Dividing by Zero in Five Levels -- Elementary to Math Major

Why can't we divide by 0? This is a common issue we encounter in elementary school when we learn to divide, but it persists through high school and even into math major courses. Let's explore why we can't (usually) divide by 0 in 5 levels, covering a range of these topics.

[simonharris]

Level 1: Don't do it
Level 2: Still don't do it
Level 3: Just don't do it
Level 4: Never ever do it.
Level 5: Go on then, do it. Look! See! You broke something.

[lellyparker]

*Differential Calculus* - _the art of how to sneak up on a divide by zero without triggering the alarm system_

[brghtley_]

My Calc 1&2 Prof once said that multiplying with zero makes everything zero, so it makes all information about a system disappear into nothing. In reverse if we could divide by zero and get a usefull answer, we could extract information about a system without any information.

I think thats a very nice philosophic approach to why it doesnt work.

[frankdaniel]

“How do you divide 6 cookies with zero friends”
That one definitely has to hurt

[insising]

What I love about the abstract algebra explanation is that, without the jargon "field", "group", etc, the explanation boils back down to a pre-calculus level of mathematics. This is the first insight for understanding that mathematicians don't use new words to talk amongst themselves as the elite, but instead define precise notions and eventually clean up what they've learned and teach it to others at a more basic level.

[soyoltoi]

Level 6: Riemann sphere
Level 7: Wheel theory
Level 8: ???

[stefansiqueland]

I went from watching basketball highlights to this

[ProactiveYellow]

I love how all these explanations are really just more careful and rigorous ways to explain that intuition we had all the way at level 1.

[matthiasbergner]

Here is level 6, graduate algebra course: You actually can divide by zero within the ring that contains only one element, because there zero is equal to one.

[Qermaq]

My go-to for explaining it is #3. Since division is just undoing multiplication, 0x can only equal 0 so if 0x = y where y is not 0 it's an impossibility. And if 0x = 0, then x could be any value at all, so it's still undefined as to what x is.

[nosuchthing]

Level 0: model division as repeated subtraction.

If a pie has 10 pieces and you keep taking away 2 pieces, how long can you do this? 5 times. Or, 10 / 2 = 5.

So if you keep taking away zero pieces, how long can you do this? You can do this operation forever.

Or 10/0 = Infinity.

This has a more intuitive feel, because if you 'never take anything away' the operation will 'never finish'

[gungy_vt]

"How do you split 6 cookies evenly among 0 friends?"
This doesn't work for explaining to an elementary student because the answer is obviously "I eat them myself".

[jefflangley]

Sean taught one of my math classes during my freshman year of college. By far one of the BEST teachers I’ve ever had and helped me enjoy math, which is a subject I usually struggle with.

[asparkdeity]

I just wanted to say, division by zero does indeed exist the null ring if anybody is interested :)

Also in the extended complex plane, the positive and negative infinities map to a singular infinity on the North Pole of the Riemann sphere, and so there dividing by 0 will give infinity, which is particularly useful in Möbius Maps :))

Also, wheel algebra defines an element called the nullity element which is kind of like a ‘void’. In this case, we define any indeterminate form (a/0, 0/0, 0^0 etc…) to be equal to the nullity (¥ say), which u can think of as being “more powerful” than 0 and infinity combined. Any operation with ¥ results in ¥, e.g. 1/0 = ¥, 0/¥ = ¥ etc… :))))

[diemonder]

3:54 to be fair, the cookie-friend analogy breaks down much earlier than that. 6/0.5 for example. how do you divide 6 cookies with half of a friend? by giving each friend 12 cookies? it doesn’t make any sense either, yet as we all know, 6/0.5=12 is very well established.

[guillaume]

Another interesting approach (for lower abstraction levels) is viewing division as repeated subtraction.

How do you share 6 cookies among 3 friends ?

You give one to each friend, leaving you with 6 - 3 cookies.

You give a second one to each friend, leaving you with 6 - 3 - 3 = 0 cookies.

The number of cookies each person has is equivalent to the number of times we substracted 3 from 6 before getting to zero, which is in this case 2.

In that sense, if we want to know what 6 divided by zero is, we would have to subtract 0 repeatedly from 6 until we get to 0.

This is where we see an issue : subtracting zero doesn't change anything, meaning the process never terminates.

This could either indicate that dividing by 0 is undefined, since the process doesn't end, or it could indicate that the answer is infinity in a certain non rigorous sense.

This gives the intuition that there is something to do with infinity without needing to introduce calculus or limits.

[MerderMarderInMyHead]

Him merely mentioning “friends” in a dividing by 0 video made me anxious and nervous. for it was true.

[ezekielanderson]

When I learned economics at community college, that’s when I finally understood why you can’t divide by zero.

So there’s a concept in economics called elasticity of demand. Basically it means how much the quantity-demanded of a good or service changes in response to a price change. For example, when Netflix raises its prices, people cancel their subscriptions because Netflix is not a necessity. When the price went up, quantity-demanded drops by a lot. So in economics terms, Netflix has a high elasticity. On the flip side, if something like water goes up in price, the quantity-demanded doesn’t go down much, because people need water to live. In economics terms, it has a low elasticity, or that it’s inelastic.

After learning all that, I thought about what would happen if something were perfectly inelastic. When you graph it, the line would go straight up and down. It would have a slope of x/0. This would mean no matter how high the price got, the quantity demanded would not change.

Then I thought, what kinds of things behave like this? Stuff like food and water. No matter how expensive food and water get, people still need it. But what happens when it gets to expensive that no one can afford it? The people starve.

So in economics, when you divide by zero, people starve. Kinda morbid, but that’s how I understood division by zero.

[jasoncrobar]

My preferred explanation is practical calculus. Look at speed. It's the rate of change of distance over time. It's a derivative. If distance is non-zero, but time is zero, speed must be infinite. The only way to be in 2 places separated by a distance, at the same time, is to be moving infinitely fast. By the same token, distance over speed equals time. 100 km divided by 50 km/h equals 2 h. So if you want to travel 1 km, at a speed of 0 km/h, how long does it take to cover the distance? Infinite time. Now, infinity can't be defined as a number, which is why whether you say undefined or infinity, both fit.

[Fallkhar]

After taking some abstract algebra and analysis the way I see it is that for most sets of numbers defining division by zero is impossible without losing some structure in the process which leads you to now not being able to do some other things. You cannot divide your cake by zero and eat it too.

Defining things is like signing a contract. You promise to follow some rules for something and it turns out, defining zero often isn't worth it.