Complex numbers A very rough guide

1
Complex numbers A very rough guide
The imaginary unit i2 -1
The complex number z a bi
Re(z) a real part
Im(z) b imaginary part
Argand Diagram
Im(z) i
z a bi r(cosq i sinq)
z a bi
bi
r
q
a Re(z) r cosq b Im(z) r sinq r z
v(a2b2)
Re(z)
a
z
Quadrant
z is the complex conjugate of z z a - bi

• 0 for a gt 0 I
• b gt 0
• for a lt 0 II, III
• 2p for a gt 0 IV
• b lt 0

b a
arctan

q
p/2 for a 0, b gt 0
3p/2 for a 0, b lt 0
Some basic Algebra of complex numbers with z1
a1 i b1, z2 a2 i b2



Addition z1 z2 (a1 a2) (b1
b2)i Multiplication z1 z2 (a1 a2 - b1 b2)
(a1 b2 a2 b1)i Division
a1 a2 b1 b2
z1 z2
a2b1 a1b2

i
a22 b22
a22 b22
Two complex numbers are identical when both their
real and their imaginary parts are identical
a1 b1i a2 b2i when a1 a2 b1 b2
2
z cosq i sinq eiq
Eulers Equation
z a ib r(cosq isinq) reiq elnr iq
r1 r2
z1 z2
Multiplication z1 z2 r1 r2 ei( )
ei( )
Division
Powers
zn rneinq
De Moivre Theorem (cosq i sinq)n cosnq
isinnq
If z r(cosq i sinq) r eiq then its complex
conjugate z r(cosq - i sinq) r e-iq
Solution of the Equation zm r(cosq i sinq) -
Roots
Recall zm r(cos(q) isin(q)) reiq
r(cos(q2pk) isin(q2pk)) rei(q2pk)
k 0,1,2,3,
The solutions of this equation are
q2pk m
i(q2pk) m
q2pk m
m
m
zk vr e vr(cos isin
)
With k 0,1,2, m-1.
sinq and cosq
e(iq)e-(iq)
e(iq) - e-(iq)
cos(q)
sin(q)


2
2i
e(iiq)-e-(iiq)
(e(iiq)-e-(iiq))i
i i
sin(iq)
The hyperbolic functions

2i
2ii
e(iiq)e-(iiq)
i (eq - e-q)
(e-q - eq)i
cos(iq)

sin(iq)

2
2
-2
e-qeq
cosh(q)
sin(iq) isinhq

2
(eq - e-q)
as sinhq
2