Zero-Product Property

A product is equal to zero if and only if at least one of its factors is zero. $$ a \cdot b = 0 \Longleftrightarrow a=0 \ \ ∨ \ \ b = 0 $$

This condition

• is necessary, because if a product equals zero, then at least one factor must be zero
• is sufficient, because if at least one factor is zero, then the product necessarily equals zero

Zero is the absorbing element for multiplication.

The product of two or more factors is therefore zero whenever zero appears at least once among the factors in the multiplication.

For example

$$ 3 \cdot 0 = 0 $$

The position of zero among the factors is irrelevant, since multiplication satisfies the commutative property.

$$ 0 \cdot 3 = 0 $$

The zero-product property remains valid even when more than two factors are involved.

$$ 2 \cdot 3 \cdot 0 = 0 $$

And so on.