Multiplying any number by zero always equals zero. Below are some examples.

To see why multiplying any number by zero always equals zero, let's take a look at the multiplication table for , shown below.

As we go down the list, the number in blue is reduced by each time. Following this pattern, we must have .

Note that, since multiplication is commutative, multiplying zero by any number (including itself) equals zero as well. Some examples are shown below.

Dividing zero by any number (other than zero) always equals zero. For example:

To understand why dividing zero by any number (other than zero) always equals zero, think about it this way: if you have zero pizzas, and you want to divide zero pizzas among ten friends, then each friend will get exactly zero pizza.

Division by zero is undefined. No number can be divided by zero, not even zero itself.

To see why, notice that if we divide by we get

Therefore, by the relationship between multiplication and division, we have

However, this argument does not work if we divide by zero. Suppose we have

Then, by the relationship between multiplication and division, we have

and this relationship cannot be true because any number multiplied by zero is zero! Therefore, is not a number.