Identity Property of Multiplication

What is the identity property of multiplication?

The identity property of multiplication, sometimes called the one property of multiplication, means that the product of any number and one is always the original number itself.

Whenever you multiply a quantity by one, the quantity retains its original mathematical identity. Understanding the properties of multiplication helps simplify complex calculations and makes solving algebraic equations much more efficient.

The multiplicative identity guarantees that multiplying by one does not change a number's value.

Why one is the multiplicative identity

In mathematics, an identity is a special value that leaves other numbers unchanged during a specific operation.

For multiplication, this special number is one. When you have one group of a certain quantity, you simply have that exact quantity, without any increases or decreases.

Because multiplication is commutative, the order of the numbers does not matter. Whether you multiply a starting number by one, or multiply one by the starting number, the final result is identical.

Formula and visual model

The identity property can be written as an algebraic formula, where the letter $a$ represents any real number.

$a \times 1 = a$

$1 \times a = a$

We can model this property visually using arrays. One row of five objects represents the same total quantity as five rows containing one object each.

Identity with different number types

The multiplicative identity applies uniformly to every type of number, not just positive whole numbers. The rule behaves the exact same way for fractions, decimals, negative integers, and unknown variables.

• Whole numbers: $42 \times 1 = 42$
• Fractions: $\dfrac{3}{4} \times 1 = \dfrac{3}{4}$
• Decimals: $1 \times 8.65 = 8.65$
• Negative numbers: $-15 \times 1 = -15$
• Variables: $1 \times y = y$

Identity versus zero

Students often confuse the identity property with the zero property of multiplication.

While multiplying by one preserves the original number, multiplying by zero completely changes the value because it always results in zero. Additionally, while the undefined nature of division by zero restricts calculations, multiplying and dividing by one are perfectly valid operations that leave quantities intact.

Worked examples

Example 1: Finding an unknown multiplier

Question: Find the missing value in the equation $14 \times \square = 14$.

Method:

1. Look at the starting number and the final product.
2. Notice that the number $14$ has remained completely unchanged.
3. Apply the identity property of multiplication, which states that any number multiplied by one equals itself.

Answer: The missing value is $1$.

Check: $14 \times 1 = 14$. The equation is correct.

Example 2: Simplifying algebraic expressions

Question: Simplify the expression $1 \times (3y + 2)$.

Method:

1. Identify the multiplier outside the parentheses.
2. Recognize that multiplying the entire expression by one will leave its value unchanged.
3. Remove the multiplier of one, leaving the internal expression intact.

Answer: The simplified expression is $3y + 2$.

Check: Distributing the $1$ yields $1(3y) + 1(2) = 3y + 2$.

Example 3: Applying identity to fractions

Question: What is the product of $\dfrac{5}{8}$ and $1$?

Method:

1. Set up the multiplication problem: $\dfrac{5}{8} \times 1$.
2. Use the identity property of multiplication. Any fraction multiplied by one retains its original value.

Answer: The product is $\dfrac{5}{8}$.

Check: Write $1$ as $\dfrac{8}{8}$ and multiply: $\dfrac{5}{8} \times \dfrac{8}{8} = \dfrac{40}{64}$, which simplifies back to $\dfrac{5}{8}$.

Frequently asked questions

Is there an identity property for addition?

Yes, but the identity number for addition is zero, not one. The additive identity property states that adding zero to any number leaves it unchanged (for example, $7 + 0 = 7$).

Does the identity property apply to division?

It works in one direction. Dividing any number by one leaves the number unchanged ($a \div 1 = a$). However, dividing one by another number changes the value ($1 \div a \neq a$), so division does not share the same full commutative identity property as multiplication.