Science & Technology

Why Can't You Divide by Zero?

Even if it's been decades since you've thought about math, you definitely remember this rule: You can't divide by zero. It's impossible. But did you ever stop to wonder why it's so impossible? After all, if I divide a pizza by nothing, I still get a pizza. It turns out that when you divide a number by zero, you do get an answer — it's just not a very useful answer.

Breaking All the Rules

To think about what you might get if you divided, say, 10 by zero, let's start by dividing 10 by 5. The answer to that is 2. What if you divided 10 by a smaller number, like 2? You'd get a bigger number: 5. What about 10 divided by 1? A bigger number still: 10. 10 divided by ½ is 20, divided by ¼ is 40, divided by 1/32 is 740.740 repeating. Every time you divide by a smaller number, you get a bigger number in return. That is, the closer your divisor gets to 0, the closer your answer gets to infinity. So if you were actually to divide 10 by 0, you'd get infinity, right?

Not exactly. Why? Well, let's see how that plays out.

10 ÷ 0 = ∞

If the above statement is true, then this statement should also be true:

10 = ∞ x 0

But we know that anything multiplied by zero is zero, which would mean 10 = 0. That's not true in the mathematics we know, so there must be something wrong with the equation we started with — 10 divided by zero must not be infinity.

But here's another problem. Remember how the smaller the number you divide by, the larger the answer you get? Well, what if instead of dividing 10 by 5, you divided 10 by -5? Then you'd get -2. Dividing 10 by -2 gets you -5, and dividing 10 by -1 gets you -10. Dividing by negative numbers that trend toward zero gets you negative numbers that trend toward negative infinity. So if you really wanted to say that 10 divided by 0 was infinity (which we've already shown you can't), you'd have to say that it was also negative infinity. An equation that equals out to both negative and positive infinity isn't much use to anyone.

That means that the answer to a number divided by zero is "undefined." There's just no value assigned to it. It isn't a thing. The answer to "What's 10 divided by zero?" is the same as "what's the sound of one hand clapping?" and "what is the universe expanding into?" The question doesn't make sense, so the answer doesn't contain any real information. It's undefined.

Put a Sphere on It

But hold on — if mathematicians just left it there, they wouldn't be worth all of their fancy degrees. It turns out that there is a way to divide by zero; you just have to get into some complex numbers to do it.

Imagine, if you will, a two-dimensional plane that goes off into infinity in all directions with zero smack-dab in the middle. Now imagine that you curved that plane into a sphere, with the zero as the South Pole and the edges joined at the top where the North Pole would be. Never mind that that should be impossible — we just did it, and now infinity is the North Pole. Ta da! Mathematics!

Now, take another infinite two-dimensional plane and slice it through your sphere at the equator. Any point you choose on that plane can connect to the sphere's North Pole via a straight line. If the point you choose is outside of the sphere, the connecting line will intersect the sphere in the Northern Hemisphere; if it's inside the sphere, it'll intersect at the Southern Hemisphere.

What you've imagined is a Riemann Sphere, and this way of associating every point on the plane to an intersection point on the sphere is called a stereographic projection. Basically, any point you can find on the plane, you can find on the sphere. That includes infinity. The closer you get to infinity on the plane, the closer you get to the North Pole of the sphere.

Here's the cool part: Any number on the plane that intersects at a given point on the sphere has an inverse (that is, 1 divided by your number) that intersects at the same point on the opposite hemisphere (for example, 30 degrees North versus 30 degrees South). If your point hit right at zero, the inverse would have to be infinity. That is, on a Riemann sphere, dividing by zero is the same thing as multiplying by infinity. You can't normally say that dividing a number by zero equals infinity, but on the Riemann sphere, you can.

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For a fun guide to tough mathematics, check out "Math with Bad Drawings: Illuminating the Ideas That Shape Our Reality" by Ben Orlin. We handpick reading recommendations we think you may like. If you choose to make a purchase, Curiosity will get a share of the sale.

Written by Ashley Hamer April 8, 2019

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