Lecture 5/12: Euler’s Formula and de
Moivre’s Theorem
MA154: Algebra for 1st Year IT
Niall Madden
Niall.Madden@NUIGalway.ie
1 Feb 2007
CS457 — Lecture 5/12: Euler’s Formula and de Moivre’s Theorem
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Outline
1 Recall...
Polar form
Euler’s formula
Binomial expansions and de Moivre’s Theorem
2 Applications of de Moirve’s Theorem:
3 Roots of Unity
4 Primitive roots
5 Geometry of the roots of unity
The 8th roots of Unity
CS457 — Lecture 5/12: Euler’s Formula and de Moivre’s Theorem
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Recall...
Polar form
Polar form of the complex number z = x + i y is
z = (r cos θ, r sin θ) = r (cos θ + i sin θ)
where
r = |z|
and
θ is the angle in the Argand plane that the line joining z to the
origin makes with the real axis. That is
y
r sin θ
sin θ
=
=
= tan θ.
x
r cos θ
cos θ
CS457 — Lecture 5/12: Euler’s Formula and de Moivre’s Theorem
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Recall...
Euler’s formula
For θ ∈ R define Euler’s formula
exp(i θ) = e i θ = cos θ + i sin θ
So can write the polar form of z 6= 0 as
z = r (cos θ + i sin θ) = r e i θ .
CS457 — Lecture 5/12: Euler’s Formula and de Moivre’s Theorem
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Recall...
Binomial expansions and de Moivre’s Theorem
The Binomial Theorem:
n X
n n−k k
n
(a + b) =
a
b
k
k=0
n
n!
where
=
.
k
k! (n − k)!
De Moivre’s Theorem is:
(cos θ + i sin θ)n = cos nθ + i sin nθ for all n ∈ Z.
CS457 — Lecture 5/12: Euler’s Formula and de Moivre’s Theorem
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Applications of de Moirve’s Theorem:
Example: Show that cos2 θ = 12 (cos 2θ + 1).
CS457 — Lecture 5/12: Euler’s Formula and de Moivre’s Theorem
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Applications of de Moirve’s Theorem:
Exercise: Show that 16 cos4 θ = 2 cos 4θ + 2 cos 2θ + 6.
(Hint: use the previous identity for cos2 θ.
16 cos4 θ
15
10
5
0
−8
−6
−4
−2
0
2
4
6
3
8
2 cos 4 θ
2 cos 2 θ
2
1
0
−1
−2
−3
−8
−6
−4
−2
0
CS457 — Lecture 5/12: Euler’s Formula and de Moivre’s Theorem
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4
6
8
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Roots of Unity
There are two complex numbers that are solutions to the problem
Find z such that z 2 = 1.
They are z = 1 and z = −1. These are the “Square roots of
Unity”.1
1
We say “Unity” when we mean 1 .
CS457 — Lecture 5/12: Euler’s Formula and de Moivre’s Theorem
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Roots of Unity
What about the 4th Roots of Unity (numbers z such that
z 4 = 1)? Clearly 1 and −1 will do. But so too will i and −i.
Definition: Roots of Unity
A complex number z with z n = 1 (n = 1, 2, 3, . . .) is called an nth
root of unity .
When trying to find them, its often best to try to solve z n − 1 = 0.
CS457 — Lecture 5/12: Euler’s Formula and de Moivre’s Theorem
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Roots of Unity
Example
Find the 3rd roots of unity. Write the complex ones in the form
e iθ .
Solution:
CS457 — Lecture 5/12: Euler’s Formula and de Moivre’s Theorem
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Roots of Unity
It turns out that every root of unity is of the form e iθ for a
suitable θ:
n
(e iθ )n = cos θ + i sin θ = cos nθ + i sin nθ = 1.
So nθ must be 0, or 2π or 4π or ...
Important: z = e iθ is an nth root of unity if
θ=
2kπ
n
for some
k ∈ Z = {. . . , −2, −1, 0, 1, 2 . . . }.
CS457 — Lecture 5/12: Euler’s Formula and de Moivre’s Theorem
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Roots of Unity
z = e i θ = cos θ + i sin θ is an nth root of unity if
θ=
2kπ
n
for some
k ∈ Z = {. . . , −2, −1, 0, 1, 2 . . . }.
Example:
The 3rd roots of unity are given by taking θ = 0, 32 π and − 23 π:
z1 = cos 0 + i sin 0 =
2 2 z2 = cos π + i sin π =
3
3
2 2 z3 = cos − π + i sin − π =
3
3
CS457 — Lecture 5/12: Euler’s Formula and de Moivre’s Theorem
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Primitive roots
Definition: Primitive root
An nth root of unity ωn is called Primitive if are all roots of unity
are powers of it.
Example
The 4th roots are 1, i , −1 and − i . Both i and −i are primitive
because:
CS457 — Lecture 5/12: Euler’s Formula and de Moivre’s Theorem
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Primitive roots
Example
The 8th roots of unity are the solutions to z 8 = 1. These are
{1,
−1,
√
√
2
2
+
i
2√
2√,
− 22 − i 22 ,
√
i,
−
−i ,
√
2
2
+
i
2√
2√,
2
2
2 − i 2 }
Which are primitive?
CS457 — Lecture 5/12: Euler’s Formula and de Moivre’s Theorem
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Geometry of the roots of unity
The nth roots of unity are the vertices of a regular polygons
(equilateral triangle, square, pentagon, hexagon ...) on the unit
circle!!
Example
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Geometry of the roots of unity The 8th roots of Unity
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