%% Seriously, don't add anything more to this. It won't help. There used to be so much natter here it was unacceptable even for an IJBM page.

* Why is it impossible to divide by 0? It's possible to ''multiply'' by 0 (with the result always being 0), and dividing is the exact opposite of multiplication, so why isn't any number/0 always 0? Even the explanation for why this is doesn't make sense to me: the dividend is the number of an item in a single group, and the divisor is the number of groups you have to break the large group of the dividend into; this is why dividing (or multiplying) by 1 will always equal to the number you're dividing (or multiplying) by 1 - the "group" represented by the divisor already contains the number of items represented by the dividend. However, it's impossible to break up the group represented by the dividend into 0 groups, so any number divided by 0 will always be undefined. But that makes no sense when you consider its opposite - multiplying by 0. With multiplication, you're combining multiple groups with the same number of items within, the number of groups being the multiplier, and the number of items within each group being the multiplicand. Again, any number times 1 will always be the first number, since there's only 1 group of that many items in the equation. However, multiplying by 0 means you're combining 0 groups of a specific number of items...which is equally impossible. Yet the answer to any multiplication by 0 is always an unambiguously defined 0. Why? In both cases, you're trying to come up with 0 groups of a specific number of items - a mathematical impossibility, since if you know how many items will be in the group, you ''cannot'' have 0 groups of that many items, you'll always have at least 1 group. So, either the answer to dividing by 0 is 0, or the answer to multiplying by 0 is an undefined number.
**  Think of it this way:  If you divide something by three, you divide into three parts, right? 12 things divided into 3 parts is 4. Into 2 parts is 6. Into 1 part is 12.  Into 0 parts...  Well you can't split anything into 0 parts.  Try cutting a piece of paper into 0 parts; you can't do it.
** For those of us who don't want to go through all this, tl;dr version ahead of time: Division is essentially saying "x/y=z because z*y=x. But, if y is zero, then because of the rules of math (anything multiplied by zero is zero) x has to equal zero. For example: 5/0=?. To solve this I say 0*?=5. But that's not possible, since  0 multiplied by anything, ever, is 0.
** A new and cleaner attempt to explain this. Zero is something people have had troubles with as a concept for a long time because the concept just bugs people. Zero started as just a book keeping notation. You were counting up the usual things you had, say fruit. On any given day while counting fruit you put a mark thru where you had nothing. All other math with zero comes from what happens when there's nothing at all. Broken up for readbility, and focusing on the history with an attempt to use concrete answers.
*** Multiplication of zero comes from adding up sums and trying to figure out profits. Adding up the number of each fruit sold and then determining how much profit was made for each one wasn't needed when you didn't sell any of it. So the mark for nothing was still nothing, a mark to literally skip on and don't pay attention. People began using the same mark, and that mark for nothing is the modern zero. Where more math comes is more complicated problems, where the amount lost and the amount gained in a deal weren't even. People lost money or gained money or were even. Negative numbers were created as just short hand for the losses involved. Zero, used before as a disregard mark was used when losses and gains were equal, because again you could ignore the amount since there was nothing there.
****Overtime folks realized how to use this math and formalized it, creating negative numbers and Zero concepts, as it was a useful technique, even if much of the times it was a matter of language, because to some people it was a quicker and easier principle to grasp. Much of algebra is just based on advance book keeping to determine problems with money. A good example of it is the profit splitting question. 2 guys want to split $10,000 from selling a car, but they owe 10,000 on the car already. Their profit is (10,000-10,000)/2, which is zero because there is no profit. They will pay no taxes on profit, they will be able to buy no shoes from their profits. Because there was nothing, there is still nothing. Zero just formalizes this concept and is useful when two items cancel each other out, like +1 and -1 which cancel out.
*** Division of Zero gets more complication. Once you except that there can be number that means 'nothing' then math questions pop up. Say you had a $100 profit that 5 guys earned and everyone wants an equal share. They kill each other off to get a bigger share, so 5 would each get 20, 4 get 25, 3 get 33.33, 2 get 50 and the last one gets all $100. But if they all kill each other, no one can collect. Since no one can collect it that money isn't split, the shares  are meaningless since there is no way to split it. But that isn't the same as nothing, as that $100 still exists, its just the way to split it up has becoming meaningless. Its not that no one knows the answer, its the answer has no meaning. There are more rigorous attempts to explain it that need more a math background before.
** Part of the (admittedly widespread) problem is that there is a semantic problem with anything that is undefined: it's actually just hard to explain to high school (and, sadly, college) students what we (math teachers) mean when we say, "Dividing by zero is not defined."  As mathematicians, this seems like a completely reasonable thing to say; however, it is often heard as, "You can't do that."  Of COURSE you can divide by zero; however, the answer is undefined, and, consequently, pretty useless.  It can't interact with anything else in the realm of mathematics ((7/0) * 2 = ... well, it doesn't equal anything, since one of the inputs is undefined).  Think about it this way: dividing by zero is very much akin to taking the square root of Walrus Dream Butter.  You can argue that it's possible all you like, but within the strictures of mathematics, it just doesn't mean anything.
** It seems to me like your problem is more with multiplying by zero, not dividing.  That's pretty straightforward to explain by example: Let's say you have 50 groups each containing 5 items, and you take away 5 items; that's the same as having 49 groups of 5 items.  Looking at it the other way around, 49 groups of 5 is the same as 50 groups of 5, minus 5.  0 groups of 5 would be the same as 1 group of 5, minus 5, and 5-5=0.  Expressed algebraically, it's a function of the distributive property: (n-1)x=nx-x, so (1-1)x=1x-x.  1-1=0 and 1x=x, so 0x=x-x=0.
*** ...that makes no sense. Again, how do you know that the group has 5 items if the group doesn't exist? Saying it's the same as "1 group of 5, but minus the 5 items" is nonsensical, since that's assuming that, if the group ''did'' exist, it would have 5 items in it - an assumption ''you cannot prove''. Since there's no way of proving how many items are in a nonexistent group (or, the inverse, proving how many groups contain a nonexistent number of items), multiplying by 0 should ''not'' give a definitive answer of 0; it should be "undefined" like dividing by 0 does.
**** Using generic terminology, 0 groups of any quantity will have a total of 0 elements, it doesn't matter how many elements such groups should have. Inversing the operands, any number of groups of 0 elemnts will yield a total of 0 elements, it doesn't matter how many groups you have. What's so troublesome to grasp in that idea?
****You, sir, are the nonsensical party here.  You don't need to prove what quantity the groups contain, since you're the one defining them.  It is perfectly valid for me to say "I have zero sheds with five unicorns each in them" because... well... I don't.
** [[{{Zeful}} This Troper]] has a similar problem with multiplication, but chooses to make the answer make sense. Multiplication is like addition, the same backwards as forwards. 2+5=5+2 and 5*2=2*5. So instead of combining 0 groups of five things (which creates existential issues), This troper combines 5 groups of 0 things, making the problem disappear.
*** An extension of the above (but from a different troper): These forms of number-bending (I don't know the real term) are opposites. Just as 5+4=9, 9-4=5. This will always be true, at least in normal mathematics. Similarly, 5*2=10, so it is both logical and correct to conclude that 10/2=5. When you multiply by zero, the answer is ''always'' zero. That's why 2*0=0 and 0/2=0; n*0=0, and 0/n=0. But if you say n/0=0, then it would follow that 0*0=n. Since this is impossible, dividing by zero is also impossible.
****...or well, undefined. It's perfectly possible for 0*0 to equal n; it just doesn't mean anything since n is all possible numbers.
** As for dividing by zero: Assume for the moment that you're allowed to divide by zero.  That means that there exists an answer for 5/0=n.  Since division is the inverse operation of multiplication, that's the same as saying 0n=5.  We know zero times any number is zero, so n can't be any number.  Logical contradiction; therefore you're not allowed to divide by zero.  (That's just one of MANY arguments by the way; you can also use division by zero to prove 1=2.  Also we're specifically talking about the reals here.)
*** See above; how can you prove that a nonexistent group has a set number of items (or how many groups contain a nonexistent number of items)? You can't, not without making an AssPull of an assumption; in other words, if you don't have at least 1 group of 1 or more items in it as a basis, you don't know and cannot prove what either number would be if the other was rendered nonexistent, and if you ''have'' that one group...then you ''cannot'' have 0 of either, can you? So, if dividing by 0 has no answer, then neither should multiplying by 0.
** The problem with your reasoning is that it is possible to have the number of items in a group be defined even if no such groups exist; I can know that each tub of ice cream contains three-eights of a gallon even if I don't have any tubs of ice cream and therefore have zero gallons of ice cream. On the other hand, if I say that I have five hundred kids, and I don't have any buses, and I want to know how many kids to put on each bus, the only reasonable response is "but there aren't any buses."
*** I'm sorry, but that logic is faulty to me; how do you know how much a tub of ice cream contains if you ''didn't have an ice cream tub at one point?'' You can't know; if you never ''had'' any tubs of ice cream, then you can't possibly know how much a tub contains. If you know how much a tub of ice cream contains, then one ''had'' to have existed, at one point; otherwise, you'd just be pulling a number out your ass, which is meaningless. Or, to put it another way, say I'm dealing with candy and kids; it's impossible for me to distribute candy equally to the kids, if either the kids or the candy doesn't exist, just as it's impossible for me to collect candy from the kids in that same scenario - either way, I come up with a "meaningless" answer. If I know how many kids I would give/receive how much candy to/from, then both the candy ''and'' the kids had to have existed at some point in the equation for me to know those numbers, even if there's just one piece of candy and one kid. So multiplying by 0 should be just as nonsensical and give just as "undefined" an answer as dividing by 0, but it doesn't, for some stupid reason.
**** But the thing is that tubs of ice cream exist ''even if I don't have any''. I can go to the grocery store and look at how big their tubs of ice cream are, I may have eaten a tub of ice cream in one sitting last week, I can Google it or look it up on Wikipedia or any of a million ways to discover this bit of information, and it still doesn't change the fact that if I go open my freezer right now I'm not going to find any tubs of ice cream there. I'm not "pulling a number out of my ass", the number already ''exists''. Or, to use a different example, if I have five empty bottles of Coke, each bottle contains 0L of Coke. Put together, all five of the bottles contain 0L of Coke. Where is the ass-pulled number there? Is it maybe the five? Because I have the five bottles right here, I can show them to you. Or perhaps the zero? But they're all empty, obviously they don't have any Coke in them, which is exactly the same as having 0L of Coke.
***** No, the numbers do ''not'' exist, at all. They only exist if you ''physically have them in your position'', otherwise you're ''just making stuff up''. For example, you say you have five empty bottles, which equates to 5x0L. What is the volume of those bottles? Are they 20 ounces? 1 Liters? 2 Liters? 3? You can't know unless you have the bottles physically with you; in other words, unless you are given how ''many'' bottles you have, and how much is ''supposed'' to be in them, you won't know ''how many of those empty bottles you have''. And, if you already have that data, ''then there is no way you can have zero of them!''. And besides that, even if you ''do'' have a certain number of empty bottles, why bother with them? There's no liquid in them to distribute or collect, so it would be pointless to do either. Imagine if I've got a keg that's empty, and my 10 friends don't have any beer to fill it; I can't distribute the contents of the keg to them, since I would be giving them nothing, and I can't collect anything from them to fill the keg, because they don't have anything to fill it with. So, even if you ''do'' have the solid numbers to make the equation with, if one of those numbers is reduced to 0, there's ''absolutely no point in continuing with the equation, because you won't get anything out of it, dividing'' '''or''' ''multiplying''. So, having a definitive answer of 0 when multiplying by zero, when there is not a definitive answer when dividing by zero, is stupid and nonsensical.
****** How much is "supposed" to be in the bottles is irrelevant, what matters in the situation is how much there actually ''is'' in the bottles. So if I tell you that I have sitting right in front of me five empty 20-oz bottles, it is absolutely clear that I have five of them, and it is also absolutely clear that each of them contains 0 liters of Coke. I'm not making a single goddamn thing up about the Coke bottles, they're sitting ''right there''. And about why you would ever do that, it's true that in general you wouldn't employ multiplication in situations that would explicitly result in multiplying by zero. But having multiplication by zero is important and useful, because it allows you to say things about general situations even when one of the values is zero. Eg. there are X cars in the parking lot, and nothing else, and each car has Y people in it. Therefore there are XY people in the parking lot. That equation isn't specifically about a situation in which we would multiply by zero, but in the situation where X is 0 or Y is zero (or both!) it returns a sensical answer (well, all the cars are empty so there are 0 people in the parking lot). That happens when dividing zero by something else too (let's give 2 pieces of candy to each kid! If I have N pieces of candy, I can give candy to N/2 kids. In the event that I have 0 pieces of candy, I don't have enough candy to give hand out to even one kid - I can give candy to 0 kids, if you will). On the other hand, when dividing ''by'' zero, [[CapitalLettersAreMagic Weird Things]] start happening. (I'll give 0 pieces of candy to each kid! How many kids does it take for me to give 5 pieces of candy away? Uh...) There are concrete situations all over the place where multiplying by 0, or dividing 0 by something, has a logical, internally consistent answer, and it's always the same one. Dividing ''by'' zero only sometimes has an answer, and if it does it's nearly always something weird like "infinity" or "any number you like" that doesn't work well when translated into everyday mathematics.
::::On the other hand I could just go the formalist route and tell you that there is an entity called 0 and an operation called multiplication which is defined such that 0X=0 for all X, and that the operation called division is defined so that for B!=0, A/B=C if A=BC, and for B=0, A/B is undefined.
****''"They only exist if you physically have them in your position, otherwise you're just making stuff up"''. If I have five 2-litre bottles of Coke, how much Coke is that in total? Well, 5x2=10, so that's 10 litres of Coke in total. But wait... I don't actually have ''any'' Coke bottles in front of me right now, so both the count that there were five of them and the volume of two litres were just things I made up. Does this mean that 5x2 is ''not'' equal to ten? Of course not, and that's because maths is about more than just counting things that are in front of you.
** I've been thinking about this, and I think the problem you're having is not with division or multiplication themselves but with the idea of a vacuous truth. Suppose I show you an empty room and say, "Every elephant in this room has 3 heads." Is what I just said true or false? From what you have said elsewhere on this page, I suspect your answer will be something along the lines of "there are no elephants in the room, so you're just pulling the elephants out of your ass and the statement doesn't make sense." However, in formal logic, which is what mathematicians use, my statement is true. This is because the statement "every elephant in the room has three heads" is (in formal logic) the exact same statement as "there is no elephant in the room which has a number of heads other than 3", they just look a bit different. To prove this statement is true, we enter the room and search for an elephant with a number of heads other than 3. We cannot find such an elephant. Therefore there is no elephant in that room that has a number of heads other than 3. Therefore every elephant in that room has 3 heads. So sure, we may be pulling numbers out of our asses when we tell you that 5 times 0 equals 0 - but the numbers we're pulling out of our asses are true, so it doesn't matter whether we pulled them out of our ass or not.
**...but...but...but...'''''There are no elephants in the room!''''' How can your sentence be true if '''''the elephants are non-existent'''''? You can't make the claim that every elephant in the room has three heads, because there are no elephants in the room to make such a claim. There may not be any elephants in the room that have a number of heads other than three, '''''but there are also no elephants in the room that have three heads, either''''', so therefore, the claim is faulty and cannot be used. It's like the thing is based on the whole "absence of evidence" fallacy -- there's no elephants for you to prove that they all don't have exactly three heads, so therefore they must all have three heads. It's absurd and stupid. So, technically, the answer to your claim would be that '''''zero''''' elephants have three heads in the room, '''''because there are no elephants in the room to have heads to begin with'''''. I still don't get where people are coming up with this "undefined answer" crap; you'd think that, if 0x1=0, 1x0=0, and 0/1=0, then it would be a no brainer that 1/0=0; how many pieces of candy can you hand out to 0 kids? '''''Zero''''', since '''''there are no kids to hand out candy to'''''. Why is that so difficult for people to understand?
*** It's weird, sure, but that's how formal logic works. A statement being true is semantically equivalent to its negation being false, and the negation of a statement that says that all items in A have property X is a statement that says that there is and item in A which does not have property X. The truth or falsehood of the statement is independant of whether there are elephants in the room, it only depends on whether there is an elephant there that doesn't have three heads. And your theoretical example doesn't show division by zero. It's multiplication by zero. (I give X candy to each of 0 kids, how much candy do I give out? Zero pieces.) For it to be division by zero you would have to say "I have X pieces of candy. I divide it evenly among the 0 kids. How many pieces does each kid have?" in which case there isn't an answer because there isn't a kid. The problem here isn't the step of finding out how much candy each kid has, but the actual act of dividing candy pieces into 0 groups; it just doesn't work. Also, division is defined in a way so that if A/B=C then A=BC, so if 1/0=0 then 1=0x0. Which, of course, it doesn't.
***** No, I'm sorry, but that's bullshit; saying that your statement is true not because there are no elephants in the room ''at all'', but because there are no elephants ''with more or less than three heads'' in the room, is a logical fallacy. Seriously, try to pull that stunt for real on a group of people; I bet you anything that no one will believe it for a second. It all comes down to whether you can know how many items go into a group if the group doesn't exist, and the logical answer to that is that '''''you can't'''''; you cannot know how many elephants have three heads in the room if there are no elephants in the room, just as you cannot know how many people are in each of the cards in the parking lot if there are no cars in the parking lot, nor how much candy to give to each kid if there are no kids to hand out candy to. It just doesn't work, and pulling random numbers out your ass doesn't make your statement true; it just means you're pulling numbers out your ass and are therefore too unreliable to be believed. If you know how many items are in a group, '''''then you have at least one group containing that number of items''''', period; otherwise, it's just unprovable hearsay that no one would believe if they actually thought about it. I will accept that you can have groups without items, but you're saying you can have items without groups, with logically '''''makes no sense whatsoever'''''. If you have no kids to give out candy to, you will be giving out no candy, period, so therefore, dividing by 0 '''''should always have the answer of 0.'''''
****** It's neither bullshit nor a logical fallacy, it's explicitly the way logic works. Now you may decide to say "well formal logic is fucked up then", but in that case you get to say goodbye to pretty much all of mathematics. As for having 0 groups, each of which is of a defined size, say I give you a piece of paper with the instructions "Go into that room. Give every child in it 2 pieces of candy." Without knowing how many children are in the room, the number of pieces of candy you give to each child is already known - it's two, that's what the instructions say. Even if you go in the room and there aren't any kids in there, it's still two. The size of a group can be defined independently of how many groups there are, even if there are 0 groups. And when you say "If you have no kids to give out candy to, you will be giving out no candy, period" the problem is that by stating that you're dividing, say, 5 by 0 you have already accepted that you are giving out 5 pieces of candy. You can't go back later and say "oh but actually you're not giving out any candy because there are no kids" because you ''already stated you're giving out 5 pieces of candy''. (And then if you decide to divide 0 by 0 you find out that every number is the answer.)
******* No, that doesn't work, I'm sorry. If you can't give out any candy because there are no kids, ''then you are not giving out any candy''. Saying that "you already stated you're giving out five pieces of candy" doesn't negate the fact that ''you did not give out any candy at all''. If there are no groups to distribute the items to, then the number of items you're told to distribute is useless and an asspull, and the fact that you don't distribute ''any'' items should take precedence; in other words, '''''it doesn't matter if you're told how many items to distribute to the groups, if there are no groups to distribute to, you cannot distribute anything, and therefore the number of items each group is given IS ZERO!''''' I'll admit that math is my weakest subject, but this is supposed to be basic multiplication and division, so why is this so convoluted and difficult for everyone to grasp? If I have nothing to give, I give out nothing; if I have no one to give to, '''''then I still give out nothing'''''. Same as if I try taking something; if there's nothing to take, I take nothing, and if there's no one to take from, then I still take nothing. It's not like I'm going to lose money if I decide to donate equally to five different businesses, and they all dry up and close shop before I can -- the check will be sent back and my money will remain safely in my bank. I'm sorry, I really am; I'm trying to understand, but the more it's explained to me, the more nonsensical and illogical it sounds to me, and the more simply having the answer to 0 sounds like what it really must be, logically.
*******The question is not "how many items did you give?", it is "how many items does each group have once you gave the items?", it is exactly because there are no groups that the answer can't be any real number, not even zero. You say the answer is an "undefined" number because there isn't any number that fits, you can't say that the groups have zero items because there aren't any groups. It is different with multiplication because if you say that one group would have 6 items (1*6) the question becomes "how many items are there after you count the items in all groups?", if you have 0 groups, you ignore the number of groups, and since you didn't count any items, the answer is 0.
******* The entire concept of vacuous truths only exists in Boolean logic (i.e. every statement is either completely false or completely true), which is the most well known system of logic isn't the only one. With Ternary logic (e.g. every statement is completely true, completely false, or neither) in a room with no elephants the statement "All elephants in this room have three heads" isn't true or false, its neither. And mathematics works just as well with Ternary logic as it does with Boolean logic. Also, consider the statement "Every elephant in this room both has and does not have three heads." if the room is empty and one accepts the idea of vacuous truths one has a true contradiction which most systems of logic aren't equipped to deal with (i.e. for most logic systems if one contradiction is true every statement is true).
**When people say that dividing by zero is impossible they are either mistaken or they mean its impossible to divide by zero using only the set of real numbers. If you simply [[http://en.wikipedia.org/wiki/Real_projective_line add unsigned infinity]] to the set of numbers you're using division by zero becomes possible.
**One of the problems I see here is that it's not always possible to apply real world situations to accurately portray mathematical concepts. For example, the candy example is closer to "Take these five pieces of candy and give an equal amount of candy to each kid in that room. If there are no children in the room, do not give any candy." The last part is implicit in real life, but not in math, where you'd have to specify that when x=0, 5/x=0. If you change the same problem to "Take 5 pieces of candy and give an equal amount to each child in the room until you have no candy left," then you see that if there are no children in the room, you can't do it. That's what undefined usually turns out to be in the real world: not a strange answer, but an impossibility. Studying theoretical math is a different monster compared to studying "practical" math. Sure, basic multiplication and division is simple as long as you take everything you're taught without questioning. It's when you start questioning that math goes from basic to theoretical. And this case is difficult not because of the multiplication or division, but because of 0, which is a very special number that does very odd things at times. I've found that studying theoretical math takes a change of perspective in order to really understand it. Saying that numbers are ass-pulls if you can't have them physically in your possession right now will not get you anywhere. For example, try to have exactly pi of anything in your possession. You can't. There is always a number that can't be physically had. Like negative numbers. Or imaginary numbers. Or even a number so large that even if you counted all the sub-atomic particles in the universe you won't come close. But that doesn't mean that pi doesn't exist, or that it isn't useful, or that it isn't meaningful. Even imaginary numbers are useful in electrical engineering (don't ask me how, all I know is that it is). In theoretical math, multiplication and division aren't always opposite of each other the same way addition and subtraction are (especially when 0 or infinity is involved). If you have 5 and add 6, you can subtract 6 to get 5 back. This usually works with multiplication and division, but not with 0. If you have 5 and multiply it by 0, you can't divide 0 by anything to get 5 back. You can make a case for 0/0 being equal to any number, but saying 0/0 is usually nonsensical. In addition, in math, there are such things as groups with no items. These are known as empty sets. It doesn't matter whether or not a set is empty for it to exist. If I say the set of real x for which x=sqrt(-1), then I have defined an empty set (since i and -i are not real numbers). Does the set exist? Yes, because I have defined it. Is it an ass-pull? Not exactly. In math, definitions merely have to be internally consistent, which basically means that you can define a set for anything, whether or not anything can actually go in it.
*** So, in essence, math is so fucked up you can do anything you want with it as long as MagicAIsMagicA. No wonder it's one of my worst subjects in school...
**** Ok, I have a question for the person who keeps going "But That's an AssPull", what grade are you in? how old are you, because your difficulty here seems less like a problem with math, and more like being either unable or unwilling to perform even the simplest of abstract conceptual thought.
***** I got my GED, thank you very much, and I made it up to the first half of Senior Year in high school before I dropped out. And how is "abstract conceptual thought" a factor in this? I just explained why multiplying by 0 should be just as impossible as dividing by 0; if you do not have anything to group or break up, then you ''cannot'' do so, regardless if you're grouping or breaking them up.
***** Zero comes from Book keeping math to explain multiplication. You sell several types of products, but that day you didn't sell any apples. When totaling up the profits per piece, say you make 10¢ an apple, your profits on apples were zero that day because the number sold was zero and the total profit on apples is the number of apples times the individual profit. You could say it doesn't exist, and that's the point it doesn't exist. Dividing by zero is more complicated, but involves more book keeping as well. See the bold explanation.
*** Well, I admit I oversimplified things a bit. Basically, the answers should follow logically from the definitions, and the definitions are themselves often answers to other definitions and rules. The rules themselves go back to number theory. It seems like you can do anything, but there are coherent rules that are so far back that it's hard to see it all. Besides, this our whole universe is governed by Magic A Is Magic A, if you think about it.
** Can it not be displayed without using any inherently nonsensical (though reasonable under [[MagicAIsMagicA Language A is Language A]]) logic, by stating that you can cut no pieces of candy into five groups, and there will still be no pieces of candy, but you cannot cut five pieces of candy into no groups, because there would be five pieces thus unaccounted for, which isn't allowed since there is no kerf in mathematical division and thus no loss no matter how many times it is divided (see .999...)?
*** With no candy to cut into pieces, you cannot cut anything into pieces, so the equation should be impossible. That's my argument, and I'm sticking to it; if cutting 5 pieces of candy into 0 groups is impossible, because you can't cut the candy into oblivion, then cutting 0 pieces of candy into 5 groups should be impossible, as well, since you have no candy to cut into any groups. Similarly, joining 0 groups of candy into 5 pieces should be impossible, because you have no groups of candy to join, and joining 5 groups of candy into 0 pieces is impossible, because you can't join the candy into oblivion. Addition and Subtraction can utilize the 0, because you're just putting new candy in or taking candy away from one lump sum, not breaking that lump sum into different groups or joining the groups into one lump sum. If I have no candy, and I add 1, then I have 1 total pieces of candy; if I have 1 candy, and I add no more candy to it or take away none of the candy I have, then I still have that one piece of candy; if I have no pieces of candy, and I take away 1...well, then we get into the negative numbers that cause the analogy to fall apart, but still. 3 out of the 4 times you can use 0 in addition/subtraction, you get a solid number, because all you're doing is putting in/removing a set number if candies from a set lump sum of candies. With multiplication/division, though, you're not doing that; you're taking a set number of groups of candy, each with an equal amount of a set number of candies, and combining them into 1 lump sum, OR taking 1 lump sum of a set number of candies and breaking them up into a set number of groups. You're not taking away from the number of candies you ultimately have, nor are you putting in any more - the number of items you have in the lump sum is always fixed by the equation, and you're just determining how many items in it you can distribute to how many people, or how many people you need to collect a certain number of items from to get the number of items in the lump sum. And, if there's no items to distribute/take to/from people, or there's no people to distribute/take the items to/from, then distributing/taking the items ''is pointless and impossible''. Hence, dividing ''and'' multiplying by 0 ''should not work''.
**''' It is easier to think about this as part of a set of problems. Say you're doing a series of bookkeeping totals where you add up the 15% tax on different types of items, but Item X didn't sell any item so there's no tax. So 0 items*15% tax=0 tax taken. If you still grumble about this, take note it took a long time for people to accept the idea of nothing and that a whole lot of nothing was still nothing. It's even easier to grasp in a problem doing income and losses and having to pay out a dividend of your profits. So say you have 5 partners each taking a share of the profits. Income was 5 million, costs were 5 million. The equation to figure out your profit share is simply (Income-Costs)/Partners= Profit share. In this case its (5,000,000-5,000,000)/5=0  because there were no profits'''
*** As to dividing by Zero, it gets a bit complex. Dividing by Zero is impossible simply because is an asymptote. What this means is if you divide say 1, by smaller and smaller numbers it will approach infinity. So 1/1=1 1/.1=10, 1/.01=100, 1/.0000000001=10,000,000,000 and so on. But at Zero it breaks down because if you count from -1 up to zero you don't go in the same direction, you go more and more negative as you approach zero where 1/-1=-1, 1/-.1=-10 and so on. So at dividing by Zero needs to give you both an positive infinity and negative infinity which is why the answer is undefined. [[LiesToChildren To make it simpler]] you could say it's infinity, but because of the whole negative infinity being true at the same time it really isn't true.
*** To expand on the partner example for x/0: lets say all 5 partners suddenly died.  You now have a profit to split over zero people.  Who gets the money?  The answer is undefined unless you asspull hither to unknown beneficiaries and inheritance law.
** You can't divide by zero because dividing by zero means you have zero divisors, which means you can't have a field, and fields are useful. To understand this requires knowledge of [[http://en.wikipedia.org/wiki/Abstract_algebra abstract algebra]], which is admittedly confusing. But if you don't understand it, please either learn it or trust those who have; trying to think about it using the concrete terms you learn in elementary school only leads to huge discussions like this that go nowhere.
** You'll are making this too complicated.  Math is supposed to be simple.  Multiplication is basically a faster way to do addition; e.g. 2 X 5 = 2+2+2+2+2 or 5+5 =10.  Division is likewise a faster way to do subtraction; e.g. 10/2= 10-2-2-2-2-2.  You can subtract 2 from 10 five times before you get to zero or cannot divide evenly again so 10/2=5.  Multiplication by zero works like this: 5 X 0 = 0+0+0+0+0 = “0” or “no” 5’s added to “no” 5’s equals nothing i.e. still zero.  Division by zero:  5/0 - you can subtract 0 from five an infinite number of times and never get to zero so the concept is impossible.
*** "Division is likewise a faster way to do subtraction; e.g. 10/2= 10-2-2-2-2-2." Erm. No. 10/2 = 5; 10-2-2-2-2-2 = 0.
*** '''Another very simple explanation (in bold because perhaps it will be understood better...one can hope). Remember in elementary school, where your teacher would phrase 10/5 as "How many times can 5 go into ten?" Think about it like that. "Zero" is nothing. Any other number is "something". So, "How many times can nothing go into something?" to which a kid might answer "A million billion gajillion times", an older person might answer "infinite times", and a mathematician would answer "undefined", because it is both infinity and negative infinity, as another troper pointed out.'''
**** [[CaptainObvious Apparently you need a young man to inform you that nothing can go into something no times at all. Otherwise it wouldn't be nothing, and the something wouldn't be something.]]
** Theoretical mathematics do not need to reference actual objects. Seriously, this is not hard to understand.
** Wow. SeriousBusiness indeed. Lets try to put this simply for those of you who skipped to the end: 'undefined' is a mathematician's answer for 'we really don't know'. That's it. By our current logic and knowledge, we can't figure out what it is and we may never be able to.
*** Actually, undefined means exactly what it sounds like - the answer is not defined. It's like asking what the square root of Mariah Carey is. It's not that we don't know the answer, the question doesn't even make sense in the first place.
*** However, we can tell you with certainty that [[TheImpossibleQuiz sqrt(onion) = shallots]].  But that's the exception rather than the rule.
** You ''can'' define 0x to be undefined and x/0 to be defined, just as you can define 1+1 to be 0. The question is then how useful this system is. It happens that the system in which 0x=0 and x/0 is undefined (or indeterminate) has much more useful behaviour in the real world than your one, for a number of reasons outlined above, so that is what we use. If you want x/0 to be 0, you are free to sit down with a pen and some paper and discover how the resulting system behaves; however, you will encounter serious problems for the reasons stated above.
** This troper hadn't even heard of that rule until Algebra I, when the teacher got to the unit about slopes and graphing lines. The formula for a slope between two points on a graph is (Ysub2-Ysub1)/(Xsub2-Xsub1). If you have the two coordinates (3,5) and (6,4), then the slope would be (4-5)/(6-3), or -1/3. However, if you have the coordinates (3,4) and (6,4), then the slope would be 0/3 or 0, because the two lines make a '''horizontal''' line. However, if you have the coordinates (3,4) and (3,6), then the slope would be 2/0. However, the slope in this case would be ''undefined'' instead of ''zero'' because those two points form a '''vertical''' line. While the slope of a flat surface is obviously zero, what can the slope of a perfectly vertical line be described as anything ''except'' "undefined"?
** In response to the claim that "Math is supposed to be simple" I would say "since when?"  If it's supposed to be simple, then my degree was a lot harder than it had any right to be.  This troper has a degree in mathematics and dividing by zero did bug me for a while until it was presented as an issue of convergence.  We get frustrated when we look at division by zero as a function of the set of integers; but when we consider the set of real numbers the problem becomes a bit less of a problem and exponentially more confusing.  The problem is what happens when we cross the threshold of 1, not 0.  Bear in mind that division is a function and we tend to represent functions as lines on a graph.  Looking at the graph of division, what happens when we divide y by 1/x?  While numerically, you remember that the process involves reciprocating your divisor and then simply multiplying, you're still dividing.  So while division results in a quotient < Y anytime we divide by anything greater than 1, in order to approach zero, we must recognize the rising trend of the quotient.  Let's say we split out divisor in half every time, so first we divide by 1, then by 0.5, then by 0.25, etc.  What we discover is that the quotient grows by an inverse proportion.  The closer we get to 0 (without actually touching it) the bigger and bigger the quotient becomes until dividing by zero essentially grants us infinity (as infinity cannot be reached via additive/multiplicative means, we'll never ACTUALLY have it, but whatever).  Now, approaching the problem from the negative gives us the same problem, except with a difference in signs.  The closer you get to 0 from the negative, the closer to "negative infinity" you get.  We can keep dividing forever and get closer and closer by halving our divisor but never ACTUALLY touch 0, and as such grow larger and larger in one direction or the other.  When we try to describe what MIGHT happen if we were to divide by zero (finding the limit of the function) then we're presented with conflicting data - the function approaches both positive and negative infinity depending on which direction we approach 0 from.  We can't say that it's "infinity" because it approaches two different types of infinity.  In essence, the graph presents us with a vertical line at x=0, and as the previous commenter mentioned, a vertical line has an undefined slope.
**Basically what that last guy said: x*y = z means that, if you were to make x groups of y objects each, there would be a total of z objects. 10*0 = 0; 10 groups of no objects each is no objects total. 0*10 = 0; no groups that have 10 objects each is no objects. x/y = z means that you can remove y objects from a group of x objects z times. You can remove 0 objects from a group of (any number here) objects as many times as you want, so x/0 is undefined. (Or, if you prefer a more mathematical proof, a restatement of a formula from above: we already know 0*n = 0. By definition of division, n = 0/0. But n is every possible number, which means that 0/0 literally has no specific value; it is all possible values at once.)
** Division by x is defined as multiplication by x's reciprocal. x's reciprocal is defined as the number that, when multiplied by x, yields 1. No number, when multiplied by 0, gives 1. See the problem?
** It is logically impossible.Example: 5 divided by 0.If it equals 0, then multiplying the end result 0 by 0 should equal 5.It doesn't.If it equals 5, then multiplying the end result 5 by 0 should equal 5.It also doesn't.You can multiply by 0 (which always equals 0), but not the other way around.
** Division is a shortcut of subtraction in the same way that multiplication is a shortcut of addition. To divide 31 by 7, start at 31 and subtract 7 over and over again until you reach a number smaller than 7. The number of times you subtract (4) is your quotient, and the number you finish at (3) will be your remainder. Now try dividing 5 by zero in the same fashion. Let us know when you're done.
** Think of it the way I taught myself: Say you're hosting a party. There's cake. [[strike: It's not a lie.]] Say 6 people show up, there's 6 people to divide the cake amongst. If nobody shows up, there's nothing to divide.
*** Yeah, but this answer is why some people suggest 0 / 0 = '''[[BigEater 1]]'''.
**** 0 people show up to divide 0 cakes = 1 person gets a whole cake?
** Because it's useful to define it that way. Trust us on this one.
*** Clarifying: it's not impossible. But doing so causes what one calls "numbers" to stop working like we're used to. There are other structures (special kind of "numbers") where division by the zero element of that structure is possible. Therefore, when working with "standard" numbers, the zero entry on the map called "division" is just declared not to exist, and the problem goes away.