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Transcript
Complex numbers
If you have not seen much about complex numbers, here are some elementary properties.
(If you have any questions, feel free to ask!)
The complex numbers C are the set of all numbers of the form a + bi where a, b ∈ R and
i2 = −1. We may define addition of complex numbers by
(a + bi) + (c + di) = (a + c) + (b + d)i
and multiplication as
(a + bi)(c + di) = ac + adi + bci + bdi2 = (ac − bd) + (ad + bc)i.
For any number z = a + bi we say the real part of z is
Re(z) = a,
and the imaginary part of z is
Im(z) = b.
Writing the complex numbers as 2-tuples instead, e.g., (a, b), they natually look like points,
or vectors on the xy-plane, where the real part corresponds to the x-axis and the imaginary
part corresponds to the y-axis, e.g., the complex number z = 1 + 2i can be thought of as
the point (1, 2) in the plane. Addition is the the same as vector addition. (Multiplication is
more subtle, see below.)
This gives an insight into some geometry in C, as to each point/vector z = (a, b) we can
associate a length or magnitude
p
√
|z| = a2 + b2 = Re(z)2 + Im(z)2 ,
which is non-negative, via Pythagoras’ theorem. (Draw the triangle!) Similarly, we can
consider the angle between the vector z and the real (i.e., x) axis in the ‘positive’ direction.
We have
b
a
and sin θ =
.
cos θ =
|z|
|z|
Thus, any complex number z = a + bi can be written as
z = |z|(cos θ + i sin θ)
for some 0 ≤ θ < 2π. Euler’s formula states that
eiθ = cos θ + i sin θ,
which follows from looking at the respective Taylor series. (Note: what are eπi , e2πi and eπi/2
?) So any complex number can be written in the form z = |z|eiθ .
Now we can address multiplication of complex numbers. Writing
z1 := a + bi = r1 eiθ1
z2 := c + di = r2 eiθ2
we have
z1 z2 = (a + bi)(c + di) = (r1 r2 )ei(θ1 +θ2 ) .
We see the magnitude of the product z1 z2 is the product of the magnitudes |z1 | · |z2 |, and
the angle between the x-axis and the z1 z2 is the sum of the angles between the x-axis and
z1 , and the x-axis and z2 . (Draw some simple examples!)
1