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Complex numbers and Euler’s formula
Calvin W. Johnson
August 18, 2014
1
Complex arithmetic
I won’t start from the very beginning, but assume some familiarity with this
topic.
Complex numbers are very useful in mathematics and physics. They are of
the form
a + ib,
(1)
where a and b are real numbers, and, famously,
i2 = −1.
(2)
You should be familiar with basic addition and multiplication of complex numbers, namely, if
z1 = a1 + ib1
and
z2 = a2 + ib2
then
z3 = z1 + z2 = (a1 + a2 ) + i(b1 + b2 )
(3)
z4 = z1 · z2 = (a1 a2 − b1 b2 ) + i(a1 b2 + b1 a2 ).
(4)
and
Note: I won’t always be quite as pedantic as this.
Example. Let z = 1 + 2i and w = −4 − 3i. Then z + w = −3 − i and
zw = 2 − 11i.
You try (1): If z = 3 − 5i and w = −4 + 3i, what is w + z? zw? (answer
below–don’t peek!)
You try (2): If z = 5 + i and w = 2 + 2i, what is w + z? zw? (answer to be
given in class)
We can also introduce division of complex numbers, but for that we need a
couple more concepts.
1
Im z
Re z
2
The complex plane (Argand diagrams)
Many people think visually, and one of the great advances in mathematics was
Descartes’ marriage of geometry and algebra into graphs, such as plotting functions y = f (x) on the x-y plane. In the same way, we can turn complex numbers
into graphs, with the real part of the number on the x (horizontal) axis and the
imaginary part on the y (vertical) axis. This is sometimes called an Argand
diagram.
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Complex conjugation and magnitudes
Because we can think of complex numbers as vectors in a plane, we can ask
about their lengths or magnitudes. While the components of a real vector, or
the real or imaginary parts of a complex numbers, can be positive or negative,
we have the following rules for magnitudes:
• A magnitude is always a real, nonnegative number. It never has an imaginary
part and is never negative. There are no exceptions.
• The magnitude of a complex number z = a + ib can be zero (0) if and only
both the real and imaginary parts are zero, i.e., z = 0 + i0. There are no
exceptions.
From looking at Argand diagrams,
it is pretty easy to see that the length of
√
a complex number z = a + ib is a2 + b2 , or
p
|z| = (Rez)2 + (Imz)2 .
(5)
One way to get to magnitude is to introduce the concept of complex conjugation. Complex conjugation changes the sign on the imaginary part, that
2
is,
z = a + ib, z ∗ = a − ib
(6)
Real number do not change under complex conjugation. This includes real,
negative numbers. There are no exceptions.
Then the square of the magnitude is particularly simple:
|z|2 = a2 + b2 = (a + ib)(a − ib) = zz ∗ .
(7)
Example. The magnitude of 1 + i is
p
√
√
(1 + i)(1 − i) = 1 + 1 = 2;
alternately, you can simply do
p
12 + 12 =
√
2.
Example. The magnitude of −3i is
p
√
(−3i)(3i) = 9 = 3.
Example. The magnitude of 4 + 3i is
p
√
(4 + 3i)(4 − i3) = 16 + 9 = 5;
alternately, you can simply do
p
42 + 32 =
√
25 = 5.
You try (3): What is the magnitude of 5 − 2i? (answer below–don’t peek!)
You try (4): What is the magnitude of 2 + 7i? (answer to be given in class)
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Complex arithmetic, continued
With complex conjugation in hand, we can now easily divide by complex numbers. Suppose we want to find w/z where w and especially z are complex
numbers. The trick is to multiply by 1 = z ∗ /z ∗ (this kind of trick, multiplying
by 1 or adding 0, is frequently used in mathematics, as we’ll see throughout the
course):
w
w z∗
wz ∗
=
=
.
(8)
z
z z∗
|z|2
As |z|2 is a real number, it is easy to divide by it.
3
Example:
4 − 3i
4 − 3i 2 − i
5 − 10i
=
=
= 1 − 2i
2+i
2+i 2−i
5
(9)
You try (5): If z = 3−5i and w = −4+3i, what is z/w? (answer below–don’t
peek!)
You try (6): If z = 5 + i and w = 2 + 2i, what is z/w? ? (answer to be given
in class)
5
Polar coodinates in the complex plane
As you know, for two-dimensional vectors in the x-y plane, we can either represent them in Cartesian coordinates, i.e. as (x, y), or we can use polar coordinates
(r, θ), with
x = r cos θ;
(10)
y = r sin θ
The same applies for complex numbers: we write
z = a + ib = |z|(cos argz + i sin argz,
(11)
where argz is the argument of z,
tan argz = Imz/Rez = b/a.
(12)
(Note, however, that the argument of a complex number is actually multivalued,
because you can always add a multiple of 2π to get the same complex number.
Refer to a full text on complex analysis for details.)
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Euler’s formula
This leads us to a very important formula, and one you must know by heart:
exp(iθ) = cos θ + i sin θ
(13)
which is Euler’s formula. This is a key formula we will have many opportunities
to use.
In this form we can write
z = |z| exp(iargz).
(14)
While Cartesian coordinates are useful adding and subtracting complex numbers, polar coordinates are useful for multiplying and dividing complex numbers
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(as well as for taking logarithms, square roots and other powers, but I won’t
cover that now):
z1 z2 = |z1 ||z2 | exp (i(argz1 + argz2 ))
(15)
and
z1 /z2 =
|z1 |
exp (i(argz1 − argz2 ))
|z2 |
(16)
That is, the magnitude of the product of two complex numbers is the product
of the magnitudes, and the argument of the product of two complex numbers is
the sum of the arguments.
A particular case of Euler’s formula is
eiπ = −1
(17)
which is truly astounding, because it relates
• -1, the fundamental number of arithmetic;
• i, the fundamental number of algebra;
• π, the fundamental number of trigonometry; and
• e, the fundamental number of calculus.
You should know by heart the following, which come out of Euler’s formula:
eiπ = −1
ei2π = +1
π
ei 2 = i
ei
3π
2
= −i
π
You try (7): What is ei 4 ? (answer to be given in class)
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Answers to selected problems
Other questions will be answered in class.
(1) (3 − 5i) + √
(−4 + 3i) = −1 − 2i; (3 − 5i)(−4 + 3i) = 3 + 29i.
(3) |5 − 2i| = 29.
11
(5) (3 − 5i)/(−4 + 3i) = − 27
25 + 25 i.
5