If you’ve been following this blog for a while, you know what this picture implies.

When I started drafting this – before noon – I had not yet been advised of a magnitude 7.0 quake that occurred in the Vanuatu region. By noon, however, I had found that my forecast for the month has been fulfilled: there has been an earthquake of magnitude at least 7 somewhere on earth in the month of August.

(Considering the potential loss of life and tsunami damage, I would be happy to have been wrong… but my forecast didn’t cause it.)

Here’s their usual picture:

And here’s a close-up:

You may think it odd, but I find teaching to be very stressful – and that’s what I was doing for the past 5 days at work. It only happens every few years, fortunately.

I’m still unwinding.

I should talk more about Koenderink’s “Color for the Sciences”… I had read through p. 66 when I wrote about it last week… and then, when I picked it up afterwards, p. 67 introduced “dual numbers”. OK, I know what dual numbers are.

By way of introduction, the most common way of dealing with complex numbers is to write them as

a + i b

and do arithmetic with them assuming that we have associativity, commutativity, and that multiplication distributes over addition. We assume that a and b are real numbers, but i is just a symbol for now.

But we assume that i^2 = -1 (corrected).

As we know, everything works out just fine.

Well, what if we used something other than that?

Suppose we had expressions of the form

a + w b

and we make some assumption about w^2. It turns out that the only three cases of interest are

w^2 = -1: the complex numbers;

w^2 = 1: called the “double numbers”;

w^2 = 0: these are “dual numbers”.

(One thing to note is that, unlike the complex numbers, the double numbers and the dual numbers are not fields: we cannot always do division.)

Here’s what that means to me. To multiply two numbers

(a + b w) (c + d w)

we just treat every symbol as a number and expand:

ac + ad w + bc w + bd w^2.

To get the complex numbers, we set w^2 = -1 and get

ac – bd + (ad + bc) w.

We could now describe complex numbers without ever introducing i, by using ordered pairs. Our multiplication is then just a rule which we impose:

(a,b) (c,d) = (ac-bd, ad + bc).

For the double numbers, we use that same expansion

(a + b w) (c + d w)

= ac + bc w + ad w + bd w^2

but then we set w^2 = 1, which gives us

ac + bd + (ad + bc) w.

Written using ordered pairs, that’s

(a,b) (c,d) = (ac + bd, ad + bc).

Finally, for the dual numbers, we have the same expansion

(a + b w) (c + d w)

= ac + bc w + ad w + bd w^2

but we set w^2 = 0, getting

ac + (ad + bc) w.

That is, written as a product of ordered pairs,

(a,b) (c,d) = (ac, ad + bc).

Koenderink’s description of dual numbers leaves much to be desired: “A nil-squared number is a number whose square is identically zero, although it is itself different from zero. The nil-squared numbers are the nontrivial solutions of the equation x^2 = 0.” (He’s talking about my “w” with w^2 = 0.)

Well, there certainly aren’t any real numbers which are nontrivial solutions of x^2 = 0. (He will proceed to treat nil-squared numbers as “infinitesimals”. It appears that he wants to use dual numbers just to simplify calculus. I have found other physicists doing the same thing. And maybe it can all be made right. But do we need it for color? And why re-do calculus?)

Now, I’m used to the idea of w^2 = 0 when w is not zero – but I don’t expect w to be infinitesimally small because its square is zero. For example, I could work in the integers mod 4 – that is, the set {0,1,2,3} with addition and multiplication done mod 4 – and we do have that 2^2 = 0. And 2 is not small.

(You’ve seen the integers mod 12 all your life.)

Or, since we’re defining strange multiplications, let’s just consider the vector cross product. For every vector v, we have

v x v = 0,

and v can be quite large.

So, the book is still interesting… but I had expected to be making the transition to infinite-dimensional vector spaces, rather than doing mathematics in strange ways.

To put that another way, his strange mathematics seems more of a distraction than a help.

I still find it ironic that I had quit reading before last week’s post just one page before he introduced dual numbers in order to do calculus.

August 20, 2011 at 8:18 pm

Hi Rip,

I believe there is a typo here: “But we assume that i^2 = -i.”

You can delete this comment after/ if you correct the typo.

Regards from finally sunny Victoria, B.C., Canada! No major upcoming earthquakes here, hopefully;) or anywhere else, for that matter…

Sper

August 21, 2011 at 7:22 am

Thanks for the correction.

It turns out that the islands were hit by two quakes yesterday, 7.1 and 7.0 about one and a half hours apart.

August 20, 2011 at 10:14 pm

I really enjoyed reading your post – as I do, generally! Clear, yet sufficiently detailed to grasp an understanding of the concept, yet not too overwhelming. … Still you say you don’t like teaching? Hmm…

In any case, thank you for this post!

August 21, 2011 at 7:18 am

Not that I don’t like it – I just find it stressful.

August 22, 2011 at 8:18 am

As a non mathematician I can confirm that his short tour to the dual numbers did distract me and that it didn’t help me in any way to better comprehend the topic he tried to explain. Moreover as far as I can see he’s not using dual numbers anywhere else in the text nor is he referring to them.

Maybe he just wants to make other people aware of the rich mathematical universe but this way it’s like telling a new chinese word almost out of context.

Again your tour did much better help me to understand the topic and if I would have had it on hand already at the time when I first read Koenderink p. 67 it would have definitely spared me a lot of time researching it.

August 22, 2011 at 4:28 pm

From p. 68 I infer that he wants dual numbers for calculus – lowest-order term of the Taylor series of a function. I haven’t read enough more to see if he actually used it. You may have recognized the answers without knowing how he got them.