what's this i number, anyhow?

Oh my, I never used them for anything "useful". This Prof was into General relativity, and he made up some of his own forma lism with quaternions... I can't really recall, a long ago math physics class. I do remember that I *didn't* get the hgihest grade in the class, 'cause on the final I screwed up this min-max problem of finding the shortest distan ce between two points on a shpere ... I knew what the answer was... but sta rted out by writing all the equations in Cartesian coordinates! (duh) And t hen proceeded to fill pages with trig equations. "Times up" Funny, how it's ones mistakes that make the biggest impression.

George H.

Huh? What's your experience of -5 apples? That's also an abstraction. To deal with all the solutions of algebraic equations we need not only positive and negative numbers, but also imaginary numbers... no big deal.

The harmonic oscillator is probably one of the most important model systems. Can you solve that without imaginary numbers*?

George H.

(*I was going to say that imaginary numbers only make it easier...)

Or even -2 fishes in the Dirac three fishermen puzzle:

I'll see if we can get that corrected.

The most famous graffiti of this kind is about Quaternions:

As a practical EE issue, complex numbers give us a way to deal with phases. Geometrically, the equivalent is 2D plane geometry, a measurement system that has orthogonal axes, where rotations move things within the orthogonal measurement system. Constellation modulation arranges information snippets in an array in the real and complex spaces, lighting up squares on a checkerboard to encode multiple bits per baud.

So, what's the 3D version of "real" and "imaginary"? The third axis? I can't think of an electrical equivalent of a 3D signal, at least on a single wire.

Sensor fusion for vehicle motion in 3D. There are a lot of different ways to deal with 3D rotation; they're all messy and highly nonlinear. The quaternion, for all that it presents the system designer and software guy with a boatload of 'i's to dot and 't's to cross, and innumerable ways to mess up, seemed to be the one that would be the most direct in the end.

Apparently they're also popular in the 3D graphics community, for many of the same reasons.

And interestingly enough, to some extent the quaternion echoes a real- world practice: it turns out that you cannot build a three-axis gimbal that will let you rotate the payload to point in any arbitrary direction without running into "gimbal lock" at some orientations. Gimbal lock happens when two gimbal axes line up, and at least one of them has to rotate infinitely fast to allow the payload to sustain a finite rotation speed.

However, there are a number of mechanically robust ways to make a 4-axis gimbal that will let you point the payload arbitrarily, and take arbitrary trajectories between points, all without gimbal lock.

Turn to math: there is no way that you can represent a 3D angle with just three numbers that does not involve singularities, where at least one of the derivatives in your vector goes to infinity at some rotational angle. This is proven (see "Hairy ball theorem"). However, quaternions will do the job just fine -- and quaternions have four numbers.

You and AlwaysWrong share more than your compulsions, Blobbs.

A mashed root?

multiplication because 2 is not an irrational.

No that does NOT follow. Properties of a set are not universally inheritable to all possible subsets.

?-)

multiplication because 2 is not an irrational.

They need to be closed under the algebraic operation in order to even have an arithmetic. Abstractly the operator ( addition or multiplication) is a f unction O: (R x R)->R , a function from the Cartesian product of R with its elf into R. Using the sqrt(2) example, O:(IxI)->I fails where I is irration als, O is x operator because 2 is not an element of I.

I guess not in general, but I believe it can be shown to follow in this particular case, because the set of irrationals is dense in the reals.

Maybe there's some confusion about what we're calling the axioms of arithmetic; I'm thinking of the following:

just

Ahem

most ^ +

the same

positive and negative numbers, but also imaginary numbers... no big deal.

systems. Can you solve that without imaginary numbers*?

know it, is

Yes, your lack of experience is not mine.

?-)

There is no precise term "axioms of arithmetic", but it would usually refer to arithmetic based on the Peano axioms. These are defined for natural numbers, not reals or irrationals. If you want to extend these to the reals R or subset S thereof, then it only makes any sense at all if "+" and "*" are functions on the subset, mapping from S x S into S. In other words, for a subset S of R to "satisfy the axioms of arithmetic", then S must be closed under "+" and "*" - otherwise the "axioms" make no sense.

And obviously the set of irrationals is not closed under either addition or multiplication - density is irrelevant.

I think you have been mislead here. There is no such thing as the "axioms of real numbers" - that page merely lists some properties of what we call "the real numbers". An "axiom" is a fundamental assumption

- we simply state that it is true, and use it as a base for building more complex mathematics. The list on your webpage is a set of desired properties for real numbers - we can use the axioms of Peano set theory to /define/ real numbers that have these operations and properties.

The webpage here also fails to give the definition of the "+" and "*" operations - had it done so, it would have said that the set was closed under these operations.

Just checking if anyone was awake :-)

So addition and multiplication for the irrationals are not binary operations. But in some sense the irrationals do obey the axioms of arithmetic for example as outlined here:

and the fact that the above properties do apply (even if the operations carry you out of the set in the end) is due to them being a subset of the reals, yes? If it was the ancient Greeks that discovered such properties of the irrationals then I doubt it was the Peano axioms that they were working with.