Title
Imaginary Numbers!
Problem Statement
Trying to explain imaginary numbers to the average Joe!
Problem setup
What are imaginary numbers (i=square root of 1) and what purpose do they serve?
Plans to Solve/Investigate the Problem
I plan to attempt to explain imaginary numbers in the simplest of terms so that
I can better understand this concept and perhaps help others like me to better
understand it. I have printed several pieces of information from various math
websites in order to try and put together an explanation that may clear some of
the confusion.
Investigation/Exploration of the Problem
First of all, imaginary numbers are not numbers that exist only in the brains of
mathematicians. If that were the case, then ALL numbers in math are “imaginary”
in the sense that you can’t touch them or experience them directly. There is
nothing more “imaginary” about i than there is about the square root of 2 or
some other “real” numbers because all numbers just exist in the heads of people
anyway! They are called imaginary because most mathematicians thought they had
all the “real” numbers already, before imaginaries were invented. It is not
meant to be used to mean “some nonexistent number that has a certain
property.” The problem is that the way we normally use the word “imaginary” is
not what is meant when we talk about imaginary numbers. Unfortunately, because
the word ‘imaginary” is associated with the makebelieve, that stirs up a lot of
confusion over the concept of this new number i. The term “imaginary”, when used
to refer to multiples of i, is a technical term and because of its pervasive use
among scientists and mathematicians, it helps to learn the term for
communication’s sake. Mathematicians should have waited before assigning this
name to this number! Most people learn the words “imaginary” and “number” in a
completely different context from that used by mathematicians, so there is
trouble. One mathematician, Hamilton, proposed explaining these numbers in such
a way that the word “imaginary” was never used. For best understanding, don’t
dwell on the term “imaginary number”! That insight helps a great deal. The
symbol i only means the square root of 1. Imaginary numbers are numbers that
can be written as a real number times i. So, what is a real number and what is
i? The real numbers are all positive numbers, negative numbers and zero. They
are the ones on the number line. Okay, so what is i? It is the square root of
1. Actually there are two numbers that are the square root of 1, and those
numbers are i and –i, just as there are two numbers that are the square root of
4, 2, and 2. It is NOT a real number. According to the popular idea of numbers,
there are two major groups of numbers and they are real numbers and imaginary
numbers. The square root of a real number is always a positive number.
i was invented because some renegade mathematicians wanted to be able to take
square roots of negative numbers, and you cannot do that if you only have real
numbers because there is no real number that can perform this operation. These
mathematicians decided they wanted to make up a new number and call it i. They
decided it would have the property that when you square i, you get –i. With this
new number, you can take the square root of negative numbers. For example, the
square root of 4 is 2 times i which can be written as 2i. The square root of 2
is 1.41421i. So we can make an imaginary number by taking a real number like 5
and multiplying it by i. That gives us 5i, an imaginary number. The square of
any imaginary number (except 0) is a negative number. A complex number is a real
number plus an imaginary number like 7 + 4i.
Extensions of the Problem
One of the aims of mathematicians is to say what is possible and what is not
possible. Having a number that is less than one and greater that than three
would imply that one is greater than three, which is just false. These numbers
are called imaginary because they are just rarely of any use to people, just as
fractions are of no use to someone counting unbroken marbles. They are very
useful in some branches of mathematics such as engineering. But, they do have
much less direct relevance to realworld quantities than other numbers do. An
imaginary number could not be used to measure how much water is in a bottle, or
how far a kite has traveled, or how many fingers one has. However, there are a
few real world quantities for which complex numbers are the natural model. The
strength of an electromagnetic field is an example. The field has both an
electric and a magnetic component so it takes a pair of real numbersone for the
intensity of the electric field, one for the intensity of the magnetic field to
describe the field strength. This pair of real numbers can be thought of as a
complex number. This next section makes more sense to me than anything so far
where complex numbers are concerned. Many properties related to real numbers
only become clear when the real numbers are thought of as sitting inside the
complex number system, so instead of having two separate worlds of numbers, real
and imaginary, the real numbers actually exist INSIDE the complex
system. So, there is one huge system, the imaginary number system, which
ALL numbers fit in. The real numbers are a subset within the imaginary
system. Then there are the smaller sets within the real number system, such as
integers, etc. That way, complex numbers aid in the understanding of real
numbers instead of being a separate, “weird” entity. It is like trying to
understand a person’s shadow. The shadow of a person’s body is in a
twodimensional world, so only twodimensional concepts are applicable to it.
But, if you think of the three dimensional object, the person’s body casting the
shadow, this can aid in understanding the shadow, even though the person’s body
which is a threedimensional object, has no direct application to the
twodimensional world of the shadow. Likewise, complex numbers may not be
directly applicable to a real world measurement any more than a
threedimensional body is directly applicable to its shadow, but they can still
help us understand it.
Imaginary numbers may not have any true meaning in the real world, i.e., it is
hard to have 3i dollars or even to be 240i degrees east. But in many
applications where the state of a model at one point in time is dependent on the
state of the model at a previous point in time, these imaginary values can
affect the final value. The velocity of the tip of an airplane wing at different
points in time is an example. This part is difficult for me to grasp, but the
explanation states that if you take only the real part of the first velocity,
you get an incorrect number. You must take the imaginary part for the first
velocity of the wing, then the real part for the second velocity of the wing in
order to get the correct output of the wing’s velocity later in the cycle. One
more example—this helps me a bit. Even though I don’t understand what it means
at all, it gives me another concrete reason to explain why we need complex
numbers in the first place. Suppose you have a snowplow that keeps piling up
more and more snow in front of it so that the farther it goes, the heavier the
load it is pushing, and the heavier the load, the slower it goes, and the slower
it goes the slower the pile of snow in front of it grows. You can solve this
equation with real numbers only, but it would be easier to solve this in the
domain of complex numbers because the equations are a lot nicer?! The solution
you care about at the end is only the real number part, but using the imaginary
numbers can help simplify the equation. Real number equations cause much
messiness in some instances!
Author & Contact
Pam Joseph.
pjoseph@rockdale.k12.ga.us
Link(s) to resources, references, lesson plans, and/or other materials
Internet References:
www.mathforum.org/library/drmath/
www.math.toronto.edu/mathnet/answers/relevance.html
www.history.mcs.stand.ac.uk/history/Amthematicians/Euler.html
www.jimloy.com/algebra/imaginar.htm
