## Professor Comes Up With a Way to Divide by Zero 1090

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!!! (Score:5, Funny)

## Re:Argh!!! (Score:5, Funny)

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!!! (Score:5, Funny)

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!!! (Score:5, Funny)

How many light bulbs does it take to change a light bulb?...One, if it knows its own Goedel number.

## Re:Argh!!! (Score:5, Funny)

None. It's a hardware problem

## Re:Argh!!! (Score:5, Funny)

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

None, we'll fix it in the driver.

## Re:Argh!!! (Score:5, Funny)

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

## Re:Argh!!! (Score:5, Funny)

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

## Re:Argh!!! (Score:5, Insightful)

## Re:Argh!!! (Score:5, Funny)

## Re:Argh!!! (Score:4, Funny)

## Re:Argh!!! (Score:4, Funny)

## Re:Argh!!! (Score:5, Funny)

## Re:Argh!!! (Score:5, Funny)

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

## Re:Argh!!! (Score:4, Funny)

--

Evan

## Re:Argh!!! (Score:4, Funny)

## Re:Argh!!! (Score:5, Funny)

## Re:Argh!!! (Score:4, Funny)

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

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

## Re:Argh!!! (Score:5, Funny)

O(1)

## Re:Argh!!! (Score:5, Funny)

## Re:Argh!!! (Score:5, Funny)

## Re:Argh!!! (Score:5, Informative)

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!!! (Score:4, Interesting)

iright in the middle of it. Complex numbers were discovered, not invented.## Re:Basic math (Score:4, Informative)

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 (Score:5, Insightful)

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 (Score:4, Informative)

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 (Score:5, Informative)

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

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

## Re:Basic math (Score:4, Informative)

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 (Score:4, Funny)

## Re:Argh!!! (Score:5, Interesting)

Reading his stuff, he's proposing an

abstractmachine 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!!! (Score:5, Funny)

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!!! (Score:5, Funny)

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

## Re:Argh!!! (Score:4, Insightful)

Honestly, it's like Cantor never existed.

## Re:Argh!!! (Score:4, Funny)

## Re:Argh!!! (Score:5, Funny)

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!!! (Score:5, Informative)

## Well, thats just nullty. (Score:5, Interesting)

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. (Score:5, Interesting)

## Re:Well, thats just nullty. (Score:4, Informative)

## Re:Well, thats just nullty. (Score:4, Funny)

were deemed so useless when first conceived that they were called imaginary numbersThose of us with an electrical engineering background prefer to call them

jmaginary.## Re:Well, thats just nullty. (Score:5, Insightful)

## Re:Well, thats just nullty. (Score:5, Funny)

## Re:Well, thats just nullty. (Score:5, Insightful)

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. (Score:4, Insightful)

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. (Score:5, Insightful)

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. (Score:4, Interesting)

mostlikely 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 (Score:4, Interesting)

(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. (Score:4, Informative)

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. (Score:5, Funny)

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. (Score:5, Funny)

## Re:Well, thats just nullty. (Score:5, Funny)

yourcountry, but I think that "appalling" describes anything involving them and nudity quite well.## Re:Well, thats just nullty. (Score:4, Funny)

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

## Re:Well, thats just nullty. (Score:5, Funny)

## How can this be a ring? (Score:4, Insightful)

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. (Score:5, Funny)

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. (Score:5, Funny)

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 (Score:5, Interesting)

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 (Score:5, Funny)

## Umm... NaN? (Score:5, Funny)

## YaNaN? (Score:3, Funny)

## Re:YaNaN? (Score:5, Funny)

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

(returns to its corner)

## Re: (Score:3, Funny)

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? (Score:5, Insightful)

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? (Score:5, Interesting)

(nullity)^(-1) = nullity

So division by nullity is just nullity.

## Re:Umm... NaN? (Score:5, Funny)

## Hmm (Score:5, Funny)

## And this is important, why? (Score:5, Funny)

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? (Score:4, Insightful)

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 ... (Score:5, Funny)

## didn't "solve" anything (Score:3, Interesting)

## Rubbish (Score:4, Funny)

## testing, exception handling etc. (Score:5, Insightful)

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... (Score:5, Interesting)

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

deservedlybe a smug American.## Re:Sad, really... (Score:4, Insightful)

## Re:Sad, really... (Score:4, Informative)

You simply don't define infinity and -infinity as numbers.Well, not Reals, at any rate:

http://en.wikipedia.org/wiki/Extended_real_number

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

## Nothing to see here, people... (Score:5, Funny)

Helpful little hint from the end of the video:

Yeah. It was that simple.

I'm just reminded of that proof from way-back-when that 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 (Score:4, Interesting)

## Even I knew this was wrong as a 10 year old (Score:4, Insightful)

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

realidea is that, for an equation 1/x = y, yapproachesinfinity as xapproaches0. 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 (Score:5, Insightful)

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

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 (Score:4, Informative)

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 (Score:5, Interesting)

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 (Score:5, Insightful)

- 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 (Score:4, Insightful)

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. (Score:5, Funny)

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

## new things (Score:5, Funny)

this whole thing is utterly stuipfluous.

## Re:new things (Score:4, Funny)

## the problem (Score:4, Informative)

## Re:Imaginary Numbers (Score:5, Informative)

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 (Score:4, Informative)

## Re:Imaginary Numbers (Score:5, Insightful)

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 (Score:5, Interesting)

completely consistent with all the old ruleswhich 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.

Shouldthere 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 (Score:4, Insightful)

## Imaginary Numbers?! (Score:5, Insightful)

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

badthing? I was under the (apparently mistaken) view that creativity was a praiseworthy trait.## Re: Limits Anyone? (Score:4, Informative)

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 (Score:5, Informative)

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

The second paper [bookofparagon.com] he says:

## Re:Dr. James Anderson's actual papers (Score:4, Insightful)

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

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. In other words, they are defined, but he doesn't like the definition.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 (Score:4, Interesting)

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 (Score:5, Informative)

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 (Score:4, Interesting)

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 (Score:5, Insightful)

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.