Identity Elements (Additive & Multiplicative)
When you add 0 to any number, the number stays the same. When you multiply any number by 1, the number stays the same.
This is because 0 is the additive identity and 1 is the multiplicative identity.
In Class 6, you will understand what identity elements are and why they are important in mathematics.
What is Identity Elements - Grade 6 Maths (Whole Numbers)?
Additive Identity:
- The number 0 is called the additive identity.
- When you add 0 to any number, the result is the same number.
- a + 0 = a and 0 + a = a (for any number a).
Multiplicative Identity:
- The number 1 is called the multiplicative identity.
- When you multiply any number by 1, the result is the same number.
- a × 1 = a and 1 × a = a (for any number a).
a + 0 = a (Additive Identity)
a × 1 = a (Multiplicative Identity)
Identity Elements (Additive & Multiplicative) Formula
Why are they called "identity" elements?
- An identity element is a special number that does not change the other number during an operation.
- Adding 0 keeps the number's "identity" unchanged.
- Multiplying by 1 keeps the number's "identity" unchanged.
These work for all types of numbers:
- Whole numbers: 5 + 0 = 5, 5 × 1 = 5
- Fractions: 3/4 + 0 = 3/4, 3/4 × 1 = 3/4
- Decimals: 2.7 + 0 = 2.7, 2.7 × 1 = 2.7
- Negative numbers: (−3) + 0 = −3, (−3) × 1 = −3
Types and Properties
Related properties:
Zero in multiplication:
- a × 0 = 0 for any number a. This is the zero property of multiplication.
- This is NOT an identity property — it changes the number (to 0).
- a − 0 = a (subtracting 0 gives the same number).
- But 0 − a = −a (not the same as a, unless a = 0).
- So 0 is identity for subtraction only from the right side.
One in division:
- a ÷ 1 = a (dividing by 1 gives the same number).
- But 1 ÷ a is NOT the same as a.
- So 1 is identity for division only when dividing BY 1.
Solved Examples
Example 1: Additive Identity with Whole Numbers
Problem: Find: 156 + 0.
Solution:
Adding 0 to any number gives the same number.
Answer: 156 + 0 = 156
Example 2: Multiplicative Identity
Problem: Find: 234 × 1.
Solution:
Multiplying any number by 1 gives the same number.
Answer: 234 × 1 = 234
Example 3: Additive Identity with Fractions
Problem: Find: 5/8 + 0.
Solution:
0 is the additive identity. Adding 0 changes nothing.
Answer: 5/8 + 0 = 5/8
Example 4: Multiplicative Identity with Decimals
Problem: Find: 3.14 × 1.
Solution:
1 is the multiplicative identity.
Answer: 3.14 × 1 = 3.14
Example 5: Identifying the Identity Element
Problem: In 45 + ___ = 45, what fills the blank?
Solution:
The number that when added gives the same result is 0 (additive identity).
Answer: 0
Example 6: Multiplicative Identity in Equations
Problem: In ___ × 78 = 78, what fills the blank?
Solution:
The number that when multiplied gives the same result is 1 (multiplicative identity).
Answer: 1
Example 7: Zero Property vs Identity
Problem: Is 0 an identity element for multiplication?
Solution:
No. 5 × 0 = 0, not 5. Multiplying by 0 changes the number to 0. So 0 is NOT a multiplicative identity.
Answer: No. The multiplicative identity is 1, not 0.
Example 8: Identity with Negative Numbers
Problem: Find: (−7) + 0 and (−7) × 1.
Solution:
- (−7) + 0 = −7 (additive identity works)
- (−7) × 1 = −7 (multiplicative identity works)
Answer: Both give −7. The identities work for negative numbers too.
Real-World Applications
Where identity elements are used:
- Checking calculations: If adding a number changes your answer, something is wrong — unless you added 0.
- Simplifying expressions: 5x + 0 = 5x, so you can remove the + 0 part.
- Algebra: The identity properties help solve equations and simplify expressions.
- Fractions: Multiplying by 1 in the form a/a helps convert fractions without changing value.
- Computer science: Identity elements are used in programming and database operations.
Key Points to Remember
- Additive identity: 0. Adding 0 to any number gives the same number (a + 0 = a).
- Multiplicative identity: 1. Multiplying any number by 1 gives the same number (a × 1 = a).
- These work for all numbers — whole, fractional, decimal, negative.
- 0 is NOT a multiplicative identity (a × 0 = 0, not a).
- Do not confuse: a + 0 = a (identity) with a × 0 = 0 (zero property).
- a − 0 = a and a ÷ 1 = a, but these are one-sided identities.
Practice Problems
- Find: 999 + 0.
- Find: 0 + 473.
- Find: 56 × 1.
- Fill in the blank: ___ + 28 = 28.
- Fill in the blank: 45 × ___ = 45.
- True or False: 0 is the multiplicative identity.
Frequently Asked Questions
Q1. What is an identity element?
An identity element is a special number that does not change the other number during an operation. For addition it is 0, for multiplication it is 1.
Q2. Why is 0 not a multiplicative identity?
Because multiplying any number by 0 gives 0, not the original number. For identity, the result must be the original number. 5 × 0 = 0, not 5.
Q3. Does the additive identity work for subtraction?
Only from one side: a − 0 = a (yes). But 0 − a = −a (not the same as a, unless a = 0). So 0 is not a full identity for subtraction.
Q4. Are there identity elements for subtraction and division?
Not true identity elements. a − 0 = a and a ÷ 1 = a work, but only from one side. True identities work from both sides.
Q5. Does this work for all numbers?
Yes. Whole numbers, integers, fractions, decimals — adding 0 or multiplying by 1 always gives the same number.
