Complex Numbers Worksheet

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1
Introduction
1.1
Real or Complex?
Definition: The complex number z is defined as:
z = a + bi
(1)
where a, b are real numbers and i =
1. (Side note: Engineers typically use j instead of i).
Examples:
5 + 2i,
3
2i,
3,
5i
Real numbers are also complex (by taking b = 0).
1.2
Visualizing Complex Numbers
A complex number is defined by it’s two real numbers. If we have z = a + bi, then:
Definition: The real part of a + bi is a,
Re(z) = Re(a + bi) = a
The imaginary part of a + bi is b,
Im(z) = Im(a + bi) = b
To visualize a complex number, we can plot it on the plane. The horizontal axis is for the real part, and the
vertical axis is for the imaginary part; a + bi is plotted as the point (a, b).
In Figure 1, we can see that it is also possible to represent the point a + bi, or (a, b) in polar form, by
computing its modulus (or size), and angle (or argument):
2
2
z =
a
+ b
φ = arg(z)
We have to be a bit careful defining φ- Being an angle, it is not uniquely described (0 = 2π = 4π, etc). It is
customary to restrict φ to be in the interval ( π, π].
Figure 1: Graphically representing the complex number z = x + iy, and visualizing its complex conjugate, ¯ z
.
1

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