Professor Comes Up With a Way to Divide by Zero

54mc writes "The BBC reports that Dr. James Anderson, of the University of Reading, has finally conquered the problem of dividing by zero. His new number, which he calls "nullity" solves the 1200 year old problem that niether Newton nor Pythagoras could solve, the problem of zero to the zero power. Story features video (Real Player only) of Dr. Anderson explaining the "simple" concept."

Argh!!!

[Travoltus]

So much for my $200 calculator.

Re:Argh!!!

[MountainMan101]

My £100 (equivalent $200) will happily divide by Zero. It displays and "E" on the screen which I take to mean 14 in hex. So anything divided by Zero is 14. Apart from Zero divided by Zero which amusingly it consider to be Zero.

In fact, using proof-by-blatant-assertion,

if 0/0=14
then 0*14 must = 0
which it does
therefore 0/0=14
so there !

Re:Argh!!!

[buswolley]

Great, a whole new class of errors just got introduced into my code.

Why is the algorithm producing that? Oh I introduced a nullity.

Furthermore, they shouldn't have called it a nullity. They should have called it a Bush.

Re:Argh!!!

[buswolley]

And a whole new class of bad CScience jokes..That reminds me:

How many light bulbs does it take to change a light bulb?

...

One, if it knows its own Goedel number.

Re:Argh!!!

[killjoe]

How many computer programmers does it take to screw in a light bulb?

None. It's a hardware problem

Re:Argh!!!

[$RANDOMLUSER]

And the corollary:

How many hardware engineers does it take to chage a light bulb?

None, we'll fix it in the driver.

Re:Argh!!!

[aichpvee]

How many flies does it take to screw in a light bulb?

Two. But I don't know how the fuck they got in there!

Re:Argh!!!

[markh1967]

How many Microsoft programmers does it take to change a lightbulb?

None - their manager just declares darkness to be the new standard.

Re:Argh!!!

[wealthychef]

Your "argument from intuition" is not a good one. Mathematics often comes up with nonintuitive results. In fact, that's the point, in a way. Mathematics is a set of rules and a language meant for re-expressing known truths in forms that lead us to realize new truths, all by rearranging things by the rules of logic. Here's an example: exponents were created to describe the number of times you multiply a number together with itself to get an example. 2^6 is 2 multiplied by itself 6 times. So what sense does it make to raise a number to a negative power? Well, ok, it's dividing one by that number a certain number of times. Or how about a fractional power, even more bizarrely? It turns out that devising rules that "make sense" often only make sense in the context of exactly the kind of discussion this guy has, purely within the realm of mathematics. I'm not saying his idea is good, rather that your argument is bad. :-)

Re:Argh!!!

[pointbeing]

How many computer programmers does it take to screw in a light bulb?

Only two - but they have to be really small.

Re:Argh!!!

[suggsjc]

It could bring a whole new meaning to being "turned on"

Re:Argh!!!

[edwardpickman]

Is the light bulb conservative? If so we all know conservatives resist change so it's likely the light bulb will ever be changed. Simply adding more conservative light bulbs will not effect change. Adding an equal or greater number of liberal light bulbs is the only way to effect change.

Re:Argh!!!

[mrogers]

Adding an equal or greater number of liberal light bulbs is the only way to effect change.

Sure, that's what they tell you before the election. Four years later you realise the electricity bill's gone through the roof and it's still fücking dark.

Re:Argh!!!

[JohnFluxx]

Surely the conservative would replace the bulb quickly to put things back the way they are.

The liberal would seek to embrace the new darkness and accuse those who complain as non-pc conservatives who resist all change.

Re:Argh!!!

[JabberWokky]

Whereas a Libertarian wouldn't change the bulb because it wasn't his, but he always carries his own flashlight just in case.

--
Evan

Re:Argh!!!

[AutumnLeaf]

Conservatives would open a no-bid contract for Haliburton, pay the contract, but never verify the bulb was changed.

Re:Argh!!!

[246o1]

Perhaps the conservatives would hire 'scientists' to declare that the light bulb was still on, could never go out, even if it did go out it wouldn't affect us, and that nothing could be done about it anyway!

Re:Argh!!!

[tgrigsby]

Close. A conservative would declare that, according to the Bible, God create light and dark, and therefore the darkness was a sign from God that man had been arrogant to create light. He would then shake his fist and declare that the burning out of the light bulb proves that technology can't evolve, and that fire is an element God never intended for man to tame. Foaming at the mouth, he'd blame the fact that light bulbs came into existence on the gay agenda, screaming that the marriage of light bulb and the socket is a violation of nature, and he'd grab up his shotgun and run around the house shooting all the other light bulbs. Once they were gone, he'd see the street lights, blame them on the terrorists, and shoot them out as well. Running out of bullets, he'd take out a massive loan to pay for more artillery. Running from house to house, he'd shoot every light emitting device in the neighborhood, catching innocent men, women, and children in the crossfire. Soon, so in debt that he'd never be able to pay it off, he'd run out of bullets and stop.

Engulfed entirely in darkness, he'd finally wind down.

Then he'd start grumbling about the darkness, blaming it on the liberals.

Re:Argh!!!

[mrogers]

How many computer scientists does it take to change a lightbulb?

O(1)

Re:Argh!!!

[Bucc5062]

Seems recursive to me. If run it you may fill up the void with light.......God? Is that you?

Re:Argh!!!

[eric76]

There is a common term that refers to the process of dividing by zero to get a nullity. It's called a "stupidity".

Re:Argh!!!

[3rd_Floo]

Computers can't deal with imaginary numbers natively...

Uhh, they sure can. GNU C, for instance, has a complex qualifier.

I think the GP was refering to the hardware level, not an abstract software layer. Where traditonal computers, even those with modern math extensions dont know what an imaginary or complex number is. Normally, two floating point values are used to represent complex arithmetic, however its not a native operation, and still requires some software logic to be accomplished.

Re:Argh!!!

[AxelBoldt]

You can't do quantum mechanics without complex numbers. The Schrödinger equation has a fat i right in the middle of it. Complex numbers were discovered, not invented.

Re:Basic math

[Boronx]

The answer to a / 0 is defined as the limit for a / x when x approaches 0.

So you've proved that f(x) = 0/x is continuous?

lim x->0 (23 / x)
lim x->0 (-5 / x)

Neither of these exist.

Re:Basic math

[Eudial]

The answer to a / 0 is defined as the limit for a / x when x approaches 0.

So you've proved that f(x) = 0/x is continuous?

lim x->0 (23 / x)
lim x->0 (-5 / x)

Neither of these exist.

It's a bad example, because even outside of R, the left and right limits are not the same (one diverges to minus infinity and the other plus infinity).

lim x->0 (23 / |x|)

is better. It is undefined because it exceeds R, one could technically define a set of numbers which includes +=infinity, in which division by zero would be defined.

Infinity is not a number

[raftpeople]

"one could technically define a set of numbers which includes +=infinity"

Technically you could not do this. Remember, infinity is not a number, it is a concept meaning an unbounded limit. There are rules for including it in algebraic equations, but it is still not a "number."

Re:Basic math

[Chowderbags]

The limit of a constant over x as x approaches zero would depend on which direction you're approaching x from. For 23/x, if you approach 0 from the left, you get -inf, and if you approach it from the right you get a positive inf. Really, though, the behavior is better defined as an unbounded number approaching positive or negative infinity.

lim x->0+ (1/x) = inf
lim x->0- (1/x) = -inf

Re:Basic math

[saigon_from_europe]

If you speak about limeses, then it depends how you go toward some value (toward 0 in this case).

For instance, both functions f1(x)=sin(x) and f2(x)=x are 0 for x = 0, but

lim x->0 (sin(x)/x) = 1, as we know.

If you take function like f1(x) = x*sin(x) and other one f2(x) = x then

lim x-> f1(x)/f2(x) = 0.

In these two cases, "0/0" have different values.

When you use division in limeses, the path you take is important, i.e. functions that describe in which way you go toward 0. That's why other posters mentioned continuity and other stuff related to functions, and not related to numbers.

Big breakthrough would be to solve lim x->0 f1(x)/f2(x) for f1(x) = 0, f2(x) = 0.

Re:Basic math

[MouseR]

I'm not a mathetician, but as, in general, any number divided by itself is one (eg 1/1=1, 1234/1234=1, 0.5/0.5=1, etc) it would seem far more sensible if zero divided by zero was also 1.

If what you say is right then I can prove you that 1 = 2.

And that black is white and ... Oh! Zebras!

Re:Argh!!!

[sg_oneill]

Actually Im going to retract unreservedly the crank comment right now...

Reading his stuff, he's proposing an abstract machine as an alternative to the universal turing machine (also an abstract machine) that solves the problem of exceptions in algebra. He's suggesting it has alot of philisophical implications somewhat aligned with the way conventional algebra does. I havent quite grokked the central thesis of it, as my maths is way rusty, but its actually quite interesting.

The 0/0 = nullity stuff is a tragic little misstatement of what he's getting at.

Re:Argh!!!

[RDW]

He doesn't stop there, either:

http://archives.nesc.ac.uk/gcproposal-5/0080.html [nesc.ac.uk]

"It is simply a technical matter to extend this compiler to deal with the
whole of C. I could then cross-compile from Pop11, Lisp, or any other
language for which there is a C source version. At that point I would be
able to produce massive neural nets that implement operating systems, word
processors, compilers and the like. It would be relatively straight forward
to compile Linux into a neural net. This opens up the possibility of doing
research on massively large neural networks. We could then move away from
our toy implementations and start examining useful systems. "

Imagine a Beow...[Error in universe.pl line 15x10^9: Division by zero]

Re:Argh!!!

[grahams]

Imagine a Beow...[Error in universe.pl line 15x10^9: Division by zero]

No wonder the universe sucks, it's implemented in Perl!

Re:Argh!!!

[Bush Pig]

Sounds a lot like the Time Cube to me ...

Honestly, it's like Cantor never existed.

Re:Argh!!!

[Bush Pig]

It's a bug, not a feature.

Re:Argh!!!

[fintler]

So much for my $200 calculator.

wait, you paid $200 for a calculator?

b = $100
a = b
a^2 = ab
a^2-b^2 = ab-b^2
(a+b)(a-b) = b(a-b)
a+b = b
since a = b
b+b = b
2b = b
$200 = $100

They ripped you off. $200 is really only worth $100

Re:Argh!!!

[liquidsin]

if a=b, then (a-b) = 0. going from the fifth line to the sixth line, when you divided out (a-b) from both sides, you were, in fact, introducing a nullity.

Well, thats just nullty.

[BWJones]

His new number, which he calls "nullity"

Well, thats just nullty. :-)

Seriously though, as I understand it, this is simply another mathematical structure that allows a different scalar much like a real projective line, right? If that is the case, then there is nothing really new here and there can be no application or definition with real numbers or integers. Alternatively by interpreting this as a commutative ring, one might be able to extend this to where division by zero does not always get you in trouble, but the precise interpretation of "division" is fundamentally altered. This too is not a new concept.

However, all of that said, I am a bioscientist and my math skills are not as strong as a formally trained mathematician, so I will defer to those here who are stronger mathematicians than I if this interpretation is incorrect.

Re:Well, thats just nullty.

[RodgerDodger]

Perhaps. OTH, complex numbers are an incredibly useful tool in electrical engineering, yet were deemed so useless when first conceived that they were called imaginary numbers.

Re:Well, thats just nullty.

[Calinous]

At first, numbers were integers - what you could count on your fingers. (N) Later on, numbers were fractional - in order to express the sharing of things. (Q) Later on, numbers were negative - in order to express debt. (Z) Even later on, some numbers were found not to be fractionar (the first proved was square root of 2). Enter R However, not every polinomial equation has its solutions as real numbers (see x^2+1=0). The solution to this equation was named i, with the property that i squared is -1. It was called imaginary because no real number had such property, and it is as real as a figment of your imagination ;) While other real numbers can be aproximated by integers, negative integers and fractional numbers (with better and better accuracy), i has no aproximation in any of the previous pools of numbers. In engineering, a useful aproximation for pi is 3. There is no aproximation of i as an integer.

Re:Well, thats just nullty.

[Dachannien]

were deemed so useless when first conceived that they were called imaginary numbers

Those of us with an electrical engineering background prefer to call them jmaginary.

Re:Well, thats just nullty.

[buswolley]

I say this report is Bullshit. What professor, after making a huge discovery, proceeds to teach it to children before presenting it at a seminar of his peers? If these children are his peers, then I suggest he merely drew a symbol and named it 0/0.

Re:Well, thats just nullty.

[somersault]

I thought that was %

Re:Well, thats just nullty.

[Anonymous Coward]

0/0 should be a special case where dividing by zero actually yields a valid real number, and all other divisions by 0 are undefined.

Wrong.

0/x gives 0. Always. And x/x gives 1. Always. Now, try for x=0... That gives 0/0 = 0 and 1 at the same time. That's why it's undefined, usually called NaN (Not a Number).

Anything else divided by zero can be defined as giving infinity or -infinity, which can be used in further calculations just fine, even coming to the correct result.
Example: The angle of the vector (1,0): arctan(1/0)*180/pi = 90 degrees. Works just fine. Not so for NaN, any calculation involving NaN will continue giving NaN.

Re:Well, thats just nullty.

[sg_oneill]

How much nothing can you fit in nothing?

none. one nothing. Ten nothings. Twenty nothings. A billion nothings. Nothing * Anything = nothing.

Its not a number. Its a nonsense.

Re:Well, thats just nullty.

[swillden]

Yes because mathematics is a discipline of arbitrary rules, right?

Yes, actually it is, and there are different sets of rules (aka axioms) that are used. For example, Euclid chose to include the Parallel Postulate [wikipedia.org] among the axioms that define his geometry, but there are various well-developed -- and useful! -- non-Euclidean geometries that assume the parallel postulate is not true. There are many branches of mathematics that modify what most would consider the "normal" rules in various ways. Many of them prove to be useful in the real world, too.

Mathematicians realized a century ago that their work is a discipline of arbitrary rules, and that none of their theorems have any inherent real-world truth or falsehood. Math is simply an abstract model. By choosing the right set of axioms one can create a model that maps well onto various aspects of reality, making it useful for physics, engineering and much, much more. Sometimes the common rule set doesn't map well, and even physicists and engineers use the alternative rule sets mathematicians have devised.

This concept of "nullity" isn't something that mathematicians would call wrong. For it to be wrong, it would have to be inconsistent with the results of whatever other axioms Anderson has chosen to use. What mathematicians would call it, however, is an old, uninteresting idea. There have been many others that postulated a placeholder "value" for infinity and explored the results of that assumption. Some of the results are even occasionally useful in simplifying useful calculations. And sometimes the alternative system produces results that don't map well onto reality, and the distinction between the cases is well-explored and well-understood.

I may be stating that too strongly, though. It's possible that Anderson has adjusted his definition in a way that makes it useful for a broader set of problems. Honestly, though, I doubt it. This is thoroughly plowed-over terrain.

I think it's most likely that Anderson has discovered some specific, important problems in optics(which involves some very high-powered mathematics, BTW, much more so than most engineering disciplines) that can be simplified by postulating a nullity, and that he published the work in an appropriate journal to an appreciative audience.

Re:Well, thats just nullty.

[smallfries]

I think it's most likely that Anderson has discovered some specific, important problems in optics(which involves some very high-powered mathematics, BTW, much more so than most engineering disciplines) that can be simplified by postulating a nullity, and that he published the work in an appropriate journal to an appreciative audience.

Not quite. It's most likely that Anderson is a crank. He has cobbled together some halfbaked assumptions and slung them past an easy audience. If there was a real application for this then a) he would have mentioned it in the paper b) put the paper in a relevant conference and c) not written the discussion section of the paper as if he had reinvented mathematics. He does compare his own paper to the invention of the concept of zero. There is no mention of an optics application anywhere. Further crank-points are earnt by postulating a solution to AI on the frontpage of his site. "Solving the mind-body problem" and whoring his "paper" before the media rather than through credible peer-review. Yes, the SOIP is a very respectable conference, but this is nowhere near their field and why are they publishing something that they are not capable of reviewing?

actually

[Transient0]

i think it is wrong, given his axioms (as defined here: http://www.bookofparagon.com/Mathematics/PerspexMa chineVIII.pdf [bookofparagon.com]).

(inf) = 1/0 [A20]

= 1/(-1 * 0) [T77]

= -1 * (1/0) [A13]

= -1 * (inf) [A20]

= -(inf) [A24]

which contradicts his axiomatic supposition of (inf) and -(inf) as unique entities [T41]

Re:Well, thats just nullty.

[Dan D.]

I only know of one proof for why 1 + 1 = 2, and I've been wondering if there are other proofs. It *is* almost too bad that those abstract concepts aren't taught more at the younger age. I asked some of my nieces and nephews why 2 + 2 = 4 and they essentially showed me the proof on their fingers (although using the whole numbers which makes sense because they haven't really been taught 0 yet...)

Anyway the proof as I know it is this: Define 0 as a number. Define a successor function which takes a number as input and produces a number as output. Then start defining some labels like 1 (doesn't really have to be 1, could be the Symbol formerly known as Prince... just a label... still the same crazy music genius... this, it would be nice if were explained more...) is the Successor of 0, 2 is the Successor of the Successor of 0, 3 and then 4 in the same way. Then finally define + as the following construction: 0 + any number = that any number and S(x) + S(y) = x + S(S(y).

2 + 2 = 4
S(S(0)) + S(S(0)) = S(S(S(S(0)))) by definitions above.
S(0) + S(S(S(0))) = S(S(S(S(0)))) by the second rule of +
0 + S(S(S(S(0)))) = S(S(S(S(0)))) again by the second rule of +
S(S(S(S(0)))) = S(S(S(S(0)))) by the first rule of +
QED

Anyway, ask some 6 year old who knows how to count on their fingers... they'll show you that (holding two sets of fingers on either hand and then counting the "successors" by dropping fingers as they go.)

Re:Well, thats just nullty.

[itwerx]

Seriously though...if this interpretation is incorrect.

Your interpretation is correct but for proper mathematical representation it should be reduced to its simplest form.
      While simpler reductions may be possible I believe the following best conveys the essence of the equation:
      "Dr. Anderson is a pompous idiot."

Re:Well, thats just nullty.

[joestoner]

I am quite sure nudity would be a more appalling number

Re:Well, thats just nullty.

[Jesus_666]

I don't know how math professors look in your country, but I think that "appalling" describes anything involving them and nudity quite well.

Re:Well, thats just nullty.

[mwvdlee]

As I understand it; you take a famous problem (e.g. division-by-zero), give it a new name (e.g. nullity) and claim you've solved the problem.

So, I hereby claim to have solved the well-known Poincaré Conjecture by naming it "frooblewompy". There, problem solved.

Re:Well, thats just nullty.

[kongit]

If you wouldn't mind emailing me your name, address, and credit card number (used only for verification and other stuff) I will send you 1 (one) Nobel prize in the field of mathematics for a limited time offer not exceed 5 days. By accepting this offer you are agreeing that I, the arbitrary nullity, will thus forth be bequeathed of all known possessions you, the numbskull who happens to be still reading this. Furthermore, without further ado, we bring you something completely differential.

How can this be a ring?

[Ayanami Rei]

He introduced a multiplicative inverse for the additive identity (0), and added it to the real number field.
Unfortunately, he just complicates things, because he doesn't define how the + and * operators map up with it (nullity + a = ?)... if he doesn't then he breaks assoc/commu/trans properties (no longer a field then). And of course that number we need additive/mult inverses which may require nullity-prime, and so on, and he's just going in circles.

Re:Well, thats just nullty.

[NoTheory]

I think you mean, "You must be new here."

Although your number is higher than his.

So, perhaps i should say:

You must be new here, because i think you mean, "You must be new here." :)

Re:Well, thats just nullty.

[mike260]

Also, if any plane ever falls out of the sky because its software was dividing by zero, the engineers should be promptly be drug out into the street and shot.

In any case, I'm not sure I see how nullity rectifies the problem.

"Good morning ladies and gentlemen, this is your captain speaking. We're nullity minutes into this flight, and we're cruising at nullity knots, at an altitude of nullity feet below sea level. We've got a nice tailwind blowing along an axis perpendicular to spacetime, so we hope to arrive at our destination (7i-4) minutes early."

Unspoken of, third sign

[salec]

The problem with trying to abstract is that 0 holds no sign. It poses no problem when you multiply with 0, because you don't need to ask about the sign of resulting 0. However, when dividing finite with 0, you know that you have two possible and distant infinite outcomes.

Therefore, if there was 0 and -0, you could claim x/0 = (SIGN(x))*infinity and x/(-0) = -(SIGN(x))*infinity.

Perhaps nullity is used to address exactly this problem of zero's "third sign". There is also similar concept, "infinite complex number", where complex plane is mapped on Riemann's sphere, where south pole is mapped to zero, while north pole is considered "complex infinity". The nullity is "real numbers' only" version of that.

Not everyone's happy

[BadAnalogyGuy]

The professors at 'Rithmetic State were non-plussed upon hearing the news.

Umm... NaN?

[The boojum]

Is it just me or does it sound like he thinks he's invented the NaN?

YaNaN?

[Marbleless]

Yet Another NaN? ;)

Re:YaNaN?

[cyrax256]

Nah... It's more on the lines of "Not another NaN"... heh heh... Not another Nan!, recursive... gettit?

(returns to its corner)

Re:

[Tablizer]

Is it just me or does it sound like he thinks he's invented the NaN?

But he gets the credit because "Nullity" sounds smarter, so Nanny Nan Na to you!
         

Re:Umm... NaN?

[El_Muerte_TDS]

Not really. NaN is: Not a Number.
He proposes to define a new number that doesn't exist (or fit for that matter) in the current system.
But still it's useless, or at least I think it is.

100/0 != 10/0 != 1/0 != 0/0

but he uses the same identifier for all of them, so that would mean:

(100/0) / (1/0) = 1

That goes against the principle of:

infinity / (infinity - 1) != 1

Re:Umm... NaN?

[saforrest]

Sure he did. He said the reciprocal of nullity was nullity:

(nullity)^(-1) = nullity

So division by nullity is just nullity.

Re:Umm... NaN?

[__aaclcg7560]

It's just a warm up before he claims that he invented the Net and comes out with a movie to prove that Al Gore didn't invent the Net.

Hmm

[mdemonic]

There's zero comments yet. Wonder how many comments that is per poster

And this is important, why?

[NETHED]

I can make up numbers too...

What he did was assign the previously "undefined" integer with a defined symbol that means the same thing. Infinity in both directions.

While interesting, the concept has little use.

From the article "Imagine you're landing on an aeroplane and the automatic pilot's working," he suggests. "If it divides by zero and the computer stops working - you're in big trouble. If your heart pacemaker divides by zero, you're dead.".
Now, instead of getting an error message, the computer give a 0 with a line through it, and THEN an error message.

Re:And this is important, why?

[jrockway]

That behavior is a good thing. NULL is not 0 or an empty string -- it means "undefined". If you want 0, write 0. If you want "", write "".

If you add a regular number and an undefined number, the result can't be defined. That's why 1 + NULL causes the entire operation to reduce to NULL. Makes perfect sense and is an important part of relational design.

mod post up by ...

[b1ufox]

mod original post up by 0/0 points :)

didn't "solve" anything

[Doppler00]

He just created a new model, a new rule set, a new abstraction of math to deal with the case of "x/0". In general, dividing by zero is bad for most algorithms. I mean, from a CPU's perspective, I don't see how adding any additional hardware would help.

Rubbish

[Mkoms]

Only Chuck Norris can divide by zero.

testing, exception handling etc.

[bananaendian]

"Imagine you're landing on an aeroplane and the automatic pilot's working," he suggests. "If it divides by zero and the computer stops working - you're in big trouble. If your heart pacemaker divides by zero, you're dead."

This is computer programming ABC: you DONT allow undefined behavious to occur in your program! (especially if your doing MIL-STD Ada for avionics etc.) This guys 'method' is just a form of exception handling that any programmer with half-a-brain could implement.

Sad, really...

[lexDysic]

It's sad that he teaches math and thinks this is a worthwhile concept.

For just one example of why it sucks, he BEGINS by defining: (infinity) = 1/0 and (-infinity) = -1/0.
My conclusion: (0)*(infinity)=1
So 2*0*infinity = 2*1
So 2 = 2*0*infinity = (2*0)*infinity = 0*infinity = 1
And once you know that 2 != 1 and 2 =1, it turns out you can prove quite a bit...

Total nonsense, and the BBC is encouraging it. *shakes head* Although, I've got to say, it's nice, for once in my life, to deservedly be a smug American.

Re:Sad, really...

[blanktek]

Parent is correct. It is truly mind boggling how terrible his reasoning is. You simply don't define infinity and -infinity as numbers. That is not what they are. Add this guy to the list of cranks http://www.amazon.com/Mathematical-Cranks-Spectrum -Underwood-Dudley/dp/0883855070 [amazon.com]

Re:Sad, really...

[weston]

You simply don't define infinity and -infinity as numbers.

Well, not Reals, at any rate:

http://en.wikipedia.org/wiki/Extended_real_number_ line [wikipedia.org]
http://en.wikipedia.org/wiki/Real_projective_line [wikipedia.org]

Nothing to see here, people...

[Lord Aurora]

...move along.

Helpful little hint from the end of the video:

You've just solved a problem we haven't been able to solve for twelve hundred years. And it's that simple.

Yeah. It was that simple.

I'm just reminded of that proof from way-back-when that 2 = 1:

a = b

a^2 = ab

a^2 + a^2 - 2ab = ab + a^2 - 2ab

2(a^2 - ab) = 1(a^2 - ab)

2 = 1

All this guy has done is provide another little fun "proof" that you can use to win bar bets. "Betcha I can divide by zero..."

I suspect

[the_tsi]

Mr. L'Hopital would have something to say against this.

Even I knew this was wrong as a 10 year old

[joe_cot]

Seriously, in elementary school a teacher of mine tried to tell us that 1/0 = infinity

Read up on the definition of division [wikipedia.org]. If for a moment we ignore the "and the divisor is not 0" part of the definition, one of the basic principles of division is:
if a * b = c
then a / c = b, and b / c = a

A fundamental part of his explanation pivots on the following being true:
1/0 = infinity
-1/0 = -infinity

So, according to that, the following would hold:
if 1/0 = infinity
then infinity * 0 = 1
which does not work, for obvious reasons. This I told my teacher in 6th grade.

The real idea is that, for an equation 1/x = y, y approaches infinity as x approaches 0. At x=0, y is undefined, and that's all there is to it.
Secondly, the story promises one thing, and "delivers" another. It promises to tell you how to divide by 0, and instead tells you how to get 0^0 (which is based on the previously mentioned false premises). And the answer he gives on how to divide by 0 is that the answer is infinity, which it isn't! I'd fire the professor that has the gall of teaching this to kids (after probably being laughed out by his colleagues).

Re:Even I knew this was wrong as a 10 year old

[Christianson]

A fundamental part of his explanation pivots on the following being true: 1/0 = infinity -1/0 = -infinity

And for him it is true; he's defined infinity to have these values. He very specifically wants a fixed value for infinity.

So, according to that, the following would hold: if 1/0 = infinity then infinity * 0 = 1 which does not work, for obvious reasons. This I told my teacher in 6th grade.

Nor does this work. Division, in his system, is not the multiplicative inverse, but the reciprocal. So, for him: 1/0 = infinity implies 0/1 = 1/infinity, which does in fact meet our expectations.

Basically, what he's done with his system is come up with a (completely consistent, as far as I can tell from scanning from his website) framework where singularities now have a defined value, which means that all functions are defined everywhere on the real line (or the transreal line, which is what he calls his infinity-and-nullity supplemented system). Which is great, as far as it goes. But there's a big trade-off for this: there is now no longer a guarantee that if both f(x) and the limit at x of f both exist, that they will have the same value. The example he himself gives is the hypebolic tangent at infinity; the limit is 1, but by direct evaluation, it ends up being nullity. To get around this, he proposes a hierarchy of value determinations; a function is defined at a point by its transreal arithmetic value only if a different value isn't suggested by analysis. So tanh(infinity) would be treated as 1, even though working through the definition of tanh requires the value to be nullity in his system.

So in summary, he's defined terms so that division by zero is consistent and workable, but the price is that even relatively simple calculus becomes a lot more complicated. Nor is it all clear that transreal arithmetic will hold up with higher mathematics at all (when infinity is valued rather than defined by limits, how does cardinality work?). So I think he's got to a better job selling it than "it's better than NaN or having values undefined," because I can't see how it is.

The real link

[albalbo]

Submitter couldn't be bothered to do the research, but there is a paper written by this guy [bookofparagon.com] about the concept.

Don't sneeze at it

[mattr]

How does James Anderson's "nullity" differ from Douglas Adams' "a suffusion of yellow"?

Seriously though this is the sort of thing that you don't want to sneeze at, it can sound both inane and brilliant. Anderson is not such a crackpot, I found a presentation [bookofparagon.com] of his on optical computing and an introduction to its underlying theory called perspex algebra ( "Representing geometrical knowledge." [nih.gov]). He seems to be a geometer stating his perspective in the first line of that presentation: "Aims: To unify projective geometry and the Turing machine".

He's a geek hero! Who knows if his nullity will end up just NaN with a British twang or the next best thing to sliced bread and i?

I was unable to hear the realaudio casts but from Book of Paragon, The Perspex Machine [bookofparagon.com] (Anderson mentions transreal arithmetic) and Exact Numerical Computation of the Rational General Linear Transformations [bookofparagon.com] (a mathematical treatise with applications to computer vision and robotics) just glancing I'd have to say the guy seems to be a real mathematician, geek and philosopher-king. I don't know if he's up there with Newton but he at least deserves an honorable mention for his wonderfully witty (and to me as yet inscrutable) naming of the Walnut Cake Theorem (see page 10 of Perspex.pdf). It seems that he was motivated to create nullity in order to make reliable advanced computers that would not barf when asked questions about the universe, and to him "Not-a-Number" is vomit. I'd say read some of his stuff before assigning him to the 9th Hell. Would like to hear what any mathematicians or other people with brain cells over the age of 12 have to think about it. It's okay if he reinvented something but it appears he is trying to make a machine that can handle infinities and other tough numerical concepts with ease, and that's worth something. Oh, that and his quantum computer looks neat.

He's just made "error" an object

[saforrest]

Wow. Looking over the guy's axioms [bookofparagon.com], as soon as you introduce "nullity" the result of all of your computations is nullity:

- the sum of anything and nullity is nullity (his axiom A4)
- the product of nullity and anything is nullity (his axiom A15)
- the reprical of nullity is nullity (his axiom A22)

So, his arithmetic is normal arithmetic, but as soon as you hit nullity anywhere, it's a black hole you can never get out of. All he's essentially done is take the "error state" and add it into the system as an object. You still can't compute anything you couldn't compute before. So yes, he has truly discovered NaN.

Re:He's just made "error" an object

[saforrest]

So yes, he has truly discovered NaN.

Now that my original comment has been modded up, I should say, before anyone jumps on me, that this is not exactly NaN in the IEEE sense. In fact, this whole exercise seems to have been inspired by his own frustrations with the IEEE NaN. Better to say nullity is like "undefined", or some such thing.

If only we'd had this 30 years ago.

[feepness]

I will never forget when I was about 8 years old going up to the adding machine in my grandfather's home office. It was about twice the size of a toaster and made of that old typewriter metal. It looked like it weighed as much as a car and had probably cost as much new. Just to see what would happen I entered '0', '/' and '0'. Without hesitation it began producing line after line of '0', '0', '0' on the paper tape accompanied by a cacaphony of mechanical gears. It became apparent to me in a split second that it had no intention of stopping. Ever. It had come alive and was angry.

I yanked the plug from the wall socket and ran from the room in terror.

new things

[yakumo.unr]

If he can make up numbers, then I cam make up words,

this whole thing is utterly stuipfluous.

Re:new things

[CopaceticOpus]

Hey, come on. Nullity is a perfectly cromulent number!

the problem

[idlake]

The problem isn't that people haven't figured out ways of dividing by zero, the problem is that there are many different ways in which you could reasonably define division by zero, and they are not mutually consistent. Wikipedia lists some of them. [wikipedia.org]

Re:Imaginary Numbers

[Alchemist253]

Uh... are you joking?

Imaginary numbers (specifically, complex numbers, which consist of a sum of a real and an imaginary number, and which comprise the "complex plane") are INCREDIBLY important in the "real world."

I'm just a chemist, not a mathematician, but I am well aware that imaginary numbers are critical in the Fourier transforms used every time I take an IR or NMR spectrum.

Ever do electrical engineering? Circuit analysis is made a great deal easier when you can treat circuit elements in terms of complex numbers. All that "impedance" stuff you hear about capacitors and the like that makes it possible to apply Ohm's Law to LRC circuits.

These also are not merely made up properties, they are fundamental to mathematics and thus (if one believes that math is the language of the universe) physics. For example, certain integrals necessarily yield imaginary results. These integrals are not of some ethereal interest, but appear throughout quantum mechanics. This is why the amplitude of a wavefunction (used, for example, in molecular modeling that allows for practical achievements like better medicines) is not the square of the wave function (or, for that matter, its absolute value) but the product of the wavefunction and ITS COMPLEX CONJUGATE.

If you'd like more examples of the utility of complex numbers and other "random rules," check out Boas' "Mathematical Methods In The Physical Sciences."

Re:Imaginary Numbers

[Koiu Lpoi]

I hate to put it this way, but "It'll make sense when you're older". And by older, I mean when you take a higher math course. What is the square root of -1 equal to then? Nothing? Something? Saying it's "imaginary" is merely a construct that allows us to muck with things. We could say they're "happy fun times" numbers, with the symbol "hft", and it'd mean the same thing.

Re:Imaginary Numbers

[RodgerDodger]

Because mathematics doesn't deal with the real world. Physics does.

People take mathematical tools and models and apply them to the real world because they are useful. However, that usefulness is a lucky accident.

Re:Imaginary Numbers

[lexDysic]

Note: IAAM(athematician). You pose a good question. The game in mathematics, though, is not to "make up random rules so that something that occurs to them suddenly works". It's (broadly speaking) to make up new rules which are completely consistent with all the old rules which allow us to understand a previously mysterious example. This is where "imaginary" numbers succeed tremendously, and "nullity" fails miserably. See my post downthread for why nullity sucks.

"Imaginary" numbers are just the "thingys" which are solutions to polynomials. I.e., mathematicians find it useful to have an answer to the question "for what values of x does x^2 + 1 = 0?" The answers are useful, even though they aren't good at measuring length or breadth or depth or other one-dimensional concepts. They're useful because they allow mathematicians to develop a theory which has answered questions which couldn't be answered before. This is true even though both the question and the answer both lie in the realm of real numbers. Should there be an answer to every question of this type that doesn't use complex numbers? Perhaps, but it certainly doesn't have to be pretty, or easy to discover. Often the shortest path to a "real" truth lies on an "imaginary" line.

Re:Imaginary Numbers

[TorKlingberg]

if I had written a/b * b = c you wouldn't think twice about canceling b out.

If you are doing stringent mathematics, you have to. You can only cancel out b/b if b is not zero.

Imaginary Numbers?!

[Mark_MF-WN]

Are you really that clueless? Complex numbers (the sum of an imaginary number and a real number) have been used in electronics engineering for a yonk's age now. Using infinity (just a symbol that doesn't correspond to any actual number) in equations is a staple of physics, and has been for centuries. Computer scientists perform very relevant proofs about how algorithms will run on very real computers using completely imaginary "Turing Machines" as a proof tool.

ALL Mathematics is COMPLETELY synthetic. That's the whole point -- that's the power of mathematics. You can define any set of rules, any set of axioms, any set of symbols, and start deducing. If the tools you need don't exist, you make them up. Nothing is more valuable in mathematics than a nice, clean, clear definition that increases the expressivity of math. Since math has no independent existence anyway, you can get away with pretty much anything so long as your new system has useful properties. Mathematicians with the guts to make things up as they go along end up with their names in textbooks and attached to great theorems, assuming what they made is conceptually useful (whether nullity is conceptually useful remains to be seen; a written description of the definitions would be nice).

Mathematicians that only do calculations that we already know about and are comfortable with? They're called accountants, and they have no friends. Seriously though -- since when did making up new ideas become a bad thing? I was under the (apparently mistaken) view that creativity was a praiseworthy trait.

Re: Limits Anyone?

[poopdeville]

Infinity isn't a real number. Ergo, it cannot be the limit of a sequence, as the definition of a limit include the priviso that it is a real number.

You can only perform the substitution lim x->a f(x) = f(a) when f is continuous at a. f(x) = 1/x is (very trivially) not continous at a = 0.

Damnit, why is this sort of thing spilling over from sci.math now?

Dr. James Anderson's actual papers

[Bananatree3]

Here's the dear professor's blog entry on this very topic [bookofparagon.com], which links to two papers (ONLY for the mathematically inclined):

The first paper [bookofparagon.com] he describes as:

describes how to divide by zero consistently in a non-trivial way. This shows that division by zero is no longer an error. Amongst other things, the paper explains why the standard model of arithmetic is not valid.

The second paper [bookofparagon.com] he says:

explains how to extend calculus so that it works with transreal numbers. This paper disposes of various counter "proofs" that attempt to show that division by zero is impossible. The paper ends with a very simple equation demonstrating the possibility of division by zero and challenges the reader to accept it.

Re:Dr. James Anderson's actual papers

[slamb]

I just read the first paper. It indeed defines a new number system, but his description is wrong:

[The paper] describes how to divide by zero consistently in a non-trivial way. This shows that division by zero is no longer an error. Amongst other things, the paper explains why the standard model of arithmetic is not valid.

Unfortunately, that explanation seems to have been replaced by gibberish in the copy I just downloaded. Check it out:

Unfortunately, IEEE floating-point arithmetic is not a valid model of arithmetic either. We cannot accept an arithmetic in which a number is not equal to itself (NaN =? NaN ), or in which there are three kinds of numbers: plain numbers, silent numbers, and signalling numbers; because, on writing such a number down, in daily discourse, we can not always distinguish which kind of number it is and, even if we adopt some notational convention to make the distinction clear, we cannot know how the signalling numbers are to be used in the absence of having the whole program and computer that computed them available. So whilst IEEE floating-point arithmetic is an improvement on real arithmetic, in so far as it is total, not partial, both arithmetics are invalid models of arithmetic.

So basically, the two NaNs have subtle semantics (much like his nullity) and don't have a catchy name or reuse a symbol that already means the golden ratio, therefore they're broken.

[NaN's] semantics are not defined, except by a long list of special cases in the IEEE standard.

In other words, they are defined, but he doesn't like the definition.

So a function with some nullity arguments may perform arbitrary processing on them, because they are just numbers. A database record with value nullity is not set to any real value. A time stamp with value nullity is not set to any real time.

Right. Now my airplane won't drop out of the sky, because the thrust calculation that used nullity as an input produced nullity as an output, in a way completely different from the one that produced NaN from NaN before. This new name and slightly different semantics magically mean the right amount of fuel will go into the engine.

Re:Dr. James Anderson's actual papers

[Ruie]

So basically, the two NaNs have subtle semantics (much like his nullity) and don't have a catchy name or reuse a symbol that already means the golden ratio, therefore they're broken.

I think the big difference is that IEEE numbers were designed for practical use (if you got x=NaN you do not want if(x=y) to work) while his definition is designed for ease of teaching - it is probably easier to explain the rule for 0/0 rather than tell the students that in this case you have think what to do.

His example with f(x)=sin(x)/x is the best illustration - his arithmetic happily produces f(0)=NULL while in practice you should never assume that a floating point number is exact and thus the best definition is where f(x) is continuous in 0 and f(0)=1 and if the code is missing this special case it should return an error.

On the other hand, I have never seen an equivalent of NaN or NULL in analytic computation, so it might be a convenient shorthand after all in the similar way how +infinity is so convenient in measure theory. Of course, one big reason for doing analytic computation is that one can use continuity arguments and since NULL or NaN has to be an isolated point this would likely just introduce a bunch of combinatorics into derivations and make everything more complex.

Re:Dr. James Anderson's actual papers

[gomerbud]

Just read his `papers'. While this sounds like it may be an interesting exercise in abstract algebra, I'm very concerned with the effect of this on people who haven't had upper division math.

Axioms of Transreal Arithmetic:
        - The majority of his proofs are done `mechanically' and not provided.
        - He makes a big fuss about the validity of real arithmetic in the `Discussion'. Not a word about validity elsewhere.
        - He seems to equate IEEE floating-point arithmetic with real arithmetic.

Transreal Analysis:
        - This is an _Analysis_ paper with no mention of continuity or epsilon neighborhoods.
        - Doesn't the isolated nullity value cause hell when doing analysis proofs with epsilon neighborhoods?
        - How exactly does one define an epsilon neighborhood around nullity?
        - A picture of the transreal `number line' does not constitute proof.
        - Attempting to disprove other people's counter proofs is not proof in itself.
        - Why not attempt all of the fun proofs and lemmas in an upper division real analysis course regarding continuity, differentiation and integration?

Re:Not just "division by zero", but 0/0 specifical

[rve]

0/0 is special, explained:

Think of a division as the reverse of multiplication:

6 / 2 = 3, which means 3 * 2 = 6

With a division by 0, this does not hold:

6 / 0 = x, there is no possible x for which x * 0 = 6
X can be no real number

However, 0/0 is different:

0 / 0 = x, but no matter what you fill in for x, x * 0 = 0
X can be any real or imaginary number, 0 * x is always 0

This is why A / 0 has no solution, unless A = 0, then A / 0 does have a solution, an infinite number of solutions in fact: all numbers are a correct solution.

This professor didn't invent it by the way. He just seems to be the first to bother explaining it to school children.

It's Not Rubbish

[nathanh]

You cannot divide by zero [wikipedia.org] by definition. It's the property.

That's why he's defined a new arithmetic - he calls it transreal - where division by zero is defined. The PDFs on his website clearly explain what he's done.

It isn't rubbish. In second year high school mathematics they had us "invent" our own arithmetic. We could define whatever operations we like (eg, a funny symbol that would multiple the left hand value by 2 and add it to the inverse of the right hand value) and then we had to prove whether the operation was commutative, distributive, etc. This guy has done the same thing but with a new "number" he calls nullity. He has defined what happens when you add a real to nullity, when you multiply a real by nullity, when you divide nullity by nullity, etc. It's an internally consistent number system.

It's interesting for grade schoolers because it gets them thinking about number theory. Instead of thinking "you can't divide by zero" they instead think "oh, well that's just a law for the real numbers, but I'm not constrained by real numbers, I can invent a number system where division by zero is allowed". That is far more insightful and creative than "you can't divide by zero". A child who grasps that concept has the potential to become a great mathematician. A child who merely parrots "you can't divide by zero" will become a bus driver or a computer programmer :-P

It's hard to explain abstract concepts such as number theory. Congratulations to him for making it look like fun.