Additive Identity Vs Multiplicative Identity

Additive Identity and Multiplicative Identity are two different identity properties of numbers. When additive identity is added to a number, it returns the original number. Similarly, when multiplicative identity is multiplied by any number, it returns the original number. Both these properties are applicable to all real numbers. Additive identity vs multiplicative identity describes here the difference between the two properties.

Zero (0) is the additive identity and one (0) is the multiplicative identity for all the numbers such as whole numbers, natural numbers, integers, etc.

What is Additive Identity?

Additive identity of numbers, as the name suggests, is a property of numbers that are applied when carrying out addition operations. The property states that when a number is added to zero it will give the same number. “Zero” is called the identity element, (also known as additive identity) If we add any number with zero, the resulting number will be the same number. This is true for any real numbers, complex numbers and even for imaginary numbers.

Suppose, ‘a’ is any real number, then

a + 0 = a =  0 + a

For example, 120 + 0 =  120, illustrates identity property of addition, where 0 is the additive identity.

What is Multiplicative Identity?

Multiplicative identity of numbers, as the name suggests, is a property of numbers which is applied when carrying out multiplication operations Multiplicative identity property says that whenever a number is multiplied by the number \(1\) (one) it will give that number as product. “\(1\)” is the multiplicative identity of a number. It is true if the number being multiplied is \(1\) itself. The multiplicative identity property is represented as:

a × 1 = a = 1 × a  (a is any real number)

Some examples:

−1 + 0 = −1 (−1 here is the number on which the operation is carried out and “0” is additive identity.  

0 + 259 = 259 

−1 × 1 = −1 (−1 here is the number on which the operation is carried out and “1” is a multiplicative identity)  

Note: (−1) × (−1) = 1     (proves that −1 is not a multiplicative identity)

Difference Between Additive Identity and Multiplicative Identity

Additive Identity Multiplicative Identity
Additive identity for any real number is 0. Multiplicative identity for any real number is 1.
It is denoted by a + 0 = a, where a is any real number It is denoted by a x 1 = a, where a is any real number
It is used in addition It is used in the multiplication operation
Example: 7 + 0 = 7 Example: 7 x 0 = 7

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Solved Examples

Q.1: Which of the following illustrates the multiplicative identity and additive identity?

  1. 45 + 1 = 46 
  2. 50 × 2 = 100 
  3. 14 × 1 = 14 
  4. −54 + 0 =  −54

Solution:

According to the identity property of multiplication, the product of any number multiplied by 1 is the number itself.  

Here, only 14 × 1 = 14 satisfies the property.  

Therefore, 14 × 1= 14 illustrates the Multiplicative identity.  

According to the identity property of addition, the sum of any number added to 0 is the number itself.  

Here, only −54 + 0 = −54 satisfied the property.  

Therefore, −54 + 0 = −54 illustrates the additive identity.

Q.2: If a + 0 = -11, then what is the value of a?

Solution: Given, a + 0 = -11

By additive identity, if 0 is added to a number, it returns the original number.

Thus, a = -11

Q.3: If n x 1 = 1100, then what is the value of n?

Solution: Given, n x 1 = 1100

By multiplicative identity, if 1 is multiplied by a number, it returns the original number.

Thus, n = 1100

To solve more problems on the topic, download BYJU’S – The Learning App from Google Play Store and watch interactive videos.

Frequently Asked Questions – FAQs

What is additive identity?

Additive identity is the value when added to a number, results in the original number. When we add 0 to any real number, we get the same real number. For example, 5 + 0 = 5. Therefore, 0 is the additive identity of any real number.

Which is the multiplicative identity, 0 or 1?

Multiplicative identity of a real number is 1. When we multiply 1 to any real number then we get the same number. Examples are 3 x 1 = 3, -9 x 1 = -9, ½ x 1 = ½.

Is -1 also a multiplicative identity?

-1 is not a multiplicative identity, because if we multiply -1 to any real number, the sign of that number gets changed. Examples are:
4 x -1 = -4
-4 x -1 = 4

What is the multiplicative identity of 5?

Multiplicative identity if 5 is 1 only, since 5 x 1 = 5.

What is the additive identity of x?

Additive identity of x is 0, since x+0 = x.

Test your knowledge on Additive identity and Multiplicative identity

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