By now, you’ve heard about complex numbers C. The word “complex” is an intimidating word. Surely it must be a complex task to learn about the complex numbers. That may be true in general, but it helps if you know about vectors.
Complex numbers are similar to two-dimensional vectors ~v ∈ R2. We add and subtract complex numbers like vectors. Complex numbers also have components, length, and “direction.” If you understand vectors, you will understand complex numbers at almost no additional mental cost.
We’ll begin with a practical problem.
Example
Suppose you are asked to solve the following quadratic equation:
x2+ 1 = 0.
You’re looking for a number x, such that x2 = −1. If you are only allowed to give real answers (the set of real numbers is denoted R), then there is no answer to this question. In other words, this equation has no solutions. Graphically speaking, this is because the quadratic function f(x) = x2+ 1 does not cross the x-axis.
However, we’re not going to take nothing as an answer. We will imagine a new number called i that satisfies i2 = −1. We call i the unit imaginary number. The
Complex numbers solutions to the equation are therefore x1 = iand x2 = −i. There are two solutions because the equation was quadratic. We can check that i2+ 1 = −1 + 1 = 0and also (−i)2+ 1 = (−1)2i2+ 1 = i2+ 1 = 0.
Thus, while the equation x2+1 = 0has no real solutions, it does have solutions if we allow the answers to be complex numbers.
Definitions
Complex numbers have a real part and an imaginary part:
• i: the unit imaginary number i ≡√
−1 or i2 = −1
• bi: an imaginary number that is equal to b times i
• R: the set of real numbers
• C: the set of complex numbers C = {a + bi | a, b ∈ R}
• z = a + bi: a complex number
• Re{z} = a: the real part of z
• Im{z} = b: the imaginary part of z
• ¯z: the complex conjugate of z. If z = a + bi, then ¯z = a − bi.
The polar representation of complex numbers:
• z = |z|∠φz = |z| cos φz+ i|z| sin φz
• |z| =√
¯ zz =√
a2+ b2: the magnitude of z = a + bi
• φz = tan−1(b/a): the phase of z = a + bi
• Re{z} = |z| cos φz
• Im{z} = |z| sin φz
Complex numbers
Formulas
Addition and subtraction
Just as we performed the addition of vectors component by component, we perform addition on complex numbers by adding the real parts together and adding the imaginary part together:
(a + bi) + (c + di) = (a + c) + (b + d)i.
Polar representation
We can give a geometrical interpretation of the complex numbers by extending the real number line into a two-dimensional plane called the complex plane. The horizontal axis in the complex plane measures the real part of the number. The vertical axis measures the imaginary part. Complex numbers are vectors in the complex plane.
It is possible to represent any complex number z = a + bi in terms of its magnitude and its phase:
z = |z|∠φz = (|z| cos φz) + (|z| sin φz)i.
The magnitude of a complex number z = a + bi is
|z| =√
a2+ b2.
This corresponds to the length of the vector which represents the complex number in the complex plane. The formula is obtained by using Pythagoras’ theorem.
The phase of the complex number is:
φz = tan−1(b/a).
The phase corresponds to the angle z forms with the real axis.
Complex numbers
Multiplication
The product of two complex numbers is computed using the usual rules of algebra:
(a + bi)(c + di) = (ac − bd) + (ad + bc)i.
In the polar representation, the product is
(p∠φ)(q∠ψ) = pq∠(φ + ψ).
Cardano’s example
One of the earliest examples of reasoning involving complex numbers was given by Gerolamo Cardano in his 1545 book Ars Magna. Cardano wrote, “If someone says to you, divide 10 into two parts, one of which multiplied into the other shall produce 40, it is evident that this case or question is impossible.” We want to find numbers x1 and x2 such that x1 + x2 = 10 and x1x2 = 40. This sounds kind of impossible. Or is it?
“Nevertheless,” Cardano said, “we shall solve it in this fashion:
x1 = 5 +√
15i and x2 = 5 −√ 15i.”
When you add x1 + x2 you obtain 10. When you multiply the two numbers the answer is
15iare two numbers whose sum is 10 and whose product is 40.
Example
Both i and −1 have a magnitude of 1. i has phase π2 (90◦), while −1 has phase π (180◦). Consider the product of these two numbers:
(i)(−1) = (1∠π2)(1∠π) = 1∠3π2 = −i.
Multiplication by i is effectively a rotation by 90 degrees leftward.
Complex numbers
Division
Let’s look at the procedure for dividing complex numbers:
(a + bi)
(c + di) = (a + bi) (c + di)
(c − di)
(c − di) = (a + bi)(c − di)
(c2+ d2) = (a + bi) c + di
|c + di|2. In other words, to divide the number z by the complex number s, compute ¯s and
|s|2 = s¯s and then use
z/s = z ¯s
|s|2. You can think of |s|¯s2 as being equivalent to s−1.
Fundamental theorem of algebra
The solutions to any polynomial equation a0+ a1x + · · · anxn= 0 are of the form z = a + bi.
In other words, any polynomial P (x) of nth degree can be written as P (x) = (x − z1)(x − z2) · · · (x − zn),
where zi ∈ C are the polynomial’s complex roots.
Before today, you might have said the equation x2 + 1 = 0 has no solutions.
Now you know its solutions are the complex numbers z1 = iand z2 = −i. Euler’s formula
You already know cos θ is a shifted version of sin θ, so it’s clear these two functions are related. It turns out the exponential function is also related to sin and cos. Lo and behold, we have Euler’s formula:
eiθ = cos θ + i sin θ .
Complex numbers Inputting an imaginary number to the exponential function outputs a complex number that contains both cos and sin. Euler’s formula gives us an alternate notation for the polar representation of complex numbers: z = |z|∠φz = |z|eiφz.
If you want to impress your friends with your math knowledge, plug θ = π into the above equation to find
eiπ = cos(π) + i sin(π) = −1,
which can be rearranged into the form, eπi + 1 = 0. This equation shows a relationship between the five most important numbers in all of mathematics: Euler’s number e = 2.71828 . . . , π = 3.14159 . . ., the imaginary number i, 1, and zero.
It’s kind of cool to see all these important numbers reunited in one equation, don’t you agree?
De Moivre’s theorem
By replacing θ in Euler’s formula with nθ, we obtain de Moivre’s theorem:
(cos θ + i sin θ)n= cos nθ + i sin nθ.
De Moivre’s Theorem makes sense if you think of the complex number z = eiθ = cos θ + i sin θ, raised to the nth power:
(cos θ + i sin θ)n = zn = (eiθ)n= einθ = cos nθ + i sin nθ.
Setting n = 2 in de Moivre’s formula, we can derive the double angle formulas as the real and imaginary parts of the following equation:
(cos2θ − sin2θ) + (2 sin θ cos θ)i = cos(2θ) + sin(2θ)i.
Links
[ Mini tutorial on the complex numbers ] http://paste.lisp.org/display/133628
Chapter 4 Mechanics
4.1 Introduction
Mechanics is the precise study of the motion of objects, the forces acting on them, and more abstract concepts such as momentum and energy. You probably have an intuitive understanding of these concepts already. In this chapter we will learn how to use precise mathematical equations to support your intuition.