Why Can't You Divide by Zero?

Enter any number into a calculator and divide by zero. The result will always be undefined. Why in the name of Turing's testicles does this happen?

And yet in the spirit of scientific enquiry, let's try to divide by zero without a calculator and see what happens.

Dividing Beans by Zero

Last night I divided 10 baked beans between 10 dinner party guests. Everyone secured 1 bean, while I secured my reputation as a crap host.

Naturally, I pushed back at the ungrateful swines. "You each have one whole bean!" I cried, sounding like my father. "There are starving children in the world who have no beans at all!"

Imagine if I hadn't shared my beans between anyone. That would have really killed the mood. And it's the same as dividing by zero.

Is there any point asking how many beans will zero people get? I mean, the answer has nothing to do with division, and everything to do with how many people aren't involved in not getting beans, which is just silly.

But for the sake of argument, let's pretend the question is a valid one. Maybe you think the answer is zero, because 10 beans ÷ 0 people = 0 beans each. There is a certain logic to that.

But we can easily break this logic. What if I had 20 beans to divide? Then 20 beans ÷ 0 people = 0 beans each. This new scenario gives the exact same result, which is an erroneous proof that 10 = 20.

You might ask: how many times can I give away zero beans? Frankly, I could do that all day, every day. I'm doing it right now, at a rate of 0 bps (beans per second, you understand).

So the answer can't be zero.

Maybe you think the answer is 10, because I had 10 guests. So 10 beans ÷ 0 people = 10 beans each. In real terms, this is stating that zero people have 10 beans each. Which is pretty non-sensical.

What's more, this logic means that zero people can also have 20 beans each. Once again, we fall into the trap of 10 = 20.

So you throw your hands up in despair and offer infinity. Because there are zero people in the universe who have a hypothetically infinite number of beans.

And you know what? This just leads us into the same trap as before. Zero people have 10 beans, or 20 beans, or infinite beans, implying that 10 = 20 = ∞.

So what the heck is ten divided by zero? Let's ditch the bean counting and try a different approach.

Sneaking Up on Division by Zero

When we can't find a direct answer to a problem, we can try looking at nearby problems in search of indirect clues.

In this case, let's divide 10 by smaller and smaller numbers approaching zero. Where does this take us?

Okay, these numbers are getting big. And as our denominator approaches zero, the solution appears to approach infinity.

That's it! Dividing by zero equals infinity, right?

Not so fast! A new problem emerges if we approach division by zero from the negative direction.

As we sneak up on zero from negative numbers, it appears that dividing by zero equals negative infinity.

Let's plot these confounding results on a graph.

Damn. We're caught in a mathematical riptide, where we're going to be washed up in two opposite directions simultaneously: positive infinity and negative infinity. How is this possible?

Coping with Infinity

Infinity is an uncountable amount. It's not a real number. It's more like a beam of light that shines on and on for eternity.

Infinity is bigger than any number you can think of, because whatever string of numbers you come up with, you can always simply add 1 more.

Visually, we can place both positive infinity and negative infinity on a number line, but it creates the illusion that infinity occurs at some specific end point. Which it doesn't.

Fortunately, mathematicians have another perspective of infinity—one that sits on a number wheel.

Not only does this help us visualise infinity as continuous, but it solves our problem of positive and negative infinity being diametrically opposed. Now these endless opposites are connected in a circle.

Just as zero evades both positive and negative territory, so too does unsigned infinity. What's more, it sits directly opposite zero, allowing us to anchor them in symmetry.

On a number wheel, zero (nothing) is the reciprocal of infinity (everything).

Here's another interesting thing about the number wheel. We can state an expression like 1 ÷ 2 = 0.5, and see that 0.5 sits directly under 2 on the wheel. Likewise, with 1 ÷ 3 =0.33, the reciprocal relationships are reflected on the wheel.

So what happens when we divide by zero using the number wheel? According to the vertical axis, the answer is unsigned infinity.

By extension, we might conclude that anything divided by zero is unsigned infinity. So why doesn't our calculator give us this answer? Why does it insist on using the term undefined?

Infinity is Special

Despite what we just saw on the number wheel, the conclusion that 1 ÷ 0 = ∞ is actually in breach of mathematical rules.

This becomes clear when we look at the multiplicative inverse of our division. For instance, if 12 ÷ 3 = 4 then it's also true that 3 × 4 = 12. We're just flipping the equation.

So what happens when we calculate the inverse of our working hypothesis 1 ÷ 0 = ∞? We get 0 × ∞ = 1.

Can zero times infinity equal one? Hell no. For a start, anything multiplied by zero is zero. And anything multiplied by infinity is infinity. Multiply the two together and you have a real problem.

The issue is in the way we handle zero and infinity. Both of these concepts demand special treatment due to their unusual properties.

For instance, both zero and infinity absorb other numbers by multiplication. Consider that 10 × 0 = 0. We just lost the 10 because it was absorbed by the 0. Likewise, infinity also absorbs other numbers: 10 × ∞ = ∞.

So we have to drop our conclusion from the number wheel. While zero and infinity are reciprocals of each other, they're not truly inverse because infinity isn't a real number. This actually precludes us from doing arithmetic with infinity.

Indeed, zero and infinity can lead us into the realm of indeterminate expressions like 0 ÷ 0 and ∞ ÷ ∞. The more you poke around, the more you find that different indeterminate expressions can produce the same answer, and you can end up with broken proofs like 1 = 2.

This is the problem with dividing by zero. It's not that maths is wrong or insufficient. It's just that certain concepts inherently succumb to special rules. And it's why your genius calculator—a veritable pocket deity in all things mathematical—is fundamentally rulebound to show that anything divided by zero is undefined.

Rebecca Casale is a science blogger based in Auckland. If you like her content, please share it with your friends. If you don't like it, why not punish your enemies by sharing it with them?

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