I'm curious as to why we don't have any useful numbers for the non-power-of-2 dimensions. E.g. 3-dimensional numbers.
I wonder if there's anything of numbers in infinite dimensions and, more interestingly, numbers in different degrees of infinite dimensions (yup, infinity comes in different degrees too - http://sites.google.com/site/degreesofinfinity/). It seems that math is only limited by the imagination.
(Infinite is often confused for "really big", but a better way of understanding it is often to use the term "unbounded". When we deal with n-dimensional vectors, we're not saying that we are always literally dealing with vectors with a millionybillionytertrillion dimensions, such that we can't even represent one in a real computer, we're saying that there isn't a firm upper limit on the number of dimensions we may encounter. It is often more the "unbounded" aspect of infinity that we are concerned with, rather than the "larger than anything else" aspect.)
5i + 2j + 3k.
Since you can assume that 2 is getting multiplied by the unit real vector (equal to 1), you can think of 2+3i as
2r + 3i
but why write the 'r' if we know 2*1 = 2?
EDIT: I guess I learned it in high school though, and the math teacher probably didn't know this either.
Point being, everyone should come across this at some point. It would be beneficial to many math students.
What blew my mind at the time was the exponential notation for the unit phasor: e^(i * x). It turns out that e^(i * x) = cos(x) + i * sin(x) because that's just the way the math works out, and it's trivial to work it out yourself by looking at the Taylor series expansions of the three terms.
It makes little sense to say that 2 + 3i
is equal to 3 + 2i
It is generally considered in mathematics that a norm is used to describe length, size, or extent.
Of course, you could just define a relation < on the set of complex numbers by "x < y if and only if norm(x) < norm(y)". This relation is obviously trichotomous, and it could be considered an ordering of the complex numbers. In my experience, however, this assumption is generally not held or used by mathematicians.
I hate to admit it but i had started using a line of argument to certain theist friends, that if god helps you, as a concept, no need to be bothered, think no more of it than a concept such as the the mathematical concept of i, its a number that does not exist but has real consequences. now I feel stupid for doubting the existence of i.
I keep trying to relearn my fundamentals, and this article does that beautifully. i would otherwise have died a disbeliever.
betterexplained also has a fantastic visual explanation of euler's identity: http://betterexplained.com/articles/intuitive-understanding-...
My goal is to get all these concepts out of awe (which I had too) into a real sense of "Ah, I get it!". It doesn't help anyone to memorize incantations.
Two other insights I love:
* e as continuous growth: http://betterexplained.com/articles/an-intuitive-guide-to-ex... * radians as the perspective of the mover: http://betterexplained.com/articles/intuitive-guide-to-angle...
I realized that I didn't really get what e and radians were about -- I memorized them, but I wasn't comfortable. Once you have the right analogies, Euler's formula starts making sense (without resorting to "Oh, take the Taylor series expansion of each and see how they match up", which is essentially an incantation).
To me, that's a huge canary in the coal mine! Why aren't we stopping here and making sure we get it? It's like reading a sentence, not understanding the key vocabulary word, and moving on. Yes, you "read" it but did you get anything from it?
Ex-fucking-actly. Math is hard. Compsci is hard. But if it remains hard, then we adults have failed to do our job.
What's funny is that we think it stops there. "Oh, we made made multiplication easy, and negatives, and decimals, but Calculus... well that needs to remain difficult forever and ever.".
Here's my followup question and answer. We know i is the square root of -1, but what is the square root of -i ? I've always thought that you'd need another dimension to describe that, and another dimension for the square root of that unit, and so on.
But no. (Let's tackle sqrt(i) as a simpler case first.) We can answer sqrt(i) in terms of rotation as described in the article. i is a rotation from the unit vector by 90°, so applying that twice turns 1 into -1. What operation applied twice would result in i? This just clicked: a 45° rotation. Thus the unit vector at a 45° angle is the square root of i: 0.5√2 + 0.5√2 * i.
The mathematical approach bears that out. Follow the rules of complex number arithmetic to square 0.5√2 + 0.5√2 * i (multiply it by itself) and you do indeed get i.
And we can solve sqrt(-i) the same way. -i is a 270° rotation from the unit vector 1. So the square root of -i is a 135° rotation, or -0.5√2 + 0.5√2 * i.
Finally, a 270° rotation is equivalent to a -90° rotation. So -45° should also be a square root of -i, and indeed it is. Multiplying 0.5√2 + -0.5√2 * i by itself also gives you -i. We've arrived back at the axiom that all numbers have two square roots of opposite signs. 135° and -45° are the same vector pointing in exactly opposite directions.
Last question: What's the cube root of i? Easy: a 30° rotation. The 30° unit vector is 0.5√3 + 0.5i, and cubing that does indeed get you i.
What's amazing about these systems is that there is usually an Euler relation that holds. Example: e^(i*t) = cosh(t) + sinh(t) for the split-complex.
http://www.youtube.com/watch?v=egIPnwcJuZ8
But the main website is
http://www.dimensions-math.org/Dim_download_E.htm
And I think it's chapter 5.
This is why I am excited about programs like Khan Academy. One of the things he's been able to do that has eluded most public schooling is explain a concept simply, and with enthusiasm.