Why is division by zero impossible?

I’m no mathematician, but I would have to ask, logically, how anything could be divided into no parts at all.

Imagine a bowl of jelly beans; or simply imagine two of them. Can you put nothing into them?

I stretch out my hand to reveal nothing, now you take half of the nothingness and I’ll take half of the nothingness. How in the world would that be possible? Zero of something basically means it doesn’t exist so how do you divide something that doesn’t exist.

@Jeruba: One of the fascinating aspects of integral calculus is dividing something (the area under an irregular curve, perhaps, into smaller and smaller rectangles and adding them together to get the area.

As you make the number of rectangles approach infinity (which they never reach) you can calculate the area. Each rectangle is almost not there.

http://www.intmath.com/Integration/3_Area-under-curve.php

@Jeruba That may be the simple explanation I’ve been seeking.

@gailcalled I can picture putting nothing into the jellybeans, basically just leaving them alone.

@earthduzt Where I get hung up is half of nothing should just remain nothing. To me it seems the answer should be one.

I can’t top @Jeruba. But I would add that the reason we rule out letting computers try to divide by zero is that it takes an infinitely long time and never produces a result. You can keep putting nothing into something forever and not fill it up.

I was just thinking of the old pies from third grade, like the ones at the bottom here. You divide by 4 and you have one pie cut into quarters. Divide by 6 and you have sixths, by 2 and you have halves. And so on. Divide by one (the whole pie is one part) and you don’t cut it at all; you’ve got one single undivided pie. But try to divide by zero—separating the pie into no parts—and your brain explodes. That’s why you don’t do it.

Take the limit of 1/x as x approaches 0. That is, plug increasingly tiny numbers in for x. The output gets larger and larger. Extrapolating, you could say that a number divided by 0 is infinity, a concept that does not exist practically.

Division is the opposite of multiplication.

If we say “6/2 = 3” that is the same is saying “3 * 2 = 6.”

Now lets say we want to divide by 0.

“6/0 = x” I don’t know what x is, but my claim is that its a numbers.

But… x * 0 != 6. No matter what number x is, x * 0 = 0.

Division in the sense relevant to your question is a mathematical operation, not a physical activity. To divide a number y by a number x is to find a number z such that x•z = y. The crucial case is that in which y is 1; given x, a number z such that x•z = 1 is the multiplicative inverse of x. Division by x is the same as multiplication by the multiplicative inverse of x. If x•z = 1, then y/x = z•y, since x•(z•y) = (x•z)•y = 1•y = y. Since 0•x = 0 for each x, 0 has no multiplicative inverse—that’s why division by 0 is undefined.

Whoops, sorry, also if you want to say “1/0 = infinity” you run into some really neat issues. I.e. there is more than one kind of infinity. Some of them are bigger than others. I can’t find a good reference that explains it clearly, but if you want to see it, you can look up the work by Georg Cantor. It takes a while to wrap one’s head around it, but its fun stuff.

@jazmina88 If you divide by 1 you get the whole.

“Divide by 2” means take a pie, and split into two parts so that each of your kids can eat half.
“Divide by 1” means take a pie, and give it to someone so that your one kid can eat the whole thing.
In each case, the whole pie gets eaten.
“Divide by 0” means take a pie, and no one gets to eat it, but the whole pie gets eaten. Its just not possible.

@Jeruba Divide [a pie] by one (the whole pie is one part) and you don’t cut it at all; you’ve got one single undivided pie. But try to divide by zero—separating the pie into no parts…

Best explanation ever.

@earthduzt Actually, what you described is 0/2, which equals 0.

I agree with the pie explanation. If there’s no one to claim the pie, then everything stops.

@cockswain asked me to take a crack at this. My answer is the same as @ivan’s answer.
Basically x/0 = infinity.

@Rarebear So if 1/0 is infinity, are you saying that infinity * 0 = 1? Or does infinity * 0 = 2?

Urg!

Infinity is not a number :) You can define it in many ways that fit with our intuitions. In fact, mathematicians do use something called the “extended reals” which is all the real numbers and positive infinity and negative infinity. And there are rules for operations in this set, but there are other ways to talk about infinity. (Sorry, infinity is one of my favorite topics so I tend to go off on it).

@roundsquare Maybe infinity * 0 = infinity? Or is it more reasonable to say something undefined * 0 = something undefined?

@roundsquare Exactly. That’s why you can’t divide by zero, because it makes no sense. I was just piggybacking on @Ivan‘s comment that by dividing by an extremely small number you get an extremely large one. For example 1/10^-10 = 10^10. If you go even smaller, say, 1/10^-100 then you get 10^100 and so on. The smaller the denominator the bigger the number—ultimately you get “infinity” if you divide by zero. But as you so aptly point out, that’s nonsensical, so there’s your answer!

“Please understand my mathematics background is currently as far as an A in college trig.” <== The most important insight I gained from this question is that colleges are handing out A’s in mathematics for memorizing rules to people who don’t understand addition, subtraction, multiplication and division.

PS: If you want to better understand how to reasonably handle such conditions as divide by zero please look up NaN.

@cockswain This is where you can have some fun actually. Its all a matter of definitions. Any definition works as long as the rules don’t create contradictions.

I’ve forgotten the details actually, but I think its usually defined as:
if x > 0, inf * x = inf
if x = 0, inf * x = 0
if x < 0, inf * x = neg inf

And it gets swapped if you do neg inf * x

This makes sense if you view “x * y” as “get x bags of y jelly beans each.” Even if you an an unlimited number of empty bags, you ain’t got no jelly beans. Also, if you don’t have any of these infinitely large jelly beans, you still got nothing. (Feel free to replace jelly beans with anything you want).

But of course, having two bags of infinite jelly beans is the same as having one such bag. You’ll never finish one, so the second doesn’t help. Hence:
inf * 1 = inf * 2

Or

inf / 2 = inf

In most of these systems, you can’t divide by 0. Or, to put it mathematically “zero has no multiplicative inverse.” (The multiplicative inverse is just the reciprocal in this case, but in other systems it can be something else, like the inverse of a matrix).

Now here’s the big question. And to this, I don’t know the answer. I can work it out to do anything I want actually:

What is inf + neg inf?

I can get this to come out to inf, any number, or neg inf. Well, sort of… now we need to get more specific about what infinity means.

This brings up the question, what about alternate universes/ parallel universes? What if you could divide by 0?

@XOIIO The thing is, the “fact” that you can’t divide by zero is not a physical problems, its a logical problem since division is inverse of multiplication.

In theory, one could create a system where you can divide by zero, however, the rules of this system would be so different that the system would not be recognizable as what we’re used to and the fact that you call it zero would be the only similarity between it and what we normally call zero.

Note: @Jeruba‘s explanation does make it seem like a physical problem, but thats only because the physical world is in line with our logical abstractions.

Simply put, the operation has no valid definition. It makes no more sense than squirrel divided by the square root of pizza. Mathematics operates according to rules and all valid operations must conform to a valid and meaningful definition.

Actually you can divide a pie into zero parts. What you end up with is nothing. 0*x=0 for any x. The first use of 0 by Hindu mathematicians was a major accomplishment. If you think about it, it is somewhat problematic philosophically. Zero of anything is the same. 0 pie slices, 0 frogs or 0 cars all amount to the same nothing. You can see how a people whose religion included the concept of nothingness as a major component would have an advantage in being able to work with 0.

The problem comes when you ask how many of those zero parts make up the whole pie. That is where the problem of dividing by zero comes in. If you take the size of the pie as 1, then dividing it into pieces of size ¼ results in 1 divided by ¼ = 4 pieces. As you make the size of each slice a smaller portion of the pie, you end up with numbers of slices that increase without end.

There is no value for infinity – infinity. infinity + infinity = infinity and infinity + any finite number = infinity, so there is no unique inverse.

@LostInParadise By your logic, 1/0 = 0? Or am I reading you wrong?

As for adding and subtracting infinities: Sure, there is no unique answer. But you can work out the logic in a cantor-esque way to get any answer you want. At least, if you fix “infinity” to be a certain kind of infinity (e.g. aleph_0 or aleph_1).

I was a bit too hasty with my answer. The problem with saying that you can not divide a pie into zero parts is that you are limited to working with whole numbers. You can also say that you can’t divide a pie into ⅓ or 1½ parts, but it is still possible to divide by ⅓ and 1½. In order to see the problem of dividing by zero, it is better to look at the size of the slice and then ask how many of them the pie can be divided into. In this case you will be limited to numbers that are less than or equal to 1, but you can approach zero as closely as you want. It then becomes apparent that as the size of the slice gets smaller, the number of them gets larger without limit.

The concept of dividing a pie into parts gives an intuitive feel as to why dividing by 0 is wrong. But mathematically, @ratboy has the best answer. Addition is really one of the only operations that is defined “in a vacuum.” Multiplication is just addition a fixed number of times, and in algebra division is defined as multiplication by the inverse of the divisor. Algebraically speaking, solving 6/0=? is the same as solving ?*0=6. But multiplication is defined such that ?*0=0 regardless of what value we have for ?. So the equation we wrote has no algebraic solution; simply put it doesn’t work. Algebraically speaking, infinity isn’t really a value. So the calculus issue of (infinity)*0 is a question of convergence, not multiplication.

One last comment. When Newton and Leibniz developed calculus, they found it convenient to use infinitesimals, which is what you get when you divide 1 by infinity. Conversely,1 divided by an infinitesimal is infinity. In the nineteenth century, this idea was replaced by the use of limits. Recently extensions of the real numbers, hyperreal numbers and surreal numbers, have been defined that allow for definition of both infinity and infinitesimals. In these systems, it is still not permitted to divide by 0, but division by infinitesimals is permitted. It is possible to develop calculus without using limits by working with hyperreal numbers. This is known as non-standard analysis. I don’t know enough about it to say whether working with hyperreals is more convenient than working with the awkward delta-epsilon definition of limit.

In a hugely simplified sense:

“Dividing” is to take a number and separate it into groups of the number you are dividing by.

ex
12 jellybeans divided by 3, means to make groups of 3 until all the jellybeans are used up. The answer is the number of piles that have been made.
In this case there are 4 piles. 12 divided by 3 is 4.

To divide by zero is different. Because each pile of jellybeans contains “0”, the initial pile of jellybeans will never be used up. The equation has no end point. You can make imaginary piles of zero jellybeans to your heart’s content, but nothing is achieved. The only answer is “infinity” or “undefined” (depending on which of my math teachers you ask).

“Our universe prohibits it” is just babble. So there’s a universe where “zero” actually means “not zero” or “divide” means “something other than divide”? It’s just one of those properties of numbers. True just because it is true.

I thought that technically it is the limit as x approaches 0 of 1/x that trends towards infinity. Is it technically correct to simply say 1/0 = inf? I thought 1/0 is technically undefined.

@lilikoi You are right. 1/x = undefined; lim 1/x where x->0 = infinite.

Thanks to everyone except that eternal prick @malevolentbutticklish for answering this question. In hindsight, it seems very simple but it was never adequately explained in 3rd grade and I just went on applying the rule unthinkingly. @noodlehead710 ‘s answer also cleared up my other thought “why does multiplying by zero = zero” as well. Adding any number zero number of times will be zero.

Thanks again! Fluth on!

@cockswain Granted @malevolentbutticklish stated his answer in his usual gruff, badgering tone. But he was quite correct, NaN actually is a correct answer to 1/0 and looking at the reference to it is worthwhile if you are interested in such things.

@cockswain Hey it was a great question. I’ve taken 2 years of college calculus and partial differential equations and have even read Zero: The Biography of a Dangerous Idea, and I still wouldn’t be able to explain it to you.

@bob_ thanks for clarifying.

because zero is more ambiguous than integers. The semantics of zero is a wild goose chase. Profoundly contradictory.

@mammal: Chasing geese generated the synonym “goose egg” for zero. And @Coloma, our resident geese herder, would not like your pejorative description of what may become a summer Olympic sport. You have been warned.

@mammal Not contradictory, just confusing, until you get used to it.

Sorry I’m late, but I have thought of this a lot myself and am fond of an intuitive (but inconsistent and therefor wrong) system that uses something like the IEEE approach: there is Infinity, -Infinity and NaN (Not a Number) (there’s also -0 which is obviously the reciprocal of -Infinity, I think it’s funny)

As far as the algebra, x/0 (where x is not 0) = Both positive and negative infinity, just like the square root of 4 is both 2 and -2 (look at the limit of 1/x taken from Each side),

This implies that in my system, (+/-)infinity*0=x, where x is All Numbers (Every real, complex, 0, infinite, etc. number is a solution). Since 1/(either)infinity = 0, this also means 0/0=x (Again, All Numbers) which is also the same as inf/inf.
The IEEE assigns such results the value NaN, but it says 1/0=Positive infinity and -1/0=Negative infinity (I say it’s wrong there, both are both values)

The inconsistency with this system is that it’s prone to things like This – But it’s still fun to play with.

What I really admire is the group of people here who can type the equations and symbols so that they are exact and comprehensible. No one ever bitches about the math KGB, I notice.

@davidgro So infinity = neg infinity?

Nope, they are separate values. For example, log(0)=-infinty (not positive infinity) but both values are solutions to 1/0=x (or -3/0=x, etc.) the same way that x^2=9 has two solutions (3 and -3) (Again, this is just the way I think of it.)

In some systems, such as a Riemann sphere all infinities are the same thing. I understand the basic idea of that one, but the math behind it is well beyond what I’ve studied. If I remember right, on the sphere the south pole is 0, the north pole is infinity, and various hemispheres are positive/negative of real/imaginary, the equator has 1, -1, i and -i (and values in between)

I do not see the benefit of giving two values to 1/0. The graph of 1/x is a continuous function for all non-zero x. I think it is better to say that if you approach 0 from the negative side that 1/x goes to negative infinity and as you approach from the positive end it goes to plus infinity and at 0 it is undefined.

@LostInParadise I think that’s the ‘official’ answer.
As for benefit, I simply like to avoid the word undefined (or indeterminate) and give it some conceptual meaning. It’s meant to be interesting instead of correct ;-)