Main Tropes Index

Troperville

Editing Help

Tools

Toys

Narrative

• Universal
• Applied Phlebotinum
• Characterization
• Characters
• Characters As Device
• Dialogue
• Motifs
• Narrative Devices
• Paratext
• Plots
• Settings
• Spectacle

Genre

• Action Adventure
• Comedy
• Commercials
• Crime And Punishment
• Drama
• Horror
• Love
• News
• Professional Wrestling
• Speculative Fiction
• Sports Story
• War

Media

• Animation (Western)
• Anime
• Comic Book
• Fan Fics
• Film
• Game
• Literature
• Music And Sound Effects
• New Media
• Print Media
• Radio
• Sequential Art
• Tabletop Games
• Television
• Theater
• Videogame
• Webcomics

Topical Tropes

• Betrayal
• Censorship
• Combat
• Death
• Family
• Fate And Prophecy
• Food
• Holiday
• Memory
• Money
• Morality
• Politics
• Religion
• School

Other Categories

• British Telly
• The Contributors
• Creator Speak
• Creators
• Derivative Works
• Did Not Do The Research
• Language
• Laws And Formulas
• Show Business
• Split Personality
• Stock Room
• Trope
• Truth And Lies
• Truth In Television
• Weirdness Isolation

• 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.
  • 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?
  • 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.
  • 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 Ass Pull 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, 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.

  • 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 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 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 Magic A Is Magic A. 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 Ass Pull", 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.
    • 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 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 logical 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
    • 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. 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 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.
  • Theoretical mathematics do not need to reference actual objects. Seriously, this is not hard to understand.
• Alright, is infinity a number or not? If yes, say 1/infinity=x, then 1-x... If no, then how many numbers are there?
  • Infinity actually refers to several numbers, the three most basic forms of infinity are unsigned (which is part of the Real projective line) and positive and negative infinity (both parts of the Extended real number line). In all three cases any real number divided infinity = 0. More complex forms of infinity also exist, but they involve different sizes of infinity.
    • You really want to go with "different sizes of infinity"? Think about this: On the interval (-∞,∞), there are infinite integers. Then think about this: Counting decimal-place numbers, there is ∞ values between [n,n+1]. And n can be any integer in (-∞,∞). So the ∞ in [n, n+1] is multiplied infinite times, once for each integer. That means that, including decimal numbers, what you have on a number line is ∞^2 numbers! Multiply by ∞^2 for each axis you want... and in traditional three-dimensional space (neglecting time in this case), space has ∞^6 points. Include time (which can also be divided infinitely like between integers, and neglecting that time is only on [0,∞) and not (-∞,∞)), and you have INFINITY TO THE EIGHTH FREAKING POWER. Kinda gives credibility to Buzz Lightyear's Catch Phrase don't it? Oh, and don't even think about ∞^∞... Oh shi-
    • You're kind of on the right lines here, but infinity*infinity is still the same size as the original infinity. If you have a two dimensional grid consisting of integers [0,∞) in both dimensions then you can put all points on that grid into one-to-one correspondence with the natural numbers [0,∞) by following a simple pattern. Start at (0,0), move right one space, then move diagonally down and left until you reach the border, then move down one space and move diagonally up and right until you reach the border, then move right one space... Each time you pass a point number it off in order and you get 0: (0,0), 1: (1,0), 2: (0,1), 3: (0,2), 4: (1,1)... you can go on as long as you want and you will never miss a point or mark one off twice, and so the two infinite sets are the same size. On the other hand, the set of all real numbers really is larger than the set of all natural numbers.
    • A great book on this subject is The Mystery of the Aleph by Amir D. Aczel (he approaches the subject assuming the reader to be a layperson in math, to maximize how understandable the text is, but of course the complexity builds the further into the text you get; so the more math background you have, the deeper you can go). It's pointed out in that book that the larger infinity is not like infinity to some power, but more like some finite value (greater or equal to 2) being raised to the power of infinity (and this larger infinity is the exponent by which you'd get an even bigger infinity).
  • Infinity is not a number, it is the limit as something increases without bounds. In fact, it is a way of saying that something increases without bounds. There are infinite natural numbers because there is no upper limit to their quantity; you can always produce a new one. There is no highest natural number because the value of the natural numbers approaches infinity if you keep adding one to the last one ("counting"), thus increasing without bounds.
    • The other issue is some things get to infinite quicker. Squaring a number versus factorialing a number will have the factorial winning. 10^2=100, 10!=3,628,800. Knowing this helps a lot with infinite series because sometimes we can figure out a limit after which any more steps are pretty much unimportant.
    • It gets stranger when you bring in set theory, but don't worry about that as it's not something most people need to deal with.

• Why the hell do they call square roots of negative numbers imaginary and not just, not real, or fake, or something? And if those are imaginary, then what do you call the other non-real numbers? like x right here:

0x=1 do we call x in this, oh shi-- numbers? That would be nice, because 'oh shi—' will be a better name for numbers then Fucking for a city.

• If sqrt(-1)=i, what is the symbolology (why did I lol?) for other numbers sqrt(-n), where n is any number [0,∞)? Sure, i might be the most important Square Root Of A Negative Number, because (in real square roots), 1 is the only number that's its own square root, but the other S.R.O.A.N.N.s must have some kinda use...
  • Any other symbols would be redundant. The square root of -n is sqrt(n)*i. 2i is the square root of -4, for instance.
• The name "imaginary numbers" started off as a dismissive name used by critics, but like "Big Bang" it caught on and is still used even now it has become clear that they have practical uses.
• More than just utility, complex numbers have an entirely rigorous construction as points on the plane with multiplication being defined by (a,b)*(c,d)=(ac-bd,ad+bc). If we then define i=(0,1), we see that i^2=-1. Everything about complex numbers is well-defined, and properly constructed. There is no such x such that 0x=1. Maybe in the limit sometimes this occurs somewhat, but it is easy to prove that for any ring (such as the real numbers) 0x=x0=0 for any x. So you would either have to cripple your arithmetic in some way (such as dropping distributivity of multiplication over addition or you would be forced with the nonexistence of such a system.

• So what's the square root of i?
  • There's more than one, but (1+i)*sqrt(2)/2 comes to mind.
  • The easiest way is using Euler's formula: e^[(pi/2)*i] = cos(pi/2) + i*sin(pi/2) = i, so sqrt(i) = {e^[pi/2]*i]]^(1/2) = e^[pi/4]*i = cos(pi/4) + i*sin(pi/4). (Sorry for all the badly written math). The reason there are many answers is because of the nature of sin and cos (they oscillate).
  • Even easier, so i*i=-1, thus i*i*i or i^3=-i and i*i*i*i=(i*i)*(i*i)=-1*-1=1. Which leads to i*i*i*i*i=i, so sqrt i=i^(5/2). Yes it references itself, but that's because i is really just 1 on a different axis. I think I may of made things worse but using the definition of i in terms of euler's number (and it's really critical to use it, it's not some fancy aside but critical to the definition of i) you can define i as e^[pi/2]*i] so i^(1/2)=e^[pi/4]*i]

All this and just think, we haven't even gotten onto the subject of .999...=1 yet!

• On that subject... 0.999 extended to an infinite number of digits equals 1, because it's impossible to add 0.(infinity-1 digits of zeroes)1, since there's no end of infinity to subtract from, but it's still a rounding error, because it's an infinitely small fraction of an integer less than 1, but it's still smaller. So little difference that it is literally impossible to tell, but it's still a rounding error the way I look at it. I haven't been able to find any books on this particular application of infinity, but it just doesn't quite work. Nine plus 9/10 is less than ten, and nine plus 9/10 plus 9/100 is less than 10, so keep going, and not only will you never reach a proper end, but it never quite reaches ten anyway. Like Achilles and the tortoise, only Achilles can in this instance take smaller and smaller steps without reaching a plateau on step smallness. Or is this a Dead Unicorn Trope amongst mathematicians, and that's why I can never find a book that takes it seriously? Or is it just usually explained in different terms and this one is not realizing the connection?
  • Consider x=0.999.... Then 10x-x=9.999...-0.999..., so 9x=9, so x=1. There's no rounding error, they're just two ways of writing the same number. The answer for the "repeated addition of 9/10^n doesn't get you to 1" problem is "infinity is weird". There are as many even numbers as there are integers, even though "obviously" there are twice as many integers as even numbers, and 0.999...=1, even though "obviously" it can never get to 1.
  • There is no rounding error. In order to truly understand this (as opposed to simply accept it from arithmetic proofs), you need to grasp the delicate ideas of limits and infinity. Most importantly, remember that infinity is a limit; it is the limit as something increases without bounds (that is, it will never stop increasing). Now, as you allude to in your post, 0.999... = 0.9 + 0.09 + 0.009 + ... which can be written in Sigma notation as follows: http://tinyurl.com/krcjwo It is here that the subtleties become important. Notice we are not saying that 1 is equal to any finite sum, but to the limit as the number of terms being summed increase without bounds (that is, goes to infinity). As you keep adding terms to the sum you keep approaching 1, so it cannot be denied that 1 is indeed the limit of the sum. The only question remains, is the limit of the sum equal to 0.999...? And the answer is yes, because 0.999... is not any of the finite sums, but an infinte number of them, and infinity is the limit as things (in this case, the sums) increase without bounds. I hope this makes things clear.
• Does everything REALLY involve numbers or are people just making that up to scare kids into getting A+'s in math?
  • Everything really important for modern civilization yeah. Medicine for example, dosing is determined by things like the rate of absorption into the body as well as body weight factors. Electricity you can maybe use without math, but to actually understand how the signal gets to your house you need to understand trigonometry at a minimum. Or your computer which is based off numbering systems. Essentially if you want to understand modern technology at all you need math, and it starts with numbers to get to varibles where the real work starts. If you don't, you're going to beholden for someone else for everything. Heck, proper cooking needs a good understanding of math and chemistry. You might be able to do ok with out it, but understanding how the heat is dissipated from pans and absorbed by different materials and the effects of a higher versus lower temp and how it's not quite the same if you double the temperate to half the time...
• Why are calculators, even basic ones, so much fun to play with?
  • 5318008.
• Seriously, guys. Who the hell came up with the term 'integer'? What is wrong with calling them 'numbers'? If they're supposed to be called 'integers', why the hell do we even use the word 'number' anyway? Let's be honest here: when I first learned the term 'integer' back in middle school, that was the moment when mathematics Jumped The Shark for me. I've never trusted it since.
  • Integers are numbers with no fractional parts, hence the name. Pi and 2.5 are numbers that aren't integers. Sad you gave up on math due to a random misunderstanding.
  • Has there ever been a more epic case of fail?
• Why doesn't 1 count as a prime number?
  • They'd have to make too many exceptions for too little gain. Prime factorizations don't need any 1s, and we don't need the definition to be changed to "divisible by itself and 1, or just 1 if it's 1".
    • But, one is divisible by itself and one.
    • Every number is divisible by itself and one. Prime numbers are defined as having exactly two factors.
    • So, first off, that statement should be "one is divisible by ONLY itself and one." Second, if one is itself, then wouldn't it only be divisible by itself?
    • Yes, thats the point. It fails to meet the definition of having exactly TWO factors.
  • 1 is a unit; its properties are VERY different from those of actual prime numbers. A better question is "Why should 1 count as a prime number?", and there is really no answer to that beyond "It looks sort of like a prime from a purely cursory view".
• Why is it that the higher up you go in math, the more useless it becomes?
  • Because it makes sense to teach the important stuff first.
  • The low level stuff is useful for everyday things like working out how much change you should get at a store, but high level maths is far from useless. If you want to design a computer, build a plane that won't fail spectacularly, or work out advanced concepts in physics you need advanced mathematics.
• ...This page bugs me.
• Infinitesimals (things like dx and dy) bug me. We're allowed to divide by them because they're not *really* zero, but when we're adding them, we can treat them as zero because they're basically zero. Yes, I know their purpose is for limits and ratios, but it still bothers me that they are treated as both 0 and not 0.
  • Me, I always assumed it was just sloppiness. Yeah, you could do it rigorously with limits, but why would you? It's hard enough work as it is.
  • The official line is that dx/dy is not division, it's just a way of writing differentiation - Leibniz's notation is still used because it's awkward to change everything.
• Anyone else ever noticed that teaching math is like one long series of lies, and then flipping back on what you said? "Oh, no, you can never subtract a big number from a smaller number..." "... unless you use negatives..." "You can't multiply fractions..." "...Until you find the least common denominator" "You can't find the square root of a negative..." "...without using i..." etc.
  • That's a problem with the way that math is taught in school, not with math itself. Also, who tells people you can't multiply fractions, and why the hell would you need the least common denominator to do it?
    • I think he meant adding fractions, which would require some common denominator, but it's usually faster to just multiply the the denominators (which rarely gives the least common denominator) since half the time you have to simplify anyway.
  • It's part of all sciences, it just comes up most in maths. Lies to children so that they don't ask questions of teachers when they won't understand the answers.
  • Besides, some of them aren't even lies. It's perfectly true that you can't find the square root of a negative number at first, because you're working with real numbers. It stops being true once you start working with complex numbers, but before you introduce the concept of i students have no business working with complex numbers anyway.
    • Frighteningly, some teachers of early-level math teach it in that annoying "you can't do this" "now you can because I said so" way because they actually don't get it either. My (tenured) sixth grade teacher happily taught the class that the area of a circle was pi*r*2. Not pi*r squared. Pi*r times two. Fortunately she was called out by some of her own students before it could stick with the others. Unfortunately, she tried to fight them to save face instead of shrugging and making sure the students got it right, and later was forced to admit she didn't know what exponents were. As a math tutor these days, I can say with great confidence that the vast majority of problems people have with math can be traced back to a teacher screwing them over somewhere early along the line instead of genuinely being "not good with numbers" — most of the people I help have to be re-taught several things that were drilled into their heads outright wrong several years prior.
• This is more of a JBM for That Other Wiki than math per se, but...why does every single Wikipedia article about mathematical concepts read like it was written by and for people with Ph.Ds in mathematics?

---