WildTrig15: Complex numbers and rotations

[samruby82]

what would adding rotations look like in this example at @5.56. Are you saying they can't be added at all or that they can't be added and still preserve quadrance?

[jazzzyxjay]

What the heck did he just say? He lost me at hello

[heeypollol]

We see this in belgium when we are 14-15 years.

now im 17 and are working on schemes.

This is suposed easy with us?

What do you people see when you're 14-15 'cause i'm interested. Grtz

[SuperGeetha]

yea yea we knw, no wonder most lunatics are 4m belgium

[mk3v1na]

really informative!!!~

I ACED MY ROTATION TEST, THANKS!!!

ADD ME FOR CYBER

[sillypuppy2]

f(a,b)=x^2 + 2ax + (a^2 + b^2)

Have you ever plugged numbers in for a and b and then tried to find the zeros of the resulting function?

[sillypuppy2]

i^a= 1,-1,i, or -i. Is what your trying to say is that squaring these numbers is the same as adding 90 degrees to their bearing?

[njwildberger]

Multiplying a complex number by i rotates it by a quarter turn (90 degrees). So the powers of i are i,-1,-i,1, and then they repeat.

[sillypuppy2]

oh yeah, thx for correcting me. I never went to college but math was my favorite subject in high school and I like to study math.

[7NTM61Ic]

The "multicomplex" class of hypercomplex numbers, with real coefficients, seem to have a directly analogous rotational property to the complex numbers.

[catalecticant]

(continued)

I like this approach, because instead of introducing a square root of -1 artificially (as most courses on complex numbers do), you identify it with something that's already "out there".

[catalecticant]

A pity you've swept the i^2=-1 issue under the carpet, especially because transformations give you a neat way to introduce i. First, identify dilations with numbers as you have done. Then consider the number -1 as a dilation: it is not the square of another dilation. Then point out that this dilation is also a half-turn: now it *is* the square of something, namely, of a quarter turn in either direction: you can then *define* i as one of them.

[njwildberger]

Perhaps I should have stressed that the rotation(s) of a quarter turn provide natural square roots of the dilation of -1. It is certainly an important point, thanks for bringing it up.