Why Can’t We Divide By Zero? Whenever I ask a student why we can’t divide by zero I always get strange answers. Most students just restate the question in another way. Some typical answers are “because division by 0 is impossible,” “because my teacher told me I can’t,” or “because my calculator gives an error when I try to do it.” But none of these “answers” explain anything. We will clear up this mystery by the end of this post. Some of the questions we will answer in this post are the following: What does it mean for an integer to be divisible by another integer? Is zero divisible by every integer? Are there any integers that are divisible by zero? This is a pretty technical article with some fairly sophisticated mathematics. We say that 42 is divisible by 7 because 42 = 7 · 6. The number “6” itself isn’t particularly important here, but what is important is that 6 is an integer. In other words, 42 is divisible by 7 because there is an integer k such that 42 = 7k. In general, an integer n is divisible by an integer d if there is another integer k such that n = dk. In practice we can check if n is divisible by d simply by dividing n by d in our calculator (or sometimes by using the divisibility tricks I mention in this video: Divisibility Tricks). If the answer is an integer, then n is divisible by d. If the answer is not an integer (it contains digits after the decimal point), then n is not divisible by d. For example, when we divide 42 by 7 we get 6, an integer. So 42 is divisible by 7. As another example, every integer n is divisible by 1. This is true because n = 1 · n, and n is an integer. What about the integer 0? Is 0 divisible by 7? Yes it is because 0 = 7 · 0, and 0 is an integer. In fact, if m is any nonzero integer, then 0 is divisible by m. This is true because 0 = m · 0. This fact is worth emphasizing since students often get confused by it. Zero is divisible by every nonzero integer! Note in the bold statement above the word “nonzero.” Isn’t 0 also divisible by 0 according to that definition? After all, if we set m = 0 in the equation “0 = m · 0,” we then get 0 = 0 · 0 which is a true statement! We will talk about this more in just a bit. Let’s now check if there are any integers n which are divisible by 0. If n is divisible by 0, then there is an integer k such that n = 0 · k. However, we have 0 · k = 0. So, if n is divisible by 0, it follows that n must be equal to 0. In other words, there are no nonzero integers that are divisible by 0! Note that we have not shown that 0 is divisible by 0. We only proved the conditional statement “if an integer is divisible by 0, then that integer must be 0.” We still need to check if 0 actually is divisible by 0. So let’s assume for a moment that 0 is divisible by 0. This means that there is an integer k such that 0 = 0 · k. But every single integer satisfies this equation. Just to pick two specific values, 0 and 1 both satisfy this equation. This means that 0/0 = 0 and 0/0 = 1. It follows that 0 = 1. But 0 and 1 are different integers. It follows that 0 cannot be divisible by 0. Thinking more deeply like we have done here can help raise your level of mathematical maturity. The question of why division by zero is not possible is one of the big holes students have in their understanding, so I hope that after reading this post you are now a little wiser! If you enjoy thinking about why things are true or you want to begin learning some more advanced mathematics, you may want to check out either Pure Mathematics for Beginners or Set Theory for Beginners. These books are perfect for anyone just starting out in theoretical math. Comments comments