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Rule

Zero Property of Multiplication

The result of multiplying any number by

0

is always

0 .

a \cdot 0 = 0

As a consequence of this, if an entire expression is multiplied by

0,

the result is

0 .

It does not matter the number of terms that the expression has.

Proof

Consider a number

a

multiplied by zero.

a \cdot 0

The number zero can be rewritten as the subtraction of any number from itself. For simplicity, the zero can be rewritten as

1 - 1 .

a \cdot 0 = a \cdot (1 - 1)

Then the number

a

can be distributed to simplify the expression on the right-hand side of the equation.

a \cdot 0 = a \cdot (1 - 1)

Distr

Distribute $a$

a \cdot 0 = a - a

DiffZero

$a - a = 0$

a \cdot 0 = 0

It has been shown that

a \cdot 0 = 0 .

By the Commutative Property of Multiplication,

0 \cdot a

is also equal to

0 .

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Proof