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### Imaginary (Complex) Numbers

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Date: 02/26/98 at 10:00:18
From: Chris
Subject: Imaginary (complex) numbers

What are imaginary numbers?  I have to do a report on them and can't
find anything understandable anywhere else.  Thank you for your time.
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Date: 02/26/98 at 15:25:10
From: Doctor Rob
Subject: Re: Imaginary (complex) numbersz

You probably are aware that for any real number a, a^2 >= 0. If one is
faced with an equation like x^2 = -9, one can immediately conclude
that there is no real solution.  Imaginary numbers were invented to
provide solutions of equations like this.

The simplest imaginary number is called i. It satisfies the property
that i^2 = -1. Using it, the solutions of x^2 = -1 can be found:
x = i is one, and x = -i is another, if we define -i properly. Since i
cannot be a real number, it must be something else. It is called an
"imaginary" number. Of course, it is no more imaginary than 7, 0, -4,
3/5, sqrt(2), pi, or any other number. They are all figments of our
imagination!

In order to be a "number," i must satisfy the axioms of arithmetic.
It must have an additive inverse, or negative, or opposite, -i, so
that i + (-i) = 0. It must have a multiplicative inverse, or
reciprocal, which it does: in fact -i = 1/i. (You can prove this
by showing that i*(-i) = 1.) Thus -i is both the negative and the
reciprocal of i. Multiplication by a real number must be defined,
so a*i must make sense, for any real number a. In particular,
(-1)*i = -i. All the other axioms of arithmetic must also be true
for the set containing the real numbers and i and whatever else you
can get by doing arithmetic with them.

Now you can solve x^2 = -9 = 9*(-1) = 3^2*i^2 = (3*i)^2. The two
solutions are x = 3*i and x = -3*i, imaginary numbers! The same trick
works for x^2 = -a, where a > 0 is a real number.  x^2 = -a = a*i^2,
so x = sqrt(a)*i and x = -sqrt(a)*i are the two solutions, imaginary
numbers.

The system of numbers you end up with by looking at the set containing
the real numbers and i and whatever else you can get by doing
arithmetic with them has a name, of course. It is called the complex
numbers. It turns out that every complex number can be written in the
form a + b*i, for some real numbers a and b. This means that every
complex number is the sum of a real number and an imaginary number.
The rules for doing arithmetic with complex numbers are all the
ordinary ones, because we insisted that that be so when we built up
the set. The only difficulty in seeing that the form a + b*i includes
all complex numbers is in multiplication and division. For
multiplication, do the ordinary multiplication of two complex numbers
of this form using FOIL, then use the fact that i^2 = -1, and combine
terms containing i and ones not containing i.  For example,

(3 - 2*i)*(-1 + i) = -3 + 3*i + 2*i - 2*i^2,
= -3 + 3*i + 2*i + 2,
= -1 + 5*i.

The result is back in the same form, a real number -1 plus an
imaginary number 5*i.

For division, we have to know one trick:
(a + b*i)*(a - b*i) = a^2 + b^2.
Multiplying these two complex numbers gives a real number answer!
If you find this hard to believe, multiply it out using FOIL and
i^2 = -1, and see what you get.

Now we can do division. To divide by a + b*i, instead multiply by its
reciprocal, which is

(a - b*i)/(a^2+b^2) = a/(a^2+b^2) + [-b/(a^2+b^2)]*i,

and use the rules given above to reduce it back to the form of a real
number plus an imaginary number. Of course, we cannot do this if
a^2 + b^2 = 0, but that only happens if a = b = 0, so the divisor is
0 + 0*i = 0. Division by zero is forbidden, but we are used to that!

As an example, since (-1 + i)*(-1 - i) = (-1)^2 + 1^2 = 2, and so the
reciprocal of -1 + i is (-1 - i)/2 = (-1/2) + (-1/2)*i, we get

(3 - 2*i)/(-1 + i) = (3 - 2*i)*(-1 - i)/2,
= (-3 - 3*i + 2*i + 2*i^2)/2,
= (-5 - i)/2,
= -5/2 - (1/2)*i.

Now you can do any arithmetic with complex numbers, and in particular,
with imaginary numbers.  You can also take square roots of any real
number, whether positive, zero, or negative.

-Doctor Rob,  The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Number Theory

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