Dividing by Zero

I’ve gotten a few emails about a math professor who claims to have solved the problem of dividing by zero.

With the caveat that I am not a professional mathematician, I’m pretty sure this is silly. For one, any time that you have a major, front-page scientific or mathematical result reported by a mainstream news organization that does not contain some version of the phrase “this discovery, which was published in [name of major peer-reviewed journal],” you are probably looking at a news organization that is not doing their job. Until I see a group of mathematicians look over his results and say that they are consistent and significant extensions of the current body of mathematics, I’m really not buying it.

The other reason I’m not buying it is that I don’t see how you can “solve” that problem. The article says that the whole idea is that this can make divide-by-zero crashes go away. This doesn’t really make sense. When I worked with LabVIEW, one thing I noticed was that the result of dividing by zero was propagated through the system as “NaN” — Not a Number. This didn’t make anything work better. It just pointed out that I had made a mistake somewhere. If you have set up an equation where you are trying to divide by zero, you have done something WRONG. You can make the system fail gracefully or not, but that’s a matter of crash handling. Just spitting out the result “Nullity” doesn’t fix things. Sure, you could then make the next part of the program handle “Nullity” as a special case. But that’s not mathematical, that’s algorithmic.

Now, it’s true that this is sounds similar to the way the mathematical community responded to the idea of the square root of minus one being treated as a number, which was only really accepted in the 19th century, or the way negative numbers were dealt with before that. But I don’t think dividing by zero is really the same thing. Until someone can do more than just use a word like “Nullity” to mean “Undefined” (I remember trying that in 10th grade), this isn’t a useful concept, and it’s not going to stop programs from crashing when the programmer writes a function with an equation that’s supposed to spit out a real number and doesn’t. We’ve been using symbols to refer to the result of dividing by zero for years now. They don’t mean anything mathematically and they don’t solve any problem.

107 thoughts on “Dividing by Zero

  1. He’s living my dream. I always wanted to invent a number…. if only I had the talent.

  2. keep in mind, that it isn’t where you end up but how you get there.

    after all, the dirac delta function is a very very useful little beast, with a spike at 0 of infinitely high magnitude and infinitely narrow breadth, however its integral has a value of 1. inf*0=1.
    you can make 2*delta which has an integral of 2, such that inf*0=2 also. 0 is 0 and inf is inf, but their products are not the same.

    of course when all you want is an effective algorithm, there’s little reason to screw around with infinite values.

  3. Any number divided by zero is zero. I’ll explain it how most division is explained, with apples.

    There are 5 apples. If there are 0 people to share the apples with, how many apples does every person get? 0, because there are no people to get the apples.

    Except for the polar bear, but how does he even know what apples is.

  4. In just about every program I’ve ever worked on, we’ve simply treated n / 0 as 0. While not mathematically correct, it’s usually the most practical value to use. (Admittedly the most common case where this comes up is trying to get the mean of an empty collection. Not exactly complicated stuff…)

  5. I tried the same thing when I was in high school. Unfortunately, as in so many aspects of adolescence, there was nobody around to walk me through it. I got confused and embittered. Ah, well.

    The biggest problem with his particular use of NaN is that, unless you’re willing to drop some rules, there’s only one number left. Every number, as demonstrated with beauty and precision by Max (about a dozen posts back), equals every other. Although gorgeously minimalistic and free of contradiction, this imparts no further information.

    You might find this interesting. I was wandering through Wikipedia, and stumbled across Wheel theory, with a link to http://www.math.su.se/~jesper/research/wheels/ . He mentions a system that, although more complicated, at least contains more than one number. He didn’t come up with it, the idea is much older than that. His dissertation, in fact, had no practical arithmetic goal – it was an attempt to reformulate some mathematics following “constructivist” principles, whatever those are.

  6. Something else occurred to me. I may have overstated things by saying “free of contradiction”. The system as he presented it is technically consistent, but it’s entirely inconsistent, even in a technical sense, with the rest of arithmetic. Counting, as formalized by things like the Peano axioms, explicitly assumes the existence of infinitely many distinct whole numbers. The order rules (greater-than-or-less-than) also tend to assume infinitely many numbers to be ordered; they further assume a total ordering, and are inconsistent with his diagram of a point “off the line”.

  7. This is useless, as anyone who has carried on simple algebra through a division by 0 will tell you. you don’t need a symbol to represent it. you do not need i for (sqrt)-1 because you can work with it that way. the i is just convenient. we can see that ((sqrt)-1)^2 is -1 because that is how roots work. we don’t need to know the value of the answer because we have a way out. with N were stuck. no way out. you can carry on algebra after you get an x/0 in there, i have a few times just to see where it would go. but you can never get rid of it, cause multiplying both sides to get rid of it will give you 0=0. that is why well call it “undefined,” because there *is no answer.* every x satisfies the equation (or in algebra, when any given x is undefined, it is because every y can be a “valid” answer for f(x)=y). additionally N has no magnitude (unlike i) so it does not behave like a third axis that compliments the real and imaginary numbers. i+5 != 5i != i, but N = 5N = N+5. in the end this is just a cleaver way of saying “now multiply both sides of the equation by 0. . . ”

    and what? you computer wont error out? duh?

    for anyone who still does not get it:
    i has a defined value equal to (sqrt)-1
    N has an undefined value equal to x/0
    you can undo multiplication (or whatever) with i
    you cannot with N (because it is the same as multiplying by 0, once done you can’t get the original equation/number back)

    he is teaching this to children as “real” math when all it is is an interesting (not really) and not horribly useful or insightful set of axioms in the realm of number theory.

  8. Back in the 70s I took a course on calculus through non-standard analysis which was based on extending the real numbers so that things like dx that went to zero were actual numbers. It was weird. First we extended the integers into hyper-integers that were larger than any possible integers. Then we extended the reals using the inverses of the hyper-integers and some extended arithmetic. We actually managed to do a derivative or two by multiplying by dx. I doubt you can actually divide by zero, but you can divide by dx which can get closer to zero than any real number.

  9. The main problem here is that, unlike i, what Anderson has “discovered” is not a single discrete imaginary concept but more than one thing. In fact, “nullity” is already known and more commonly referred to as “the set of all real numbers.”

    When you multiply any number by 0, you get 0. So where R = any real number, and assuming that division by 0 is possible:

    R * 0 = 0

    R = 0 / 0

    This is usually ignored, since an operation that gives you not one number but every number (except, possibly, i) is not of any practical use in mathematics – so we hold it as a postulate that it is something that simply Must Not Be Done.

    What’s funniest about all this is “perspex space” – Euclidean geometry including a “point at nullity”. When you consider that what nullity actually IS is the set of all real numbers, a “point at nullity” is nothing more than a line in a dimension perpendicular to the graph. Good job, Anderson, you’ve invented both the set of all real numbers and 3D space! I have high hopes for this Anderson, maybe in the future he’ll go on to “invent” concepts like calculus and tetration, but define them in an obscure way and then rename them in order to claim credit for them.

  10. That is assuming, of course, that he relies that he is doing it. rediscovering calculus on your own is actually quite a personal accomplishment, if you can figure out that thats what you have done. of course this is unlikely to take you anywhere others have not gone before. especially when one is marketing it in a way that i would describe as fraudulent. that plus the fact that i don’t think he made the connection that “nullity” is not actually a discrete value, as he seems to be describing it as ( “a point at nullity” is absurd). like i said before, nullity is just x/0, and that is all it ever can be.

  11. So what if dividing by zero would be handled like square root of -1, as proposed? Every number n divided by zero would be nk, where k is imaginary unit vector just like i in normal imaginary units, except in third dimension. So you would have real numbers, imaginary numbers (sqrt of -1) and second imaginary numbers (div by 0). And instead of complex plain, a complex space. This way the numbers divided by zero could be used in calculations just like normal complex numbers.

  12. Me again. After some thought, that link doesn’t really answer the question. If 1/0 = k, it doesn’t necessarily mean that 0*k = 1. Multiply anything with zero, and you get zero, so 0*k = 0. Someone explain!

  13. I once asked my Maths teacher what is the square root of -1. She looked thoughtful for a moment, and said “I’ll get back to you on that one.”

    Is it actually solvable? Google’s calculator spits out “square root(-1) = i”, which I gather means “Stop it, that’s not a real question!” ;)

  14. In response to above, the square root of negative 1 is i, its a conceptual number that you can do all kind of pointless maths with (ok, not pointless, but certainly not everyday adding up the coins in your wallet type math).

    I have to say, Anderson was a lecturer of mine at the University of Reading, and he taught me C programming (a language I have never used, but then again I was also taught Delphi, which has no practical uses at all, aleast C had some). He doesn’t have a lot of support within the Comp Sci department on this whole “nullity” stuff (Or the maths department either), but his perspex machine is an interesting concept. Its worth having a good read of the concepts, then being grateful the Turing was move popular.

  15. I have no delicious pies

    My fictitious pie is multiplied by 0, ergo I have no pie

    pie x 0 = 0

    lets divide by 0 and see what happens :D

    pie= 0/0

    but i also have no cake

    cake x 0 = 0

    cake=0/0

    cake=pie

    so nullity does what to my cake/pie hybrid?

    I don’t really grasp the concept of nullity

  16. But don’t worry, a bunch of students from year 10 at Highdown can get their head around what millions of mathematicians can not!

    But seriously, who’s smart idea was it after “discovering” nullity at Reading University to go off to highdown to teach their theories? Did the mathematicians at Reading School and Kendrick mock him?

  17. To James: You actually seem to have the concept of nullity down pat. It doesn’t make sense.
    To Illarane: In one formal construction of the complex numbers, the symbol i is just a letter, and the clever bit is to take the quotient of the polynomial ring R[i] with respect to the ideal generated by i^2+1. In another, complex numbers are ordered pairs of reals with two binary operations defined on them that satisfy the field axioms. These constructions produce isomorphic fields, which are algebraically closed. (In other words, it’s possible to describe the complex numbers in a very detailed and specific way that leaves no room for fuzzy logic.) Viewing the complex numbers as a two-dimensional vector space over the reals, with the euclidean absolute value, defines a geometry and topology on them that make calculus remarkably convenient. (In other words, they’re really really useful.) The philosophical problems with complex numbers, and people’s feeling that there’s something not quite right about them, comes from the fact that they really don’t solve any problem in ordinary arithmetic. You have to take a step to the left and view the problem from a different angle before they really become useful, sensible, and obvious.
    To Stupid Chemist: You’re not far off the mark.

    There are variations on this idea that do work. Wikipedia has articles on Algebraic Geometry and the Real Projective Line. (Following links leads to more.) It’s possible to jigger the books a bit and avoid the problems with a direct insertion of division by zero into arithmetic. Doing so actually turns out to be a really good move. For instance, drawing graphs on the projective plane allows you to say that “an asymptote is the line tangent to a curve at infinity,” and be very literally right. For an area of algebraic geometry with practical applications, read about elliptic curves and elliptic curve cryptography. For one that’s just pretty, read about the conic sections as extended to the complex projective plane. Bezout’s theorem never looked so good.

    (Micah mentioned something related, but got ignored, so I figured I’d add a bit more.)

  18. 1) Can we stop with the proofs that the system is inconsistent? It’s very, very likely to be consistent, and there’s no simple way to prove it inconsistent (i.e. you can’t fit it in a few lines). You’re just making a fool of yourself when you show your lack of rigor in mathematical proofs.

    2) It makes “sense” in the sense that any provable theory makes sense under its axioms. It can be perfectly valid. The problem isn’t that it’s wrong or stupid, it’s that it’s useless, and it doesn’t seem that it would be useful in the near future.

    3) According to his school’s website, he’s not a math professor. He’s a lecturer in what seems to be computer engineering.

  19. 4) It’s not just useless, but since he’s defining his own axioms, it doesn’t work well with regular calculus, where the limit of x/x as x->0 is equal to 1 and the limit of 0/x as x->0 is equal to 0 (in other words, it would make the two functions discontinuous, which isn’t nice). This isn’t a problem that would make the theory useless, since things like the one-point compactification of the real line doesn’t work with normal calculus at all, and he did say he would define his own calculus. The fact that he can’t give us a practical reason to use it (read: something besides “because he hates not being able to divide by zero”) makes the theory useless.

  20. I was playing around with the audio feature.

    And instead of saying “I am” it says “I A-M”.

    It made me giggle.

  21. I know, it’s inconsistent. You can do stuff like this:

    2*N = 2 * 0/0 = (2*0) / 0 = 0/0 = N
    2N = 1N
    2 = 1

    However, you can do this with i too:

    -1 = i*i = sqrt(-1)*sqrt(-1) = sqrt( (-1)*(-1) ) = sqrt(1) = 1
    -1 = 1

    But, this still doesn’t mean that it’s useful.
    We have limits which do 0/0 perfectly fine.

  22. On second thought, try this:

    0/0 = x
    0 = x*0
    0 = 0

    This is true for every x in |R (the set of real numbers).
    Basically, what he does is give |R a different name and do calculations with it as if it were a number itself, which is just… impossible, leading to all the contradictions above. In short, nullity =/= nullity….

    Quoting himself: “stretching from negative infinity, through zero, to positive infinity”. Definitely |R.

  23. The most intuitive result for n/0 for all n except zero is: infinity
    the problem is: there are two possible results: positive inf and also negative inf.

    if you look at the graph of y=1/x you’ll see what i mean

    but then, i should check if there is a meaningful solution within the range of rubtsov’s new “delta”-numbers :)

  24. Back in the 70s I took a course on calculus through non-standard analysis which was based on extending the real numbers so that things like dx that went to zero were actual numbers. It was weird. First we extended the integers into hyper-integers that were larger than any possible integers. Then we extended the reals using the inverses of the hyper-integers and some extended arithmetic. We actually managed to do a derivative or two by multiplying by dx. I doubt you can actually divide by zero, but you can divide by dx which can get closer to zero than any real number.

  25. Defining 1/0 as a number k, so that 0*k = 1 is not useful. See:

    If 0 * k = 1, then (0 + 0)*k = 1, then you would use the distributive law 0*k + 0*k = 1, but that’s 2!

    So adding k to the reals, you obtain no field.

  26. In the video, he multiplies 1/0 by 0/1, and he calls 0/1 the reciprocal. The definition of reciprocal is a number whose product is 1 when multiplied by a certain number. 1/0 x 0/1 does not equal one. He fails.

  27. What can you achieve with nullity that you can’t with an error message on a calculator?
    Whiteboard showing the symbol for nullity (bottom)
    Dr Anderson’s symbol for nullity (bottom)

    “Nullity has a precise arithmetical value. The trans-real arithmetic is total, and complete, and contains real arithmetic as a sub-set.

    “You can calculate values with nullity and those are meaningful. The arithmetic is simpler than IEEE-float.

    “Trans-real numbers I have defined to be the real numbers augmented with plus infinity, minus infinity, and nullity.

    “What I have done is to take algorithms from arithmetic that happen to work for division by zero, collected them together, developed them as algorithms, proved that they’re consistent, then axiomatising it and proving it by computer.”

  28. It took me an hour to understand it, but what he’s pretty much saying, (and this is brought up in the comments) is that nothing/nothing=everything. It doesn’t make sense, but it makes perfect sense, because dividing nothing by nothing also doesn’t make sense. Just like how physics will slowly find out that nothing is real.

  29. I, for sake of my awesomeness, use a self made explanation:
    1.)since when dividing 2/2 you get 1…
    (
    2.)and dividing 2/0,5 gets 4…
    (
    3.)2/0.25=8
    (
    4.)MAGIC
    (
    5.)2/0.[0]1 would give [9]
    (
    6.) and 2/0= infinity
    wait, what? am I so stupid or so awesomely bright?
    maybe I’m simply missing something

  30. a commentary to my own commentary:
    now I see it’s similar to wikipedias explanation,
    and is supported by physics
    not in the apple/men way
    but what happens if we stop something moving in 0 milimeters?
    the g-force for it would reach infinity

  31. You can make up numbers to solve unsolvable problems.

    The square root of negative one is i. Well, if THAT’S mathematically correct, it sets a precedent for pulling other numbers out of your arse to solve unsolvable equations. He didn’t do anything that isn’t already accepted by mathematics.

  32. For example, division by zero and some rise to power are impossible to perform in arithmetic and algebra, but can be made ​​more accurately, can avoid making impossible their “classic” in another frame axioms, ie a another area of mathematics.

  33. The real error with it is in treating division as if it were a binary operator, which it is in fact not. We’ve all heard it before, but division is only a shorthand for multiplication by the multiplicative inverse of a number, the same way subtraction is just shorthand for adding the additive inverse of a number. The reason that some people wander astray of this is that Real Numbers satisfy the Additive Inverse Property but NOT the Multiplicative Inverse Property, because it instead has the Multiplicative Zero Property. All this is doing is giving zero’s nonexistant multiplicative inverse some name, and it really doesn’t matter whether we call it “nullary”, “NaN”, “throw ArithmeticException”, or anything else.

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