The identity property states that performing an arithmetic operation using a certain number leaves the original value unchanged. It is one of the key properties in arithmetic, along with the commutative, associative, and distributive properties.

- Adding or subtracting zero to any number gives the original number.
- Multiplying any number by 1 gives the original number.
- Dividing any non-zero number by 1 gives the original number.

### Identity Property Definition

**Identity Property**: The **identity property** states that when combined with another number in a specific arithmetic operation, the identity element leaves that number unchanged. The property is valid for natural numbers, whole numbers, integers, rational numbers, irrational numbers, and complex numbers.

**Identity Element**: The **identity element** is the specific value that, when combined with another number in a specific operation, leaves the number unchanged.

For addition and subtraction, the identity element is 0. For multiplication and division, the identity element is 1. Just remember that you can’t divide by zero, so for division the property only applies to non-zero numbers.

### Identity Property for Addition

The identity property for addition states that when you add zero to any number, the result will be the number itself.

**Identity**: *a* + 0 = *a*

**Examples**:

- 5 + 0 = 5
- −8 + 0 = −8
- 0 + 123 = 123

**Word Problem Example**:

Mary has 5 apples. If no apples are added to her collection, how many apples does she have?

Answer: 5 + 0 = 5 apples

### Identity Property for Subtraction

The identity property for subtraction states that when you subtract zero from any number, the result will still be the number itself.

**Identity**: *a* −0 = *a*

**Examples**:

- 7 − 0 = 7
- −3 − 0 = −3
- 0 − 0 = 0

**Word Problem Example**:

John had 10 books. If he gave none away, how many books does he still have?

Answer: 10 − 0 = 10 books

### Identity Property for Multiplication

The identity property for multiplication says that when you multiply any number by one, the result is the number itself.

**Identity**: *a* × 1 = *a*

**Examples**:

- 6 × 1 = 6
- −4 × 1 = −4
- 0 × 1 = 0

**Word Problem Example**:

A factory produces 500 toys each day. If they don’t increase or decrease production, how many toys will they produce in one day?

Answer: 500 × 1 = 500 toys

### Identity Property for Division

The identity property for division expresses that when you divide any number (other than zero) by one, the result is the number itself.

**Identity**: *a* ÷ 1 = *a* for a ≠ 0

**Examples**:

- 9 ÷ 1 = 9
- −7 ÷ 1 = −7
- 100 ÷ 1 = 100

**Word Problem Example**:

Emily has 45 chocolates. If she divides them equally among 1 friend, how many chocolates does that friend receive?

Answer: 45 ÷ 1 = 45 chocolates

### Example Problems

Test your understanding of the identity property with these example problems and answers:

7 + ___= 7

a. 1

b. 7

c. 0

d. -7

5 × ___ = 5

a. 0

b. 5

c. 1

d. -5

12 ÷ ___ = 12

a. 12

b. 0

c. 1

d. 2

−3 − ___ = −3

a. 3

b. 0

c. -3

d. 1

**Answers**:

- c. 0
- c. 1
- c. 1
- b. 0

### References

- Byers, William (2007).
*How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics*. Princeton UP. ISBN 978-0-691-12738-5. - Copi, Irving M.; Cohen, Carl; McMahon, Kenneth (2014).
*Introduction to Logic*(14th ed.). Essex: Pearson Education. ISBN 9781292024820. - Durbin, John R. (1992).
*Modern Algebra: An Introduction*(3rd ed.). New York: Wiley. ISBN 978-0-471-51001-7. - Hungerford, Thomas W. (1974).
*Algebra*(1st ed.). Springer. ISBN 978-0387905181.