Where the author talks about imaginary numbers being completely "made up" and suggests you shouldn't bother with trying to understand them, I think that's selling them short.
Imagine, if you will, trying to explain to the ancient Greeks the idea of a number that can't be written as a division of integers (the irrational numbers). That would have seemed completely "made up" to them, but we don't really see them that way, they just "are". That concept is has since become normalized, in terms of everyday concepts (like the area of a unit circle). Similar situations arise with fractions or negative numbers to some indigenous tribes, etc.
I guess what I'm saying is that complex numbers only as fictitious or imaginary as any other set of numbers that we otherwise feel like we have a good handle on.
And with a little bit of trigonometry intuition on top of this, powers of i and Euler's formula become much easier to understand as well. Great ways to think about it!
I'm sure the author acknowledges the importance of complex numbers - the part where he says "don't bother..." is why is i^2 = -1, which makes sense, as i(or j) was indeed chosen to represent sqrt(-1) and not the other way round(no one "discovered" a magical number that, when squared, gives -1) :)
Yeah, I actually really do like the rest of it. It's just the part:
"Don’t try to actually understand this term as there is no logical reason why it exists. We just have to accept that is just something that squares to -1."
that rubbed me the wrong way. But in terms of the goal of this guide, I suppose waving hands and saying "deal with it" could be chalked up to a necessary evil.
But isn't the definition of the imaginary unit axiomatic? If so it's indeed true that there is no logical reason why it exists other than it being useful.
That is so awesome....
(Yes, so this probably counts as the kind of "+1" post that should be downvoted. But check out the link and if you don't think it's the most amazing quaternion explanation, then I will humbly accept your downvote.)
Wow. Thanks for this. But I guess quaternions are much more well known because of their application in Computer Graphics/Vision (which is how I stumbled upon this article)
I think of it as the opposite: there exists a Lie group of spatial rotations, so we want to use it, and we found a relatively convenient notation.
There are many other kinds of 'complex' numbers (http://en.wikipedia.org/wiki/Hypercomplex_number) - but you probably won't hear about them outside of mathematics and physics because they're less useful.
The same applies to 2d - the SO(2) group exists; we map it to the complex numbers because it's convenient to do math with.
As much as anything, it is a chosen property of the system.
If you want to insist that mathematics is discovered (rather than invented), you can still say that this particular decomposition of vectors is prevalent because it has useful properties.
This recently posted YouTube video by UNSW Professor Norman J. Wildberger discusses the discovery of the quaternions by Hamilton and the subsequent discovery of the octonians. It's 59 minutes, 30 seconds long, and it was published on March 5, 2014:
Imagine, if you will, trying to explain to the ancient Greeks the idea of a number that can't be written as a division of integers (the irrational numbers). That would have seemed completely "made up" to them, but we don't really see them that way, they just "are". That concept is has since become normalized, in terms of everyday concepts (like the area of a unit circle). Similar situations arise with fractions or negative numbers to some indigenous tribes, etc.
I guess what I'm saying is that complex numbers only as fictitious or imaginary as any other set of numbers that we otherwise feel like we have a good handle on.