Zero property of multiplication

The zero property of multiplication states that the product of any number and 0 is equal to 0. Given that a is any number, the zero property of multiplication can be generalized as:

a × 0 = 0

0 × a = 0

It doesn't matter what order the numbers are multiplied in (commutative property), the result of multiplying 0 by anything (or anything by 0) is 0.

No matter the type of number, whether it be an integer, fraction, decimal, or even imaginary, the product of that number and 0 is 0. One way to conceptualize this is to think of multiplication as a number of groups of objects. Whether you have 0 groups of 3 objects, or 3 groups of 0 objects, you will still have 0 objects.

The figure above depicts a multiplication problem where there are 5 groups, each containing 3 groups of objects. The figure also depicts 5 groups that contain 0 objects, and a blank space to represent there being 0 groups of 3 objects. It is worth noting that even though division is the inverse operation of multiplication, it does not share a similar zero property. Dividing by 0 does not result in 0, as in multiplication. In fact, it is not possible to divide any number by 0.

The zero property of multiplication is one of a number of other multiplication properties such as the commutative, associative, distributive, and identity properties of multiplication. Learning the various properties of multiplication (and other operations) is important for building a foundation that enables a student to work with more complex mathematical concepts. Particularly in algebra, knowing number properties enables us to simplify and solve a wide variety of algebraic equations.