Complex Numbers

1
Complex Numbers

• Adding in the Imaginary i

By Lucas Wagner
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The Domain of the Square Root
We might think of as a parabola on its side,
with the following equivalent statement
So we can see that negative values of x do not
yield any real y-values.
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The Quadratic Formula
In a mathematically thorough and rigorous manner
we can apply the quadratic formula,
to any equation that can put in this form
But we can see the possibility of problems
occurring there is no mathematical requirement
that the number under the radical,
, be positive, as the following example shows.
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Girolamos Problem
In The Great Art, published in 1545, Girolamo
Cardano discusses the following problem.
No Intersection!
To find x and y, use substitution.
Apply the Quadratic Formula.
Due to the symmetry in the problem, x and y take
on values.
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Bombelli works with Imaginary Numbers
Rafael Bombelli in the 1560s figured out a way
to work with imaginary numbers. We write this in
modern notation as
Using these rules, Bombelli worked Cardanos
cubic solutions (Sketch 11 HW) to arrive at real
results, so he wasnt just interested in
imaginary numbers for themselves.
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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
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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,
Using FOIL and Bombellis rules, we can find the
product of z and w.
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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, has the
graphical interpretation,
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Trigonometric Formulas and Complex Numbers
One can derive Eulers formula from the Taylor
series representations of sine, cosine, and the
exponential, and the rules developed by Bombelli.
The result is the following
One can derive many familiar trigonometric
formulas using Eulers formula and the properties
of the exponent, i.e.
These formulas can be used to derive Abraham De
Moivres formula (though in history De Moivres
formula came before Eulers)
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Polar Form of Vectors
The complex number z also has a polar form. It
uses Eulers formula as its backbone.
The r gives the length of the vector, and ei?
gives the direction.
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Physics ApplicationCentripetal Motion
Consider an object moving in a circle of radius r
with an angular frequency of ?. What is its
velocity and acceleration?
Create a parametrization of the position in the
complex plane,
Assuming that we can take derivatives like usual,
Multiplication of complex numbers is a rotation
in the complex plane. In this case, the iei?t
gives us the direction tangent to the circle.
The direction of the acceleration here is ei?t,
which is in towards the center of the circle.
Thus we can establish the following