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Write-up

 

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 non-existent 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 make-believe, 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 real-world 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 numbers-one 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 two-dimensional world, so only two-dimensional 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 three-dimensional object, has no direct application to the two-dimensional world of the shadow. Likewise, complex numbers may not be directly applicable to a real world measurement any more than a three-dimensional 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.st-and.ac.uk/history/Amthematicians/Euler.html
www.jimloy.com/algebra/imaginar.htm