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Complex Numbers
Adding in the Imaginary i
1
By Lucas Wagner
The Domain of the Square Root
We might think of y   x as a parabola on its side, with the following
equivalent statement:
x  y2
2
1.5
1
0.5
-4
-3
-2
-1
1
2
3
4
-0.5
-1
-1.5
-2
So we can see that negative values of x do not yield any real y-values.
The Quadratic Formula
In a mathematically thorough and rigorous manner we can apply the
quadratic formula,
 b  b 2  4ac
x
2a
to any equation that can put in this form:
ax 2  bx  c  0
But we can see the possibility of problems occurring: there is no
mathematical requirement that the number under the radical,
b 2  4ac, be positive, as the following example shows.
Girolamo’s Problem
In The Great Art, published in 1545, Girolamo Cardano discusses the
following problem.
No Intersection!
x  y  10
xy  40
To find x and y, use substitution.
x(10  x)  40
 x 2  10 x  40  0
Apply the Quadratic Formula.
12
10
8
6
4
2
5
10
15
 10  10 2  4  (1)  (40)
x
2
 5   15
Due to the symmetry in the problem, x and y take on ± values.
20
Bombelli works with Imaginary Numbers
Rafael Bombelli in the 1560’s figured out a way to work with imaginary
numbers. We write this in modern notation as:
i  1
i 2  1
i 3  i    1
i4  1
Using these rules, Bombelli worked Cardano’s cubic solutions (Sketch
11 HW) to arrive at real results, so he wasn’t just interested in imaginary
numbers for themselves.
Fundamental Theorem of Algebra
Any polynomial of degree n, with n greater than zero (a non-constant
polynomial), has n roots. In other words, pn(x) = 0 has n solutions.
René Descartes and Albert Girard in the 1600s had their suspicions that
this was the case, if they allowed for three different kinds of roots:
•Positive (considered “real”)
•Negative (considered “false” at the time)
•Imaginary (Complex numbers)
Various mathematicians have tried their hand at proving this theorem:
Leonhard Euler (1749), Pierre-Simon Laplace (1795), and Carl
Friedrich Gauss (1799), to name a few.
Wikipedia – Fundamental Theorem of Algebra
Real and Imaginary Parts
of Complex Numbers
When working with complex numbers, it is shown that breaking the
number into the sum of a real and imaginary part maintains a good
algebraic field. For example,
z  a  ib
w  c  id
a, b, c, d  R
Using FOIL and Bombelli’s rules, we can find the product of z and w.
zw  (a  ib )(c  id )
 ac  i 2bd  iad  ibc
 ac  bd  i(ad  bc)
Complex Numbers as Vectors
Jean-Robert Argand in 1806 came up with the idea of a geometrical
interpretation of complex numbers. Replace the x-axis with the real
part of complex numbers, and the y-axis with the imaginary part.
Thus, z  a  ib has the graphical interpretation,
Trigonometric Formulas
and Complex Numbers
One can derive Euler’s formula from the Taylor series representations
of sine, cosine, and the exponential, and the rules developed by
Bombelli. The result is the following:
ei  cos   i sin 
One can derive many familiar trigonometric formulas using Euler’s
formula and the properties of the exponent, i.e.:
e x y  e xe y
(e x ) n  e nx
These formulas can be used to derive Abraham De Moivre’s formula
(though in history De Moivre’s formula came before Euler’s):
(cos   i sin  ) n  cos(n )  i sin( n )
Polar Form of Vectors
The complex number z also has a polar form. It uses Euler’s formula
as its backbone.
z  re i
The r gives the length of the vector, and eiθ gives the direction.
Physics Application:
Centripetal Motion
Consider an object moving in a circle of radius r with an angular
frequency of ω. What is its velocity and acceleration?
Create a parametrization of the position in the complex plane,
z (t )  re it
Assuming that we can take derivatives like usual,
z(t )  ireit
Multiplication of complex numbers is a rotation in the complex plane. In
this case, the ieiωt gives us the direction tangent to the circle.
z(t )   2 reit
The direction of the acceleration here is –eiωt, which is in towards the
center of the circle.
Thus we can establish the following:
a   2r
v  r