Complex Numbers
The complex numbers are an extension of the real numbers containing
all roots of quadratic equations. If we define i to be a solution
of the equation x2 = -1, then the set C of complex numbers
is represented in standard form as
We often use the variable z = a+bi to represent a complex number. The
number a is called the real part of z (Re z) while b is
called the imaginary part of z (Im z). Two complex numbers
are equal if and only if their real parts are equal and their
imaginary parts are equal.
We represent complex numbers graphically by associating z = a+bi with
the point (a,b) on the complex plane.
Basic Operations
The basic operations on complex numbers are defined as follows:
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a+bi c+di
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= |
a+bi c+di
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· |
c-di c-di
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= |
ac+bd c2+d2
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+ |
bc-ad c2+d2
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i |
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In dividing a+bi by c+di, we rationalized the denominator using
the fact that (c+di)(c-di) = c2 -cdi +cdi -d2i2 = c2 + d2. The
complex numbers c+di and c-di are called complex
conjugates.
If z = c+di, we use |
_ z |
to denote c-di. |
Viewed as a vector in the complex plane, z=a+bi has magnitude
which we call the modulus or absolute value of z.
Examples
- (2+3i)(2-3i) = 4-6i+6i-9i2 = 4+9 = 13.
- |2+3i| = |2-3i| = √(4+9) = √13.
Polar Form
For z = a + b i, let
from which we can also obtain
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Let: |
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θ = φ if x > 0, θ = φ+π if x < 0. |
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Then
and so, by Euler's Equation, we obtain the polar form
Here, r is the magnitude of z and θ is called the
argument of z (arg z). The argument is not unique; we can add
multiples of 2π to θ without changing z. We define Arg
z, the principal value of the argument, to be in
(-π,π]. The principal value is unique for each z but creates
unavoidable (yet interesting!) complications due to its discontinuity
across the negative real axis where it jumps from π to -π.
This jump is called a branch cut.
Examples
- eiπ = cosπ+ isinπ = -1
- 3eiπ/2 = 3(cos[π / 2] + isin[π / 2]) = 3i
- 2eiπ/6 = 2(cos[π / 6] + isin[π / 6]) = √3 + i
Multiplication and division of complex numbers is amazingly simple in
polar form! If z1 = r1eiθ1 and z2 = r2eiθ2,
then
If z = reiθ, then |
_ z |
= re-iθ (Do you see
why?) and so z |
_ z |
= (reiθ)(re-iθ) = r2. |
Example
To calculate (1+i)8, we can first rewrite 1+i as
√2eiπ/4. Then
Roots of Unity
The equation
has n complex-valued solutions, called the nth roots of unity.
Since we know each root has magnitude 1, let z = eiθ. Then
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(eiθ)n = einθ
, together with Euler's Equation, gives us deMoivre's Formula:
(cosθ+ isinθ)n = cos nθ+ i sin nθ |
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so the nth roots of unity are of the form
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There are n distinct roots, after which we start duplicating roots
already found.
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These are evenly spaced around the unit circle.
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Example
The 3rd roots of unity are
You can verify that (-1/2+ i[(√3)/ 2])3 = 1 and
(-1/2 - i[(√3)/ 2])3 = 1.
This tutorial has reviewed the basics of complex arithmetic. The
methods of complex analysis, which build on this background, are both
intriguing and powerful!
Key Concepts [index]
Euler's Equation,
provides the connection between these two representations of complex
numbers.
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