# Arithmetic properties: Identity property

### Arithmetic properties: Identity property

#### Lessons

In this lesson, we will learn:

• What is the additive identity property of zero
• What is the multiplicative identity property of one
• How to write the general formulas/equations for the identity properties
• What are the three properties of zero?
• How the identity properties are different from the properties of zero
• How to write the general formulas/equations for properties of zero

Notes:

• The identity property is observed when the identity of the original number does NOT change after the equal sign. The answer will be the same number that you started with.
• The numbers can be any real number (whole numbers, fractions, decimals, integers, etc.)
• The word “identity” can mean who you are or what you are

• The identity property only happens for TWO cases in math:
• For addition: adding zero to any number will NOT change that number
• Ex. 8 + 0 = 8
• Ex. 0.5 + 0 = 0.5
• Ex. $\large \frac{1}{2}$ + 0 = $\large \frac{1}{2}$
• For multiplication: multiplying any number by one will NOT change that number
• Ex. 8 × 1 = 8
• Ex. 1.47 × 1 = 1.47
• Ex. $\large \frac{3}{4}$ × 1 = $\large \frac{3}{4}$

• The general formulas for the identity property (where a is $a$ variable that represent a real number) are:

•  Arithmetic Property Of Addition Of Multiplication (Additive) Identity property of 0 $a + 0 = a$ $0 + a = a$ * (Multiplicative) Identity property of 1 $a × 1 = a$ $1 × a = a$

• It is important to know the properties of zero – *what happens when you multiply and divide by zero so that you don't get them confused with the identity property of zero
• There are three properties of zero:
• (1) when you multiply by zero, the answer will always be zero
• (2) when zero is divided by any number, the answer will always be zero
• (3) when you attempt to divide any number by zero, the answer will always be undefined

• The general formulas for the properties of zero are:

•  Arithmetic Property Of Multiplication Of Division Properties of Zero $a × 0 = 0$ $0 × a = 0$ $0 ÷ a = 0$ - - - - - - - - - - - - - - - - - $a ÷ 0 = undefined$
• Introduction
Introduction to the identity property of addition and multiplication (and properties of zero):
a)
Showing that $a + 0 = a$

b)
Why is it called the "identity" property?

c)
Showing that a × 1 = a

d)
The general formulas for the identity property

e)
The three properties of zero involving multiplication and division (a × 0 = 0; 0 ÷ a = 0; and a ÷ 0 = undefined)

f)
The general formulas for the properties of zero

• 1.
Additive identity property of 0
Use the additive identity of 0 to fill in the blanks.
a)
287 + __ = 287

b)
__ + 0 = 0.39

c)
0 + __ = $\large \frac{517}{1000}$

• 2.
Multiplicative identity property of 1
Use the multiplicative identity of 1 to fill in the blanks.
a)
657 × __ = 657

b)
1 × __ = 8.914

c)
__ × $\large \frac{832}{900}$ = $\large \frac{832}{900}$

• 3.
Multiplying and dividing using properties of 0
Use the properties of 0 to fill in the blanks.
a)
$\large \frac{13}{25}$ × 0 = __

b)
__ × 1 = 0

c)
0 ÷ 25 = __

d)
35 ÷ __ = undefined

e)
7.6 × 0 = __

f)
439 ÷ 0 = __

• 4.
Identity properties and all four operations
What happens to the identity of number 46 when:
a)
46 + 0 =

b)
46 - 0 =

c)
46 × 0 =

d)
46 × 1 =

e)
46 ÷ 1 =

f)
46 ÷ 0 =

g)
0 ÷ 46 =

• 5.
Identity properties word problem
If $a$, $b$ and $c$ are real numbers with secret identities:
a)
What happens to a when it is added to 0

b)
What happens to b when it is multiplied with 1

c)
What happens to c when it is multiplied with 0

d)
What happens to a when it is divided by 0