Division by Zero

A lot of people in Algebra 1 know that you can’t divide by zero, though you can divide zero by a number and the answer will be zero. This is confusing for a lot of people because they don’t understand how or why this happens. Here we’re going to explain how this happens, why it happens so later you can see why it’s relevant to solving equations when rational expressions are involved.

You learn early on that the opposite of multiplication is division, and vice versa. So consider the equation 3 * 2 = 6. The opposite would then be 6 / 2 = 3. So what if we have an equation with a variable in it? We can do the same sort of thing, so something like x * 5 = 30 becomes 30 / 5 = x, showing us that x is clearly 6.

We can reverse division problems in the same way. Since 8 / 2 = 4, we know that 4 * 2 = 8. Similarly, if x / 6 = 3, then 3 * 6 = x, and we’ll know that x is 18.

With all of this in mind, consider the equation 5 / 0 = x. The opposite of this equation would be x * 0 = 5. What you’ll notice is that there are no values that will work for x because anything times zero has to equal zero. There are no numbers when multiplied by zero equal five. Therefore, x doesn’t exist, and this shows us why division by zero doesn’t work.

Now let’s look at a similar example with the equation 0 / 5 = x. The opposite equation would be x * 5 = 0, and we know from common sense that the only value for x that works is zero. Therefore, we see that dividing zero by a non-zero number is acceptable, and always equals zero.