### Nullity and dividing by zero by professor James Anderson

Recently this was highlighted at Slashdot and Digg.com both.

A math professor James Anderson has made a new 'number' or entity that he calls NULLITY, in order to solve problems such as 0/0, which traditionally is left undefined.

Basically he first defined 1/0 as infinity, -1/0 as negative infinity, and 0/0 as nullity, using Φ (Phi) as a symbol for it. He said this nullity lies outside our normal number line, not on it.

I watched the video shown on BBC news, and there he went on to show what happens with the "age-old" problem of 00, (zero to zeroth power) using just normal rules of arithmetic plus these definitions:

 00 = 0(1 − 1) = 01 × 0-1 =(0/1)1 × (0/1)-1

Now, for every number, the 1st power is the number itself, while the -1 power is its reciprocal:
= (0/1) × (1/0) = (0 × 1) / (1 × 0 ) = 0/0 = Nullity.

Anderson said in the comments following the main article on BBC that two other professors have helped him develop axioms for this new theory, and one of them checked them for consistency.

So.... a new theory. It doesn't sound that earthshaking to me; in fact I wonder if somebody in times past hasn't already tried this...?
(And yes, there exists a system for an "extended real number line".)

In the BBC 'divide by zero' article, you can leave comments, and lots of people have. Most of those seem to be on the mocking angle, putting down the theory.

That truly disappointed me! The same attitude seemed to prevail in people's comments at Digg.com.

Haven't we learned? The best I can remember from math history, negative numbers were certainly disliked and resisted for a long time, becore accepted by mathematicians.

And similarly when complex numbers came on the scene - Descarteds coined the term 'imaginary' as a derogatory term.

It seems that no one knows an application or use for this idea - at this time. But so what? It's no reason for ridiculing someone's theory. He has even consistent axioms written for it.

People have found uses for negative and complex numbers; they've proven very useful. Time will tell if this theory is of use or not; that's not my expertise or business at all. But I just wish people wouldn't be so quick to judge.

Anonymous said…
Nullity is silly, but triviality is very, very important.

Math will always suffer from blunders like this until a formal theory of triviality is adopted.

First, "The Existence of a Trivial is Indeterminate". This is very important and easy to prove. It says that you cannot prove whether an object is really itself, or if it might in fact be an identical clone of itself. That this is strictly "indeterminate". This is also true of numbers, etc. There are many simple proofs of this.

It is clear that there are 3 distinct existential types to consider:

1) That which exists.
2) That which is nonexistent.
3) The trivial, where "existence is indeterminate".

James Anderson was after something but he missed it, and this is what it was. Triviality.

Triviality is an extremely useful concept and thus far is not part of mainstream math. Eventually, this will change.

Here is a neat thing to consider, a clever trick using triviality :

I have zero bananas, and Bob has zero kiwis. We have the same thing, we have zero. So, when is a banana a kiwi ? When you have zero of each !

This is triviality in action. I think it's pretty cool.

Cheers -
Becky
Markov Chaney said…
Maria, I would just point out that actually 0^0 and 0/0 are not undefined in traditional mathematics, whereas x/0 where x is any real non-zero number IS undefined. The former, however, are "indeterminate." I'm not a mathematician, but here is my best shot at explaining the difference.

If in the case x/0, there were a real number, z, that this equalled, we would be saying (by the fact that multiplication is the inverse operation for division), that 0*z = x, where x is non-zero. There is no real number z that has this property (all real numbers times 0 are 0, by the multiplicative property of zero). So the operation of division of a non-zero real number by 0 must be undefined.

However, with 0/0 (and 0^0, which can be shown to be the same thing as 0/0), let 0/0 = y. By the same argument, we are claiming that 0*y = z. What is y? ANY real number, by the multiplicative property of zero, again). WHICH ONE? It can't be determined.

So this, I hope, highlights a key difference between undefined and indeterminate.

As to the mocking of the nullity concept, that seems all-too-typical. I've not thought about it, but I would't dismiss it out of hand without a lot of thought and conversation with others. Lots of wise fools out there seem awfully quick to make fun without considering that history may see them as the fools in the long run. Thanks for posting and bringing attention to Anderson's idea.
Anonymous said…
Suppose you define 1/0 as lim (a -> 0) 1/a, the limit does not exist. lim (a -> 0+) is +inf and (a->0-) is -inf. So, it does not really make sense to say 1/0 is +inf instead of -inf. For this definition, 1/0 is indeterminate too.
Anonymous said…
Maria,

From mthforum.org, it seems that Dr. Anderson's derivation of 0^0 is invalid.

The fatal step is at
0^1* 0^-1 = (0/1)*(1/0) = (0*1)/(1*0)

0^-1 is the multiplicative inverse of 0 which he has already defined as +infinity. Therefore what we really have is 0^0 = 0 * +infinity.
And 0 * +infinity is also an indeterminite form so we cannot proceed onto the 0/0 step.

Therefore the assertion that 0^0 = nullity is fallacious.

Check out the 'extended real number line' for a system similar to Dr. Anderson's system that is very careful in defining what is allowed and what is not.

Other problems include:
Real numbers are not closed under division (no zero divisors). By introducing +infinity, -infinity and nullity, Dr. Anderson makes his system not closed under addition, subtraction, multiplication and division as well!
Maria Miller said…
OK, that sounds right! I'm learning! Classifying some things as indeterminate and some as undefined makes sense.

I hope Dr. Anderson gets to read this last comment himself to see the 'fatal' error...
Markov Chaney said…
I need to correct my second equationn. Letting 0/0 = y, we are claiming that 0*y = 0, but we can't specify y, as all real numbers satisfy the equation.
Anonymous said…
Actually by the axioms of transreal arithmetic, derived in the papers, the proof holds. But the application of nullity remains a mystery...
Anonymous said…
How can you claim that by the axiom of transreal arithmetic, the proof for 0^0 holds when it the flawed step has been pointed out?

'0 * +infinity' has not been established yet from his axioms therefore it cannot be used sneakily in his proof.

I suppose you can say that he can define '0 * +infinity' as another axiom in his system but that becomes another new system that also has to be proven consistent. Ie. you should not be able to derive a contradiction from just the axioms eg. 1=2 or +infinity=nullity, etc.
Anonymous said…
By the way Dr. Anderson claims to have fixed the fatal step in the first 0^0=nullity proof through another way.
This supposed new proof is not presented on the BBC website.
http://www.bbc.co.uk/berkshire/content/articles/2006/12/12/nullity_061212_feature.shtml
Anonymous said…
0*infinity = nullity is an AXIOM (A16) of transreal arithmetic.

According to Dr Anderson, the system has been checked for consistency by hand by himself and by machine at another unnamed university. I have a Mathematics degree from Oxford and am studying for a Masters in Mathematical Modelling. So far I can't see anything immediately wrong with the system.

The 'flawed' step is only a problem for real numbers, not transreal numbers.

The difference between real numbers and transreal numbers is the different axioms each system uses.

0*infinity is indeterminate only in the real numbers. This proof is not in the real numbers, it is in the TRANSREAL NUMBERS.

Reference : http://www.bookofparagon.com/Mathematics/PerspexMachineVIII.pdf
Anonymous said…
Mr. Math from Oxford: then the derivation for 0^0 = nullity comes from one of his definition of nullity not from his derivation using rules in the real number system. Fine, say so rather than do an incorrect derivation to give it a sheen of respectability. A bit like arguing that I'm right because I say so in my axioms which I admit is perfectly acceptable in math.

So 0^0 = nullity is a theorem in his system but his derivation is still incorrect (hint: his system is not a field or even a ring)
Stephen said…
Hi! Quite a few people now have incorrectly assumed cancellation rules when trying to find flaws in Andersons's system. That is an easy mistake to make because we are so used to working in a field and the guarding conditions are simpler. Anderson's arithmetic does not form a field though, which is the tradeoff for admitting the transreal numbers.

Anderson's transreal arithmetic is more similar to wheels (see wikipedia's entry under Wheel_theory) which are not fields either. The key difference between those two is that wheels define multiplicative inverses everywhere, which in turn implies that there is no difference between +infinity and -infinity. Anderson's arithmetic preserves that distinction.

Anderson's arithmetic shows a lot of influence from the work on IEEE floating point numbers. The principal difference between those two is that nullity is a number but NaN is not. As a consequence, nullity = nullity but NaN /= NaN. NaN is more like an error state but nullity is a sensible value.

The most interesting aspect of this unfolding story has been the social one. The hostile reaction to Anderson's work is all too often personal and frequently accompanied by "counter-proofs" with elementary mistakes.

It seems to me that we have become over-sensitised to quackery. And when something comes along that doesn't fit the standard, we challenge it reflexively as if it was quackery, without pausing for consideration. It is a kind of intellectual laziness in an overloaded world.

For me, the moral of this story has been that debunkers should always be polite - just in case, as appears to be true here, that the "nonsense" is nothing of the kind.
Anonymous said…
Mr anonmymous,

If you have a reason why his derivation is wrong, I think you should write to Dr Anderson and tell him why. And post the reason on this blog. We want to know!

Maths from Oxford
Anonymous said…
By the way on your other point in his derivation he is not using rules from the real number system in his derivation. He is using rules from the transreal number system as I said already.

(Example : 1/0 = infinity is not a rule of real arithmetic. 1/0 is usually undefined in real arithmetic.)

When you say he's arguing I'm right because I say so in my axioms you're right. That's why I think the application of nullity remains to be seen.

I also think that it's irresponsible to start on a proof in front of a group of kids without first explaining how and why your system is different to the normal rules they learn.

Maths from Oxford
Zero vs Nothing: I would suggest this
post by Lawrence Gage.
Anonymous said…
The reason why 99.9% of the people put down this guy is that his work has been done over and over and over again by hundreds of people in the past and this "nullity" concept is a known fact.

In programming, it's called NaN (not a number), for example.

In mathematics, there is also something called the Extended Real system which incorporates the infinities as numbers.

Basically, once you get into the "nullity" concept in an equation, there is no way out, so it doesn't help anyone produce anything useful.

The ridicule was the fact that he had not consulted other mathematicians who would have easily known such previous facts. The other major ridicule was directed at BBC for publishing this. News reporters always jump on subjects that say "ooh, zero reinvented" or "speed of light broken" without consulting other scientists first as to whether or not the work has been TRIED and TESTED in the past. it has nothing to do with ridiculing new theories. his work is just plain not new.

Great new concepts in any scientific field tend to show up in research journals first. Even the great geniuses of the past published to journals about their findings, not the BBC.
Anonymous said…
http://phebus.journalintime.com/forum/2006-05-02-zero

0phi=1
i^2=-1
i^2=+1
e^x=-1
Anonymous said…
I solved it! I solved it!

Let's say 0/0=nullity (where nullity is an alien number )

0/0=(0-0)/0=(0/0)-(0/0)=nullity-nullity=0 !

p.s. I'm an alien, not an idiot !