what's this i number, anyhow?

Sure that works. I like phasor diagrams for understanding some stuff.

Absurd? Do you dislike negative numbers too? (-1 apple.. I mean how can you have a negative apple? :^)

Let's face it mathematics is absurdly useful in science. I had a professor who just loved quaternions. And then there are those spinor matrics in QM.. (just thinking beyond i.)

George H.

Well, I can "see" a negative number - If you steal my shoes, I could feel a negative pair of shoes. But...the SQRT of -1, what's that? :) jb

Pretty easy - orthogonal to your shoes ...

The invention of a 13yo boy.

Ahem, not able to imagine an apple?

A lot of the development in mathematics of the more practical sort has consisted in finding additional models for existing sets of axioms. Pythagoras discovered irrational numbers, for instance, which satisfy the axioms of arithmetic. Later it was discovered (by Dedekind and Cantor iirc) that algebraic irrationals were countable, i.e. you can put them into a 1:1 correspondence with the integers.

Complex numbers satisfy all the axioms of the real numbers except for the Archimedean order property, i.e. that any list of reals can be uniquely sorted by value, whereas of course complex ones can't. (It doesn't make sense to say either that i > 1 or that i < 1, for instance.)

The connection between sines and cosines and complex numbers is pretty natural, because of the Argand diagram (i.e. the complex plane)--polar and rectangular coordinates, and all that.

The Euler formula is the really unexpected thing. The identity between the steep monotonic exponential function and oscillatory trigonometric functions is really very surprising. (Euler lived a long time ago.)

Cheers

Phil Hobbs

The thing that I dislike most about quaternions is that they are useful for certain limited areas of engineering which I occasionally must visit. When I get there, I find myself having to write, debug, and maintain code to do quaternion arithmetic.

Thank goodness that octonions have no use in the real world!

Right up your alley.

Here's a more concrete example:

Examples or instances, not models.

They do? sqrt(2) * sqrt(2)= 2, so irrationals are not "closed" under multiplication because 2 is not an irrational.

What was the significance of that little gem?

Ummm, sounds like you're confusing Archimedes order with what mathematicians would call a total order. The Archimedes order is something entirely different *as far as I can tell* from a quick scan of this:

Total order simply means any two elements can be compared.

Actually it was not all that profound. Euler was the analytical series guru and realized quickly that substituting ix in the known series for exp(x) resulting in the sum of series for cos(x) and isin(x).

Cheers

You probably think 1^sqrt(2) =1 and only 1?

I like this part :

"Warning: this comic occasionally contains strong language (which may be unsuitable for children), unusual humor (which may be unsuitable for adults), and advanced mathematics (which may be unsuitable for liberal-arts majors)."

They don't have to be closed under multiplication to satisfy the axioms of arithmetic. The set of real numbers satisfies the axioms of arithmetic, and so the set of irrational numbers does as well, since it is contained in the former.

They would need to be closed under multiplication to be a field, like the field of rational or real numbers, but the irrationals aren't a field (they aren't any nice algebraic structure).

If you delve a bit deeper into complex functions, things get quite spooky, like the effect of singularities on line integrals. Bit heavy for armchair reading, but can be engrossing if you make the effort.

Managed to get some way into elliptic curves after a fair struggle, that throws up some amazing results as well.

Functions on the complex plane have the nice property that if they are analytic (I.e. expressable by a power series) in some open region of the plane, then they are also holomorphic in that region (infinitely differentiable.) This is opposed to real analysis, where you can have all sorts of strange constructs like functions which are continuous everywhere and differentiable nowhere (Weierstrauss functions), or functions which are differentiable everywhere but nowhere analytic.

Arithmetic:

" The basic calculations we make in everyday life: addition, subtraction, multiplication and division.

It also includes fractions and percentages (related to division) and exponents (related to multiplication)."

From

I prefer to think of it as e^(i . pi) - 1 = 0, as it combines the 5 most important mathematical numbers into one equation.

Some sort of generalised rotation?

Just be glad you don't need to use Clifford algebras...