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 + biRe(z) = a; Im(z) = b

Just like vectors!

|z| = (a2 + b2)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: eix = cos x + i sin x

cos x = (eix + e-ix)/2

sin x = (eix - 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 = ei2x

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!

z1z2= r1 (cos1 + i sin1) r2 (cos2 + i sin2)

= r1 r2 (cos1 cos2 - sin1 sin2) +

i r1 r2 (cos1 sin2 + cos2 sin1)

= r1 r2 [cos(1 + 2) + i sin(1 + 2)]

Multiplication:

the easy way!

1 2

1 2

1 2 1 2

( )1 2

i i

i

z z re r e

r r e

“Neither dot nor cross do you multiply complex numbers by.”

Multiplication: by i

(cos sin )

sin cos

iiz ir e

ir i

r ir

Rotate by 90o and swap Re and Im

i ‘s all over the Unit Circle!

Note i4 = 1 does not mean that 0 = 4

i ‘s all over the Unit Circle!

Did you see i½?

Square root of i?Find the square root of 7+24 i.

(Hint: it’s another complex number, which we’ll call u+vi).

2

2 2

2 2

i 7+24i

( i) 7 24i

2 i - 7 24i

- 7; 2 24

u v

u v

u uv v

u v uv

Which can be solved by ordinary means to yield 4+3i and -4 - 3i.

Complex Conjugates

22 2

z a bi

z a bi

zz a b z

Complex Conjugates

1

1

; tan ( / )

; tan ( / )

z a bi b a

z a bi b a

Complex conjugates reflect in the Re axis.

Complex Reciprocals

22 2

2

1

zz a b z

z

z z

The reciprocal of a complex number lies on the same ray as its

conjugate!

Powers of z

The graph of f(z)=zn 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 nth power is multiplying its

angle by n.

How about a slice of :

Roots of z

/1231

9 /1232

17 /1233

4.24

4.24

4.24

i

i

i

z e

z e

z e

Each successive nth root is another 2/n around the circle.

If z3 = 3+3i = 4.24ei

then

Find the roots of the complex equation z2 + 2i z +

24 = 0Sounds like a job for the

quadratic formula!

22 (2 ) 4(24)

2

4 96

25 6 ,4

i iz

i

i i i i

Was that so complex?

And never forget, ei = -1