Additive Identity vs Multiplicative Identity

Additive Identity and Multiplicative Identity are the two basic algebraic identities widely used in mathematics. Additive identity refers to a unique number which when added to any other number, yields the same number. Multiplicative identity refers to the number which when multiplied by any number, gives back the original number.

In this article, we will discuss about additive and multiplicative identities, their properties, and the difference between additive and multiplicative identities.

What is Additive Identity?

Additive identity is a number that leaves the number unchanged when added to any other number. Additive Identity states that when any number is added to zero, it yields the same number. Hence Zero is known as the identity element of additive identity. This property is true for all complex numbers, imaginary numbers, integers, rational numbers, and real numbers.

Additive Identity Property Definition

Additive identity, denoted as 0, is a unique element in arithmetic operations that, when added to any number, leaves the number unchanged.

Any real number when added to zero returns the original number only, hence zero (0) is the additive identity of all real numbers. In algebraic terms, if a is any real number, then a + 0 = a.

For Example:

• 2 + 0 = 2
• i + 0 = i
• ∛3 + 0 = ∛3
• -5 + 0 = -5

From the above example, we can observe that addition of any number with zero, results in original number.

What is Multiplicative Identity?

Multiplicative identity is a number which when multiplied by any other number leaves the number unchanged. Multiplicative Identity states that when any number is multiplied to one, it returns the same number. Hence, 1 is known as the identity element of multiplicative identity. This property is true for all complex numbers, real numbers, integers, whole numbers, and natural numbers.

Multiplicative Identity Property Definition

Multiplicative identity, denoted as 1, is a special element in arithmetic operations such that when any number is multiplied by 1, the result is the number itself. Formally, for any real number a, a × 1 = a.

Any real number when multiplied to one returns the original number only, hence 1 is the multiplicative identity of all real numbers. In algebraic terms, if a is any real number, then a × 1 = a.

For Example:

• 5 × 1 = 5
• -i × 1 = -i
• √2 × 1 = √2
• -2.5 × 1 = -2.5

From the above example, we can observe that multiplication of any number with one, results in original number.

Significance of Multiplicative Identity

Multiplicative identity is essential in multiplication because it serves as the base element that preserves the value of any number when multiplied by it. Without the multiplicative identity, the basic principles of multiplication would not hold true.

The multiplicative identity is important in multiplication because it serves as the neutral element, ensuring that the operation does not change the value of the other operand.

Additive Identity vs Multiplicative Identity

Additive and Multiplicative Identities are essential in various mathematical operations and serve as the foundation for algebraic manipulation and problem-solving. The key difference between additive and multiplicative identity is given below:

Feature	Additive Identity (0)	Multiplicative Identity (1)
Definition	Adding 0 to a number doesn’t change its value.	Multiplying 1 by a number doesn’t change its value.
Property	a + 0 = a	a × 1 = a
Identity Element	0 is the identity element.	1 is the identity element.
Used in	It is used in addition operation.	It is used in multiplication operation.
Example	4 + 0 = 4	4 × 1 = 4

Important Notes on Additive Identity and Multiplicative Identity

Following are few important points related to additive and multiplicative identity:

• 0 is the additive identity.
• 1 is the multiplicative identity.
• For any real number a, a + 0 = a and a × 1 = a.
• -1 is not a multiplicative identity.

Read More,

• Additive and Multiplicative Inverse
• Additive Identity and Inverse of Complex Numbers
• Multiplicative Identity and Multiplicative Inverse of the complex number

Examples on Additive and Multiplicative Identity

Example 1: If x + 0 = 20, what is the value of x?

Solution:

Since 0 is the additive identity, adding it to any number doesn’t change the value. Therefore, x = 20.

Example 2: If y × 1 = 35, what is the value of y?

Solution:

Since 1 is the multiplicative identity, multiplying it by any number doesn’t change the value. Therefore, y = 35.

Example 3: Simplify the expression: (2x + 0) – (3x × 1) ?

Solution:

We have (2x + 0) – (3x × 1)

Apply the additive and multiplicative identities we get:

2x + 0 = 2x

3x × 1 = 3x

Solving we get

2x – 3x = -x.

Example 4: What is the result of -4 + 0 and -4 × 1?

Solution:

Apply the additive and multiplicative identities we get:

-4 + 0 = -4

-4 × 1 = -4

Practice Questions on Additive and Multiplicative Identity

Question 1: Which of the following equation is an example of additive identity and multiplicative identity? Give an explanation for each.

• -2 + 0 = 2
• b × 1 = b
• 4/5 × 1 = 4/5
• -7 + 1 = -6
• 2 × 2 = 4
• 5 + 0 = 5
• 3 × -1 = -3

Question 2: What are multiplicative and additive identity elements for real numbers?

Additive and Multiplicative Identity FAQs

What is Additive Identity?

Additive identity refers to the element 0 in arithmetic operations, where adding it to any number leaves the number unchanged.

What is Multiplicative Identity?

Multiplicative identity refers to the element 1 in arithmetic operations, where multiplying it by any number results in the number itself.

Is -1 a Multiplicative Identity?

No, -1 is not a multiplicative identity as a × -1 ≠ a.

How are Additive Identity and Multiplicative Identity Different?

While both identities preserve the value of numbers in their respective operations, additive identity involves addition (with 0) while multiplicative identity involves multiplication (with 1).

Why is 0 not an Identity of Subtraction?

0 is not an identity of subtraction as a – 0 = a but 0 – a ≠ a

Is Additive Identity Applicable to Negative Integers?

Yes, additive identity is applicable to negative integers as – a + 0 = -a.