basics.htm Basics of Complex Numbers
Basics of Complex Numbers
A complex number is a formal expression x + i y, where x and y are real numbers, i is a formal object which satisfies i·i = −1 = −1 + i 0. The real part of z = x + i y, denoted ℜz, is x; the imaginary part of z = x + iy, denoted ℑz, is y.
•
Addition:
If z1 = x1 + i y1 and z2 = x2 + i y2, the sum is z1 + z2 = (x1 + x2) + i (y1 + y2). Thus
ℜ(z1 + z2)
= ℜz1 + ℜz2,
ℑ(z1 + z2)
= ℑz1 + ℑz2.
•
 Multiplication:
If z1 = x1 + i y1 and z2 = x2 + i y2, the product z1 z2 is
z1·z2
= (x1 + i y1)·(x2 + i y2)
= (x1 x2 − y1 y2) + i (y1 x2 + x1 y2).
Here we use the relation i ·i = i2 = −1. We also write i = √{−1}.
•
Complex Numbers, Points, and Vectors
The complex number z = x + i y can be identified with a point in the x-y coordinate plane P with coordinates (x,y). another useful view is to identify the point P(x,y) [complex number x + i y] with the vector or arrow O→P from the origin to P [z].
Addition of complex numbers is best understood in terms of addition of vectors: The point [vector] corresponding to z2 added to z1 is the point z1 shifted by the vector z2.
•
Modulus and Conjugate
The modulus or absolute value of a complex number z = x + i y is defined as
|z| =
√
 
x2 + y2
 
.
The conjugate of a complex number z = x + i y is defined as
-
z
 
 =
x + i y
 
 = x − i y.
Note that
z
-
z
 
 = |z|2 = x2 + y2
and
ℜz
= 1
2
 ⎛
⎝ 		z +
-
z
 
⎞
⎠ 		,
ℑz
= 1
2 i
⎛
⎝ 		z −
-
z
 
⎞
⎠ 		.
•
Polar Coordinates
In the plane, a point (x,y) [complex number z = x + i y] (not O) is completely determined by its distance from the origin r = |z| = √{x2 + y2} and the angle θ from the positive x-axis to the ray Oz from the origin to z.

basipic1.gif

The pair (r,θ) are polar coordinates of the point P(x,y) or the complex number z = x + i y. We have:
x
= r cosθ,
y
= r sinθ,
z
= r (cosθ+ i sinθ),
z
= |z| (cosθ+ i sinθ).
Note that r is the modulus of z. The angle θ is called an argument of z, written arg(z) .
For convenience we introduce the notation
cisθ = cosθ+ i sinθ
so that for z ≠ 0,
z = |z| cis(arg(z)).
It is important that arg(z) is not uniquely determined. If θ is an argument of z, then θ + any integer multiple of 2π is also an argument of z.1
Note that
arg(
-
z
 
 ) = − arg(z).
•
Multiplication and Polar Coordinates
Geometrically, multiplication by a nonzero z is best understood in terms of polar coordinates.
If z1 = |z1| cis(θ1), z2 = |z2| cis θ2, verify that
z1 ·z2 = |z1| |z2| cis(θ1 + θ2).
Thus:
•
The modulus of the product = the product of the moduli.
|z1 z2| = |z1||z2|.
•
An argument of the product = the sum of the arguments.2
arg(z1 z2) = arg(z1)+arg(z2).
•
Reciprocal
If z ≠ 0, the reciprocal of z, [1/z], can be calculated as
1
z
=
-
z
z
-
z
 
=
-
z
|z|2
= x
x2 + y2
 − i y
x2 + y2
= 1
|z|
 cis(−arg(z)).

Footnotes:

1A particular choice for arg(z), for example, the unique arg(z) that satisfies 0 ≤ arg(z) < 2π is written Arg(z).
2Give an example of complex numbers z1 and z2 such that
Arg(z1 z2) = Arg(z1)+Arg(z2) − 2π.


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On 16 Nov 2013, 21:06.