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.
December 7th, 2006 at 4:43 pm
Well, it’s clearly nonsense, but perhaps some pissed-off pop musician could popularize the “Nullity” symbol by adopting it as his name.
December 7th, 2006 at 4:50 pm
Allow me to echo your sentiments. There are already several programming languages that have a “nullity” value (NaN), and they don’t make the situation better, they make it worse. If a divide by zero occurs, no useful data can be produced. It’s better to throw an error, thus pinpointing the location of the problem, than to create a junk value and propagate it up to where the user can see it. For instance, see the bottom of this Daily WTF.
And, as you point out, this is not similar to complex numbers. If I square i, it comes out to -1. If I multiply “nullity” or NaN by 0, I get nullity/NaN.
So basically: stupid and useless.
December 7th, 2006 at 4:52 pm
Yeah, I’m having a hard time seeing anything here that hasn’t been thought of before. I mean, it just doesn’t make a damn bit of sense.
How does it help with computer programs. I mean, it’s great if somehow your program doesn’t crash on that operation… I mean, how about a try/catch block or something! But what about what happens when you try to perform another operation with that number:
a / 0 = b;
b++;
Really, it’s context sensitive what needs to happen in that case, so error handling seems always more appropriate, and throwing an error instead of silently returning a non-numerical (to a computer) result for use in other operations is a no-brainer.
Who could anticipate how to overload the mathematical operators to use a NULLITY datatype in a way that would be useful for everyone, and not create even deeper data integrity problems.
How embarrassing for them.
December 7th, 2006 at 4:58 pm
For what it’s worth, a little searching around turns up the following.
http://www.cs.reading.ac.uk/people/J.Anderson.htm
http://www.bookofparagon.com/
http://www.bookofparagon.com/News/News_00012.htm
I’m somewhat confused as to what the two papers have to do with the venue in which they’re being published, but then I haven’t really read through them.
December 7th, 2006 at 5:01 pm
Here’s a good example of the power of NaN
Go to the bottom of the page:
http://thedailywtf.com/forums/post/100851.aspx
December 7th, 2006 at 5:21 pm
[...] The response I wish I could have written. [...]
December 7th, 2006 at 6:10 pm
Apologies to the math professor, and assorted and sundry geeks, but all this did was put this song in my head…
Nothin’ from nothin’ leaves nothin’
You gotta have somethin’
If you wanna be with me
December 7th, 2006 at 6:15 pm
While it’s definitely not the same as taking the square root of negative one, it’s analogous. Before the concept of imaginary numbers came about, taking the square root of a negative number was just as bad as dividing by zero - i.e., undefined. Then somebody said, hey! let’s define this undefined value; we’ll call it i for “imaginary”. Nobody knew what could be done with it at that time, of course, but now many years later we have the whole “multiple roots” concept, cos+i*sin, etc. Who knows what uses we’ll find for “nullity”?
http://xkcd.com/c171.html
December 7th, 2006 at 6:27 pm
Proceedings of SPIE (Society of Photo-Optical Instrumentation Engineers) probably doesn’t count as peer reviewed or major mathematical journal, but… http://spiedl.aip.org/vsearch/servlet/VerityServlet?KEY=PSISDG&smode=strresults&sort=rel&maxdisp=25&threshold=0&pjournals=PSISDG&possible1=james+anderson&possible1zone=article&SMODE=strsearch&OUTLOG=NO&deliveryType=spiedl&viewabs=PSISDG&key=DISPLAY&docID=2&page=0&chapter=0
December 7th, 2006 at 6:37 pm
The stupid thing is, mathematicians *already* work with the result of dividing 1 by 0, in situations where it’s relevant: for instances of this, google “meromorphic function” and/or “Riemann sphere”.
December 7th, 2006 at 7:10 pm
This one, though, might have more serious peer review:
James A. D. W. Anderson
Philosophical Transactions: Biological Sciences, Vol. 352, No. 1358, Knowledge-based Vision in Man and Machine (Aug. 29, 1997), pp. 1129-1139
December 7th, 2006 at 11:59 pm
Hi,
I’m from Brazil and i’m having problems to access your site. I tried from various PCs over different internet providers and did not have success. I can only access trough some web proxy, like i’m doing now. Maybe the IP range from us (200.*.*.*) and (201.*.*.*) is blocked in your host?
Thanks!
December 8th, 2006 at 12:28 am
It’s not really about dividing by zero. He claims he’s developing a mathematical system (I think it’s a field, but I’m way out of my league) over a superset of the real numbers which is completely closed and consistent. And then he goes further and creates a theory of computability around this system and ends up with something that’s supposedly more powerful than a Turing machine.
It’s the kind of thing that would be completely awesome if it checked out, but which I can’t help being skeptical about until it gets a lot more review. To his credit, it looks like he’s gotten farther than most of the pseudomath crackpots.
December 8th, 2006 at 1:55 am
To put it simply, this guy is a moron. The idea that x/0 is nullity is useless without an application for this supposedly-breakthrough-number nullity.
December 8th, 2006 at 2:22 am
As a total math nerd, I can tell you that it’s completely ridiculous.
Using his argument style, with N = Nullity: 0^1 = 0^(1 + 0) = 0^1 0^0 = 0 * N. This should not surprise, because he has claimed that N = 0/0, so it should not surprise that, if we allow that operation, then 0 N = 0.
But now notice that this means 0 (0/0) = 0. If multiplication is still associative, then (0 * 0) / 0 = 0, and we all know that 0*0 = 0, so 0/0 = 0, leading to the statement that N = 0.
That’s the guy’s own math, speaking. N becomes a real number, and that number is zero, based on the operations that we’ve observed him to perform to dance around this number.
Of course, here I’ve simply assumed that N and 0/0 are interchangeable and multiplication is associative. If I assume that N can be added, subtracted, multiplied, and divided, I get even weirder results. Consider:
0 N = 0
0 N N = 0 N = 0
0 (NN - N) = 0
I’ll take the liberty of multiplying by any constant k I wish, yielding:
k 0 (NN - N) = k 0 = 0
k(NN - N) = 0/0
k (NN - N) = N
NN = (1/k + 1) N
N = (1/k + 1).
Evidently, N is equal to any number I want it to be, except maybe 1, if you can add, subtract, multiply, and divide with it as if it were a normal number.
Et cetera. The whole idea is bogus. The limit of x^y is undefined as both x and y go to zero, but is taken to be 1 for most applications (out of utility in certain contexts).
In short, we solved this problem a long time ago, and it wasn’t by saying that 0^0 was an actual number. It was by L’hopital’s Rule plus extras in more advanced mathematics.
December 8th, 2006 at 2:42 am
Actually, he specifically says that (if nullity = N), N * 0 = N, not 0. He also says that N + A = N, for all A, and that N^-1 = N. So basically, once you get into N, you can’t get out. This is how this differs from i: you can have i, 2i, -5i, 4+7i, etc. You can’t do any of that stuff with N. You either have a real number or you have N. It’s not precicely the same as the IEEE NaN, but it’s approximately equally useful: it’s better to have this than to have nothing of the sort in real-world computation, but is completely uninteresting mathematically. And since we already have NaN, we don’t need N.
December 8th, 2006 at 3:13 am
I will admit I don’t have any real number or set theory, but I have enough higher math to read through his actual .pdf of his “Division by Zero” paper (link to it here: http://www.bookofparagon.com/), and it in fact does not look nearly as crazy. Particularly, his system is based around zero times infinity is nullity. Other axioms (to prevent the kind of abuse in the post above me): k N = N, k+N=N, N to the -1 = N, 0 to the -1 = infinity, infinity to the -1 = 0.
December 8th, 2006 at 4:46 am
NaN - Not a Notion (of brain)
xkcd berichtet von einem Mathe-Prof, der das “Problem” des durch-Null-dividierens “gelöst” hat. Lust auf more Blödsinn? Go here!
December 8th, 2006 at 5:33 am
Seems this isn’t exactly a new thing. He published papers on it in 2002 and 1997, and it is yet to take the world by storm.
http://www.bookofparagon.com/Mathematics/SPIE.2002.Exact.pdf
Its getting publicity now because he has been confusing… er… ‘teaching’ 13 year old pupils his theories at some ukian school.
http://www.bbc.co.uk/berkshire/content/articles/2006/12/06/divide_zero_feature.shtml
I doubt nullity is on their GCSE exam syllabus.
December 8th, 2006 at 7:32 am
Making up a number isn’t really a solution, otherwise I solved this years ago with the solution “Twelfty”.
December 8th, 2006 at 10:32 am
The best part of it all is the prof. is extremely self-satisfied. I agree with the point that it isn’t rt(-1), because 0 / 0 is not 1, and as we see in calculus, it ends up being anything we want it to be (well, actually 0/0 is still undefined, but lt x->0 f(x)/g(x) where lt x-> 0 f(x) = 0 and lt x->0 g(x) =0 is what we want it to be). So it’s still undeterminate, and the proposal doesnt’ achieve anything.
This absolutely calls for a comic.
December 8th, 2006 at 11:01 am
[...] Via XKCD I see an English Computer Science prof thinks he’s solved the divide by zero issue. [...]
December 8th, 2006 at 12:47 pm
Trutle gave us a few sites, i looked at one of the papers, it didnt’ make sense, the paper states that 0^-1 is infinity, but shouldn’t it be nullinity being as that would be 1/0? I think there is an obvious flaw in how this works.
December 8th, 2006 at 2:27 pm
It’s certainly ‘possible’ to come up with whatever axioms you want, but is it ‘useful’?
Here’s my axiom: whosoever divides by zero has to send me a cake.
December 8th, 2006 at 4:42 pm
from http://www.cs.reading.ac.uk/people/J.Anderson.htm : “He invented the perspex machine and continues to develop it. The perspex machine is a theoretical computer that is more powerful than the Turing machine.”
Um, no? The Turing machine is universally accepted as the most powerful model of computation known; if this were true, it would have taken the computer science world by storm. And it would have been published in ACM or Theoretical Computer Science, not the Society of Photo-Optical Instrumentation Engineers. He’s a crank.
December 9th, 2006 at 7:35 am
I fail to see how his idea is any different to Hyperreal numbers which are a better and older implementation of a similar system to deal with infinitesimals and infinite numbers
December 9th, 2006 at 1:36 pm
The “perspex machine” can supposedly handle real numbers which Turing machines can only approximate.
December 9th, 2006 at 10:52 pm
Division by zero is absurd, but triviality is quite useful.
http://sciphysicsopenmanuscript.blogspot.com/
There are 3 existential types, not just 2.
1) That which exists.
2) That which does not exist.
3) That for which existence is indeterminate.
Any trivial is of the third type because “The Existence of a Trivial is Indeterminateâ€.
This last statement can be proved quite easily. It says that given any unique object, there is no way to determine if the object is really itself, or if it is in fact a trivial clone of itself. This is indeterminate.
One can exploit this existential indeterminacy of the trivials to make all kinds of unusual models.
http://sciphysicsopenmanuscript.blogspot.com/
But division by zero ? I dont think that this will ever work.
Huang
December 10th, 2006 at 9:22 am
I don’t really see the problem with this. Why define ‘nullity’ when you have the perfectly acceptable ‘infinity’?
Answer: you want to have more rigorous mathematics, or, as I like to call it, you’re into ‘anal-maths’.
I mean, what IS the difference between saying that ‘nullity’ plus any number A (as used above) equals ‘nullity’, and infinity plus any number A equals infinity?
Nothing, that’s what. You can use all the ‘roughly equals’ or nick-names you want, but it is still the same.
So what is the big deal?
1/0 ‘=’ Inf(). If you want to define it so it is something else, sure. Go ahead! But really, why bother?
December 10th, 2006 at 2:00 pm
I would actually like to take credit for that… in high school, I figured that since i = sqrt(-1), and there was that whole i, j coordinate system, that it should be that j = 1 / 0.
Stupid idea now, but if he gets any money form this, I deserve royalties.
December 10th, 2006 at 5:28 pm
=O! XD That’s so far off, from what I’ve read I’m surprised even a GCSE class couldn’t pick up some flaws. It’s like the problems you see where 1=2 or 0=1. It looks good when you first glance at it, but when you study it you can just see it’s a neat little trick.
With sqrt(-1) you can use a symbol to represent it, there’s nothing wrong with using a consisten symbol to make things easier. With his idea though there’s no such usage.
For one thing, taking a/0, I can see that it’s undefined, yet he wants to say that’s a*N which is N yes? Well if a/0=N then a=0*a*N… a*0 is 0 so a=0*N, which from what was said before means N=a…I mean, there just seem to be too many points where his ideas are not working. Simply because an equation with a/0 can be rearranged. That’s the difference between his and sqrt(-1). We can’t rearrange sqrt(-1) in the same way we can a/0, we know that anything multiplied by 0 is 0, so, we can’t just make up things like he has. Ach, well, good try on his part though anyway. =)
December 10th, 2006 at 11:54 pm
I’m surprised that the full text of his articles are publicly available. It has been my experience that any significant, refereed article that one might want to cite is only available through my university library website because my university pays money so that i can access them.
I haven’t read through the papers but I do agree with two things.
1. The publicity is due to the fact that this is being taught to kids.
2. That nullity is just another word for NaN, which is probably useless to a computer program that needed a number. Surely it is better to produce an error.
Who knows, there may be some well defined operations on the reals union {nullity} but either they are not useful or they contradict other things. In any case, the mathematicians are not impressed yet.
December 11th, 2006 at 8:15 am
Good gravy: ah, so the perspex machine is more powerful than a turing machine in that it is not a model of effective computation… Not useful, no reason to be interested.
There is interesting work on real number computation, but of course one has to approximate reals somewhere (and hence be turing-equivalent), otherwise how could you ever (conceptually) build the damn thing?
December 11th, 2006 at 2:03 pm
I agree with you.
Something else that’s amusing, he calls it nullity, I’m forced to wonder if he does or doesn’t realize how confusing that is since in linear algebra/matrix theory nullity is the dimensions of the null space of a given matrix…
December 11th, 2006 at 5:25 pm
“Something else that’s amusing, he calls it nullity, I’m forced to wonder if he does or doesn’t realize how confusing that is since in linear algebra/matrix theory nullity is the dimensions of the null space of a given matrix…”
Given that his entire Perspex Machine theory of computation, in which nullity figures, is based on matrix multiplication, I suspect that this is not an accident.
The Perspex looks cute, it’s basically a spacetime (3D + 1D where t can only go forward) geometrical embedding of a Turing machine with properties of a neural network. His ‘nullity’ is used in its compare/jump instruction / truth table as the ‘don’t care’ outcome. (Every instruction is a read, matrix multiply, store, compare with zero, with four outcomes: 0, =0 and nullity, which seems to be always true.) Its theoretical ‘more powerful than a UTM’ properties come from that it uses true reals, not rational approximations, but as he points out, it is an open question in physics as to whether it is even possible for such a device to exist in our universe. But since when has that prevented a theoretical abstraction from being a useful tool?
The whole theory seems to have grown out of a machine vision / cognition hack, trying to figure out what is the simplest representation of 3D visual elements that can be recognised by a machine under perspective transforms, and then generalising that as a computing apparatus in the same way that the human visual cortex might function as a universal pattern recognition system.
He might be wacky, but then again he might be onto something. Buckminster Fuller and Christopher Alexander also read like crazies with geometry-as-thought obsessions, but they seem to be thinking along similar lines.
December 11th, 2006 at 10:59 pm
Amid many posts saying “haha how silly,” Snipergirl points out that mathematicians have devised various systems for reasoning about infinitesimals and infinites, so whatever the merits or demerits of Nullity, the idea of such a system at least isn’t silly.
Gabe said, “The Turing machine is universally accepted as the most powerful model of computation known;” This is not strictly true. While they are equivalent to all other models which can be built/run, Turing himself went on to develop models of more powerful machines called Oracle machines. Just assume a machine can answer the halting problem. What else could it do? What could it still not do? Repeat that sequence indefinitely.
The models may not be practical but they are of interest mathematicians and theoretical computer scientists who like to climb what I believe they call the Borel hierarchy. But now I am over my head, or under my feet as the case may be.
December 12th, 2006 at 4:09 am
Ha! I *knew* I was ahead of my time! Take *that*, Miss Savitske! In BC Calc in high school I used to annoy my math teacher by insisting that infinity was a number (she insisted that infinity was not a number and as such, did not exist).
Anyway, figure that if R / 0 = INF then INF * 0 = R (where R is any real number). Introduce a new, magical symbol called ZAN, written as a three-way figure-eight (or a zero lazily mushed into an infinity).
ZAN is infinity * zero, and it is equal to any real number. Thus it can be used as a correct answer for ANY math problem!
She really started getting pissed at us when we’d actually write the symbol for any problem we didn’t know the answer to. ;-)
December 18th, 2006 at 4:35 am
Well, this really is silly. Division by zero is nothing new, and is an absolutely integral part of basic calculus. No calculus would be possible if we didn’t know how to properly divide by zero. This guy is basically attempting to solve a non-problem in mathematics.
December 21st, 2006 at 9:44 pm
The problem is that this guy’s sloppy. He says in his discussion of trig functions, “We accept nullity unless there is good reason not to. For example, we accept a limit where it exists.” However, just about everywhere else, he takes nullity as the answer in place of established limits.
December 29th, 2006 at 12:22 pm
I came up with that when I was thirteen and first learned about the square root of negative one. It seemed obvious that you could do the same thing when you divided by zero, i.e. assign the result a letter or symbol and work with it that way. Then my dad had me try to actually do anything useful with my discovery and it all fell apart. :D
Now that I think about it, maybe I was just pissed about the square root of -1, and trying to prove that it was stupid. (”If you can do that, you could do the same thing for 1/0!”) I did mention I was thirteen, right?
January 2nd, 2007 at 10:48 am
Seems to me that nullity is, in and of itself, utterly worthless.
i is useful (practically) only when it cancels out, and nullity’s NaN-like propagation through equations makes it impossible to cancel out, even by dividing it by itself.
If his magic Perspex Machine (c.f., any Great Glass Clock in literature/bad made-for-tv films with Paul McGann) works, then that’s different. But I doubt if 13 year olds will understand it anyway, so teaching them nullity is self indulgent and irresponsible, and for that reason alone we should mock him.
January 2nd, 2007 at 10:50 am
Oh, and a good way to mock him is by repeatedly asserting that nullity is “not a real number” around non-mathematicians who don’t know what the strict definition of “real number” is. He’ll have a hard time explaining his way out of that, I reckon.
January 10th, 2007 at 1:45 am
If you want to save yourself the time of reading his articles, here’s a summary:
Nullity = NaN
1/x is not a function, but rather a relation (it equals infinity AND negative infinity, but not at the same time)
Nothing he says is technically wrong, but that’s because he defined any problem areas as being equal to nullity in his axioms.
He basically took anything that was undefined, and renamed it nullity and said that it was now defined.
The rest is my personal rant about this:
The closest thing to useful he gets is differentiating between infinity and negative infiniity. Whether or not that’s actually useful is really quite debatable, but since infinity can’t really come up physically, it’s probably not problematic.
That and using his nullity as an error checking to prove why the old x^2-x^2=x^2-x^2 –> 2x=x doesn’t work could be called useful. (Saying division by zero isn’t allowed works too however, just a different way of saying the same thing really)
As many have stated, the main problem is that once you “enter” the nullity values, you’re stuck. No operations reverse nullity, and until an operation turns it to either 0 or 1, then it’ll be useless.
Consider the usage of i in phasors. Phasors come in handy, but this is because a*e^b*i lets us take b as an angle. For nullity, as long as f(nullity)=nullity for all functions (as far as I’m concerned, he demonstrated this fairly satisfactorily), then we can’t use nullity for a conceptual approach the same way we do with i.
Not sure why I’m bothering, but in his defence, anyone who claimed that he has contradictory statements (i.e. the a/0=N therefore a=0*N=N) didn’t follow the axioms he set properly (mostly by skipping the steps in rearranging). His math is all right… just useless.
(i.e.
a/0=N
0*a/0=0*N
N=N)
So remember, when your weather program can’t handle a small exponent, instead of getting NaN degrees outside, this guy wants you to say Nullity degrees outside. So much more meaningful….
Steve
January 21st, 2007 at 8:12 pm
1) 0=0^1=0^(1+0)=(0^1)*(0^0)=0*(nullity)=(nullity), so 0=(nullity).
2) a+0=a
3) a+(nullity)=(nullity)
so EVERYTHING EQUALLS NULLITY! Am I missing something here?
January 26th, 2007 at 5:06 pm
Lets explore this
0/0 = x
0 = x * 0
So logically thinking, what times zero equals zero.
The answer is every number.
Yes “Nullity” equals any number.
all numbers real or imaginary satisfy the equation.
This is why its called undefined. Because IT IS… slapping a title on it
should come with some real definitions.
ie. Nullity equates to any and every single number. Meaning Nullity is One and it is also two.
its not a variable because that would imply it ISNT particular numbers. But Nullity is every number. (1 and 2 and 10 and 50 and 12^5434 and 1.24×10^99283 and i and -i, and all the other numbers too)
If they plan to define Nullity, its a lot more than just 0/0 …
But the special mathmatics means it cannot exist in any normal equation,
Nullity = 1… Nullity = 2. But 1 != 2. Thus nullity is kinda useless.
January 27th, 2007 at 3:42 am
Now that’s what you call a perfect response to this idiocy. The guy has so little credibility in this field that its scary that he gets this much attention for what he said. Say something that somebody who either doesnt care or isn’t paying attention and you can get some coverage on the BBC it seems. Nullity is just a fancy word for “undefined.” Blatant idiocy. Sigh.
February 10th, 2007 at 11:48 pm
I was mucking around with this sorta of thing earlier this year (I’m sixteen)
I was able to get as far as
The relation y = x/0 + c graphs as a vertical line, on x=-c
As well as some extensions of that before I got bored.
Called 0/0 ‘om’ (or that hindu symbol with the 3 in it) meaning ‘everything’ or something like that.
Still insisted that n/0 where n 0 wasn’t possible.
Basically, this guy is doing the same thing as every ambitious teenage maths student ever. (I don’t even do the highest maths, my friend does and he taught me about i)
February 11th, 2007 at 1:32 pm
As a computer science student at the University of Reading I had the pleasure to listen to him directly at one event. He was discussing his theory and is facing objection even by his colleagues at the university.
So far “Daniel Axelrod” got it best. He’s developing a so-called “Perspex machine” which is a completely new way of computing. Using it a division by zero won’t be an error anymore.
You can read more at his website http://www.bookofparagon.com where you can also find papers of his theory.
All best,
Misel
February 28th, 2007 at 12:16 am
Hey,
There’s nothing wrong with leaving it undefined. The typical real number multiplication group omits 0. There are plenty of groups with undefined inverses. Try multiplication modulo 8, besides the obvious 0, the number 2, 4, and 6 also have undefined inverses (try it/prove it if you don’t believe me, see if any number mod 8 will result in 2, 4, or 6 =1).
Do we try to create a new map that would include this element and have inverses? Maps of such can be created (since, for example, the set of rational numbers is an infinitely countable group, the rationals combined with {0} is countably infinite, therefore there is a bijection between the rationals and integers, and the integers are isomorphic to any infinite cyclic group and we can go from there), but they aren’t as useful as the maps we already know. In fact, in order to create the complex group, we didn’t just add…..
You know what, I have to go, let me just say that mappings are possible. BELIEVE IN MAPS. (btw, my first thought when I saw your spam protection was, “which moduli?”)
March 5th, 2007 at 5:46 pm
seems like extreme abuse of notation to me…
May 30th, 2007 at 10:43 pm
He’s living my dream. I always wanted to invent a number…. if only I had the talent.
July 3rd, 2007 at 12:39 pm
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.
August 4th, 2007 at 11:05 pm
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.
August 7th, 2007 at 8:59 am
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…)
September 24th, 2007 at 1:20 pm
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.
September 24th, 2007 at 3:01 pm
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”.
October 4th, 2007 at 11:13 pm
Turing
October 18th, 2007 at 7:08 pm
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.
November 22nd, 2007 at 12:27 am
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.
December 3rd, 2007 at 11:13 pm
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.
December 5th, 2007 at 4:04 am
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.
December 19th, 2007 at 5:58 am
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.
December 19th, 2007 at 6:01 am
I shall immediately answer my own quwstion:
http://www.nutters.org/log/div-zero
December 19th, 2007 at 8:15 am
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!
January 12th, 2008 at 8:58 am
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!” ;)
February 18th, 2008 at 8:58 am
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.
June 4th, 2008 at 6:49 am
you can divide 0 by 0
June 27th, 2008 at 10:13 pm
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
July 6th, 2008 at 7:37 pm
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?
July 24th, 2008 at 5:22 am
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.)
July 31st, 2008 at 3:31 am
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.
July 31st, 2008 at 3:40 am
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.