Complex Numbers Notes

Numbers!FROM
Complex SCRATCH (1.1) 10/10/2022

=- 1 Ex1.1 x 2x + 2 0 Ex1-2 2 2 + 2i

Towriteinmedsingshere
=
=
=




=
i F 22 2x
-
+1 10
+
=
121 5 252
= =




(n 1) -
-
=
1 1z1 is the modulus Im



2 - 1 i
=




argument
arg(z) is the
sir
arcan()
+H
x 1 In
=
↳
in
-
T




If the complexnumber I in the
lies


left side of the axes, you and to t i

z* is the
conjugate
complex Ex1.5 Repeated Multiplication and -
I
respectively
zx
cy
1zI(cos(arg(z)
=

+




*
z
x-in
=




Multiplication by : causes a rotation isin
+

larg(z1))
by I.

Ex1.6
Multiplying by z*
multiplying by conjugate a causes


something
similar to
rationalising in surds,
leftwith only real numbers. (n+iy)(n-iy) ni+y2
=




=Iz12


12110122
zz* 1z1 =




-ercluswz"ut?
zi (x iy)(n iy)
siye
=

+ -




=n"-y+ zxy:
zt ib roots!
Think about
sinncusy+ansusing a
simply
=
w a
=
+ -




w a 2-b2+labi
=




exponential form!
complex (1.2) Euler's formula can use -P, hella
waffl
iD
ein_cos cosP-isink
-



+ isine e =




Greig
-




phasor isjust
A a sinusoidal wave as a time-dependentfunction.

u(H Acos (wt 3)
=
+




FM?
remember on
from
Also:
x(H Acos(nt+2) + iAsin(nt + E)
=




cosO-
well sink= eio_eio
-
Just
functions!
hyperbolic like