Wikipedia:Reference desk/Archives/Mathematics/2007 August 9

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August 9

Dividing by Zero

I have a Math question, but with a caveat. The caveat is this: please answer this question as if you are explaining to, say, a typical high school or middle school student ... as opposed to a Math Ph.D. student. Thanks. The question: why exactly can we not divide by zero? (Joseph A. Spadaro 16:49, 9 August 2007 (UTC))

Well, what would you get when dividing by zero? For instance... we know that 12 / 4 =3, and we can reverse that to say 3 * 4 = 12. But, when dividing by 0... if 7/0 = x, then x * 0 = 7. What could x possibly be? That's why you can't really divide by 0; because what number, when multiplied by 0, will give you back the number you divided? Does that make sense?

Also note that in other number systems, etc, division by 0 is possible... but in the reals and complex numbers it is not, and I'm assuming that's what you're talking about. That sentence is just a disclaimer. Gscshoyru 16:58, 9 August 2007 (UTC)

Good answer. Nowadays, kids in school (at least one school I know of, anyway) learn math with "fact families"; i.e., 6*2=12, 2*6=12, 12/6=2, and 12/2 = 6 are all one fact family. Good approach, I think. So, from that point of view, since you don't have fact families where 0*x=anything but zero, you can't have other members of that fact family, where something/0=something. Gzuckier 17:15, 9 August 2007 (UTC)

Thanks... I think that was the best example I've read, wherever it was I read it. The only problem with it (I seem to have forgotten to mention) is that it seems to allow 0/0 = 0... which also doesn't work. Why? Because, in another sort of "fact family" (I like that term). You know that if 12 / 3 = 4, then 12 / 4 = 3, right? Well... 0/0 = 0 makes sense... but we know that 0/1 = 0, so 0/0 must be 1, right? Same with 0/2 = 0 meaning that 0/0 = 2, etc. This is rectified by the fact that you can't divide anything by 0, even 0 itself. I just needed to add that, because I forgot it... and I was under the impression 0/0 = 0 after reading that first explanation for quite a while, and I don't want to give anyone else the wrong impression. Gscshoyru 17:20, 9 August 2007 (UTC)

Bringing the disucssion to a slightly higher level, the standard definition for division of real numbers is ${\tfrac {a}{b}}=c$, where c is the unique real number such that $b\cdot c=a$. It can be shown that this definition is valid whenever $b\neq 0$. However, when $b=0$ we run into problems; If $a\neq 0$ then such a number does not exist (there is no real number c such that $0c=a\neq 0$), and if $a=0$ then such a number is not unique (for every real number c, $0c=0=a$). There is therefore no sensible way to define division by zero within the context of real numbers.

Of course, you should take a look at division by zero for more information.

Personally, I think this "fact family" thing (about which I have now heard for the first time) is a pedagogic disaster, of the kind that leads to generations of students who can solve math problems perfectly without the slightest understanding of what they are doing. -- Meni Rosenfeld (talk) 18:22, 9 August 2007 (UTC)

How so? What about that means they don't know what they're doing? Gscshoyru 18:28, 9 August 2007 (UTC)

I have drafted some attempts to answer this question in a way that might make sense to anyone other than myself, but failed. Take it as a personal opinion without an accompanying justification. -- Meni Rosenfeld (talk) 18:39, 9 August 2007 (UTC)

When speaking to a young child, just say We can't divide by zero because we do not know what the answer is. 202.168.50.40 01:48, 10 August 2007 (UTC)

Yes. And if the child is anything like I was, he'll want to know why we don't know the answer, and be pissed off because the adults are taking him for a dummy, and are brushing him off like an idiot. But that's just me. Maybe most kids don't mind that. Gscshoyru 01:58, 10 August 2007 (UTC)

In any case, we do know what the answer is - we know that the answer is "there is no (sensible) answer". Confusing Manifestation 03:53, 10 August 2007 (UTC)signed some time later so timestamp is wrong

A problem with "why" questions is that the word why has too many meanings. Why Johnny can't have another ice-cream right now is a whole nother kind of why than why the sky is blue, or why a year has 365 days. The most important insight we might want to give to a potential fledgling mathematician between the ages of 4 and 104 is that the why of why we can't divide by zero is more like the ice-cream why than like the blue sky why or the 365-day year why. As mathematicians, we define things, and if mathematical questions have an answer, it is mostly because it follows from how we choose to define the meaning of the ingredients that make up the question. And we do not pick definitions randomly, but prefer those that give rise to nice mathematics. We might have chosen to define 1/0, and if we choose to do so, we have several alternatives, but in return we also lose something (in particular that the domain we're working with is a field (mathematics)), and the gain does usually not outweigh the loss.  --Lambiam 02:49, 10 August 2007 (UTC)

If I may suggest a slightly different style of answer for a "typical high school or middle school student", it would be a more Socratic approach. It could be a lot of fun, very educational, more persuasive and memorable than a canned answer, and much more difficult to pull off successfully.

The strategy is this: Ask back, "What do you think the answer should be?" Then explore the implications of that choice. Try to explore more than one choice. Each leads to interesting mathematics. Each has benefits and drawbacks.

The problem, of course, is that you need to be willing and able to work with their choice. It's OK to not have all the answers at your fingertips, but you probably don't want to appear totally bewildered.

An easier approach is for you, yourself, to suggest some alternatives, with answers prepared in advance. So long as you anticipate the answers your student would propose or accept, that may also work well. But they may insist on proposing something different, and remain unconvinced if their idea isn't explored.

Finally, let me suggest a resource, where teachers and professors share their insights: the Ask Dr Math forum at Drexel University. --KSmrq^T 04:27, 10 August 2007 (UTC)

• I'm quite surprised that nobody has offered my instinctive answer to the original question - albeit I admit mine probably doesn't hold up to the advanced mathametician's answer. The term 'divide'/'division' is normally taught that in 'a / b = c', the meaning is "a, distributed amongst b, equals c'. Example: "12 sweets distributed (read: divided) amongst 3 kids = 4 sweets each". When you put zero into the equation: '12 / 0 = ?' is "12 sweets distributed/divided amongst 0 things = ?" - there is no place to put the sweets. A further way of putting it is, that when dividing, we have to know in what way we are dividing something or else we can't divide. So by dividing by zero, it either opens an infinite number of ways of dividing something, or, it doesn't divide at all. Rfwoolf 01:18, 14 August 2007 (UTC)

Thanks to all! (Joseph A. Spadaro 03:17, 15 August 2007 (UTC))

Graphical representations of complex functions.

Hi all. This might seem a bit of a weird question, but does anyone know of a page with lots of pictures in the form of [1], together with the functions used to create them? I guess I have no reason for asking other than I like the look of the pictures, and want to see how they compare to the graph of the same function of a real variable. 80.169.64.22 21:21, 9 August 2007 (UTC)

Have you looked at mathworld.wolfram.com? —Tamfang 23:26, 9 August 2007 (UTC)

If you have access to a Mac, there's a program called Grapher in the Applications:Utilities folder. If you ask for a 3-d graph, you can get the graphs shown for any function using, e.g., $z=|f(x+iy)|$. It also has Re and Im functions. It's more fun than a barrel of monkeys. Donald Hosek 00:02, 10 August 2007 (UTC)

Hi. As the creator of those images, I thought I'd explain how I did it. First of all, I'm using MuPAD [2]. Those particular images are those of the LambertW function (quite obviously). There isn't actually much special with the images, as those type of surfaces are basic in MuPAD. Maybe the only subtleties are the height lines.

Here would be an example code for a LambertW graph:

z := x + I*y:                                                 // Here I define z as x+iy, so I can use it later

f := plot::Function3d(Re(lambertW(z)),                              //This is the function I'm plotting, Re(W(x+iy))

x = -5..5, y = -5..5, Mesh=[20,20], Submesh=[1,1], AdaptiveMesh=0,  //Here I'm defining the domain of the function, as well as the accuracy of the plot (Mesh and Submesh)

FillColor=[0.6, 0.8, 1, 1], FillColor2=[0, 0, 0.3, 1]):             //These are the two fill colors, following [Red,Green,Blue,Opacity] (all between 0 and 1)

plot(f, //This plots the function

plot::Transform3d([0, 0, 0], [1, 0, 0, 0, 1, 0, 0, 0, 1],    //This is for the height lines : basically, it's the same function but not filled and with different colors
             plot::modify(f, ZContours = [Automatic,15],     // This defines the number of coutours
                             LineWidth = 0.3,                //
                             LineColorType = Dichromatic,    //The rest is self explanatory
                             LineColor = RGB::White.[0.2],   //
                             LineColor2 = RGB::White.[0.2],  //
                             XLinesVisible = FALSE,          //
                             YLinesVisible = FALSE,          //
                             Filled = FALSE)),               //
       plot::Transform3d([0, 0, -1], [1, 0, 0, 0, 1, 0, 0, 0, 0],  // This is for the projection on the bottom plane. 
                                                                   // [0,0,-1] corresponds to a translation (as the bottom plane is at a height of -1)
                                                                   // And  [1, 0, 0, 0, 1, 0, 0, 0, 0] can be viewed as a transformation matrix (the height is flattened)
             plot::modify(f, ZContours = [Automatic,15],           //
                             LineWidth = 0.5,                      // Self explanatory again.
                             LineColorType = Dichromatic,          //
                             LineColor = [0.6, 0.8, 1, 0.4],       //
                             LineColor2 = [0, 0, 0.3, 0.4],        //
                             XLinesVisible = FALSE,                //
                             YLinesVisible = FALSE,                //
                             Filled = FALSE)),                     //     
GridVisible = TRUE, SubgridVisible = TRUE, GridLineWidth = 0.2, SubgridLineWidth = 0.1, 

GridInFront = FALSE, GridLineColor = RGB::SlateGreyDark.[0.2], SubgridLineColor = RGB::SlateGreyDark.[0.1],

TicksNumber = Normal, TicksBetween = 4, TicksLabelFont=["Cambria",10,RGB::CornflowerBlue], AxesTitleFont=["Calibri",12,RGB::SlateGreyDark],

ViewingBox = [-5..5, -5..5, -1..2],

Height=150, Width=150, 

AxesTitles = ["x", "y", "Re(W(x+iy))"])

Sorry for the very messy code, for a strange reason it doesn't seem to display well. (You can always edit the page to see how I wrote it, hopefully it's clearer.)

Hope this is helpful. --Xedi 15:38, 10 August 2007 (UTC)

Thanks for all your answers. Unfortunately, I don't have MuPad, or any other maths software, but I might download the free trial to have a play around. I've found this [3] page, which has pretty much what I'm looking for, although the diagrams aren't quite as aesthetically pleasing as Xedi's. 80.169.64.22 19:03, 10 August 2007 (UTC)