Chapter 2: Euler's Formula
The Story…
In the first chapter we asked the question: When does a polynomial have all roots real? The reason we asked it is that it is easy to write a polynomial equation for which a solution cannot be found among "ordinary" numbers. For example, if we try to solve: for x, we quickly see that there can be no ordinary number (number representable in principle as a possibly unending decimal) that satisfies this equation.
We call
numbers of this "ordinary" sort: real numbers. In order to
deal with the nonsolvability of such simple equations, mathematicians invented
a new number called i. This number is defined to satisfy the
equation above. So by definition! This number i is
in some sense independent of all the real numbers, and so when i
is adjoined to the real numbers, we must represent the new numbers in the form:
where a and b are ordinary real numbers, and i
is the special new one. Then admitting that the usual rules of arithmetic
should still hold (commutative, associative, distributive laws, and so on) it
is easy to prove that the "numbers" in this special form: form a perfectly good number system.
This
system of numbers is called the complex numbers, or the imaginary
numbers. They are far from imaginary, however, because their use is
absolutely essential in formulating all of the most fundamental laws of modern
physics, that is, Quantum Theory.
We should
state a few of the most important properties of complex numbers. First of all,
every cubic equation (and indeed every polynomial equation at all)
where the coefficients (a, b, c) are fixed complex or real numbers, has a
solution among the complex numbers. Thus, we need no other special numbers to
solve polynomial equations. This is the so-called Fundamental Theorem of
Algebra.
Next,
given complex number ,
we say that the "absolute value" or "norm" of z is
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This norm is denoted: and obviously, if and only if a = 0 and b = 0. In the explorations, we will denote as . If w and z are two complex numbers, then
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Further,
if is an arbitrary complex number, then there is
another complex number called the "conjugate of z" which is defined
to be and is denoted or in the explorations, .
Obviously, the conjugate of the conjugate of z is z itself. Also,
in general, and .
From this it follows that if is not zero, then
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and so the inverse or reciprocal of z is .
Now there is a
remarkable relationship between the number i and another famous
number in mathematics called e, for Euler's number. Euler
discovered that if t is a real number, then one could consistently write
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and all the usual rules of algebra and trigonometry would still hold.
This is a
truly remarkable fact, and it is called Euler's Formula. It follows
easily that for all real t. The complex numbers are
sometimes denoted ,
just as the real numbers are denoted .
Complex numbers in are often represented as points in a plane.
The usual convention is to represent as the point
. But there is in light of Euler's Formula, another convention that uses
polar coordinates. For r > 0, and
arbitrary real numbers, the complex number is the number z such that ,
and such that the ray from the origin to the number makes an angle (measured in radians) from the positive real
axis. We will frequently use this polar representation.
On the Complex Arithmetic page is an exploration
that will help you become familiar with the representation of complex numbers
in the plane and the algebraic operations on them. After that, we will discuss
the roots of unity, which play a critical role in the algebra of polynomial
equations.
