Multiplicative Identity — Definition, Formula & Examples

The multiplicative identity is the number 1, because multiplying any number by 1 gives back that same number. No matter what value you start with — positive, negative, fraction, or decimal — multiplying by 1 leaves it unchanged.

The multiplicative identity is the unique real number, denoted 1, such that for every real number $a$ , the equations $a \times 1 = a$ and $1 \times a = a$ both hold.

Key Formula

a \times 1 = 1 \times a = a

Where:

$a$ = Any real number
$1$ = The multiplicative identity element

Worked Example

Problem: Simplify the expression

-7.5 \times 1

Apply the identity property: Multiplying any number by 1 returns the original number.

-7.5 \times 1 = -7.5

Answer:

-7.5

Why It Matters

Recognizing the multiplicative identity helps you simplify algebraic expressions quickly — for instance, knowing that

\frac{x}{x} = 1

(when

x \neq 0

) relies on this property. It also underpins how we create equivalent fractions by multiplying by

\frac{n}{n}

, which equals 1.

Common Mistakes

Mistake: Confusing the multiplicative identity (1) with the additive identity (0).

Correction: Multiplying by 0 gives 0, not the original number. The number that leaves a value unchanged under multiplication is 1, while 0 is the identity for addition.