Complex Numbers? What’s So Complex?

Complex numbers are vectors represented in the complex plane as the sum of a Real part and an Imaginary part: z = a + bi Re(z) = a; Im(z) = b

Just like vectors! |z| = (a 2 + b 2 ) 1/2 is length or magnitude, just like vectors.  = tan -1 (b/a) is direction, just like vectors!

Just like vectors! For two complex numbers a + bi and c + di: Addition/subtraction combines separate components, just like vectors.

Useful identities Euler: e ix = cos x + i sin x cos x = (e ix + e -ix )/2 sin x = (e ix - e -ix )/2i

Things named Euler

Sure, he’s French, but we must give props: DeMoivre: (cos x + i sin x) n = cos (nx) + i sin (nx) cos 2x + i sin 2x = e i2x cos 2x = (1 + cos 2x)/2 sin 2x = (1 - cos 2x)/2

What about multiplication? Just FOIL it!

Scalar multiples of a complex number: a line

Multiplication: the hard way! z 1 z 2 = r 1 (cos  1 + i sin  1 ) r 2 (cos  2 + i sin  2 ) = r 1 r 2 (cos  1 cos  2 - sin  1 sin  2 ) + i r 1 r 2 (cos  1 sin  2 + cos  2 sin  1 ) = r 1 r 2 [cos(  1 +  2 ) + i sin(  1 +  2 )]

Multiplication: the easy way! “Neither dot nor cross do you multiply complex numbers by.”

Multiplication: by i Rotate by 90 o and swap Re and Im

i ‘s all over the Unit Circle! Note i 4 = 1 does not mean that 0 = 4

i ‘s all over the Unit Circle! Did you see i ½ ?

Square root of i? Find the square root of 7+24 i. (Hint: it’s another complex number, which we’ll call u+vi). Which can be solved by ordinary means to yield 4+3i and i.

Complex Conjugates

Complex conjugates reflect in the Re axis.

Complex Reciprocals The reciprocal of a complex number lies on the same ray as its conjugate!

Powers of z The graph of f(z)=z n for |z|<1 is called an exponential spiral.

This shape is at the heart of the computation of fractals!

The basic geometry of the solar system!

It shows up in nature!

And the decorative arts!

The rotation comes from our old buddy DeMoivre: (cos x + i sin x) n = cos (nx) + i sin (nx) Raising a unit z to the n th power is multiplying its angle by n.

How about a slice of  : Roots of z Each successive n th root is another 2  /n around the circle. If z 3 = 3+3i = 4.24e i  then

Find the roots of the complex equation z 2 + 2i z + 24 = 0 Sounds like a job for the quadratic formula!

Was that so complex? And never forget, e  i = -1