0}.\)

Αποκάλυψέ την.

undefined
Simplify: \(\frac{3}{4}\cdot \frac{4}{3}(6x+12).\)

Αποκάλυψέ την.

We cannot combine the terms in parentheses, so we multiply the two fractions first.
\(\frac{3}{4}\cdot \frac{4}{3}(6x+12)\)
Multiply; the product of reciprocals is 1. \(1(6x+12)\)
Simplify by recognizing the multiplicative identity. \(6x+12\)
Simplify: \(\frac{2}{5}\cdot \frac{5}{2}(20y+50).\)

Αποκάλυψέ την.

20y + 50
Simplify: \(\frac{3}{8}\cdot \frac{8}{3}(12z+16).\)

Αποκάλυψέ την.

12z + 16
\(101+0=101\)
\(\frac{3}{5}(1)=\frac{3}{5}\)

Αποκάλυψέ την.

identity property of multiplication
\(-9\cdot 1=-9\)
\(\frac{7}{12}\)

Symbols used here

\neq

not equal
The two sides are different.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\approx

approximately equal
Equal to the precision shown, not exactly.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Properties of Identity, Inverses, and Zero

Recognize the identity properties of addition and multiplication
Use the inverse properties of addition and multiplication
Use the properties of zero
Simplify expressions using the properties of identities, inverses, and zero

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

Δοκίμασε μόνος σου.

Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

Περισσότερα σε Arithmetic

Order of operations Fractions Percentages Powers and roots

ⓐ
	\(0\div 5\)
Zero divided by any real number, except 0, is zero.	\(0\)

ⓑ
	\(\frac{0}{-2}\)
Zero divided by any real number, except 0, is zero.	\(0\)

ⓒ
	\(0\div \frac{7}{8}\)
Zero divided by any real number, except 0, is zero.	\(0\)

ⓒ
	\(\frac{4}{9}\div 0\)
Division by zero is undefined.	undefined