Imaginary Numbers

We are working with radical expressions this week, which means we have strayed far from the earth's surface to talk about imaginary numbers. If there were ever a good time to ask x=why?...

I have some questions about these imaginary numbers.

First off, why is the square root of -1 "imaginary"? If nothing times itself can equal a negative number, than how can these numbers exist at all?

Second, Why do we need them? When, on earth, would you ever want to use them?

By Michael Alison Chandler  |  December 15, 2008; 11:26 AM ET  | Category:  Class Time
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Comments



Try reading pages 19 and 20 of the book previewed here for a partial answer to your second question.

http://books.google.com/books?id=lvhET_Mbu18C&printsec=frontcover&source=gbs_summary_r&cad=0

Posted by: davidcash | December 15, 2008 12:34 PM | Report abuse

The term "imaginary numbers" just lead to confusion. Unfortunately, we're stuck with it. The square root of -1 isn't a "fake" number, it's just not an element of the set of "real" numbers (another poor choice of a name).

From http://www.math.toronto.edu/mathnet/answers/imaginary.html, "The name dates back to when they were first introduced, before their existence was really understood. At that point in time, people were imagining what it would be like to have a number system that contained square roots of negative numbers, hence the name "imaginary". Eventually it was realized that such a number system does in fact exist, but by then the name had stuck."

What are they good for? Well, they make a convenient 2-dimensional space - you can think of a complex number like 2 + 3i as being the same as the ordered pair (2,3), with real numbers on the x axis, and imaginary numbers on the y axis. In engineering, this is a common use when you're working with waves - frequency is represented by a real number, and phase (shifting the wave) is represented by an imaginary number. There are probably many more that I'm not aware of.

But all that is probably a few years away for your average 8th grader. For the moment, the most useful thing for imaginary numbers might be to eliminate exceptions to rules. Things like "ax^3 + bx^2 + cx + d = 0 has 3 solutions" works all the time only if you allow complex solutions.

Posted by: tomsing | December 15, 2008 1:09 PM | Report abuse

Imaginary numbers do exist and are usually called complex numbers even though these numbers are not more difficult to use than real numbers. The term "imaginary" is unfortunate as is the term "complex." It is best to ignore these terms and think of i as just a new kind of number whose property is that i*i=-1.

Without complex numbers, our modern technology would not exist. Complex numbers are essential for electrical engineers. The famous inventor Tesla used them to design the devices that are used to generate our electricity.

In algebra, complex numbers are important because they "complete" the real numbers. What this means is that any polynomial with real number coeficients may be factored completely using the complex numbers. For instance, the polynomial x^2+1 may not be factored further using the real numbers, but using the complex numbers, it factors as (x+i)(x-i). This factoring theorem (the fundamental theorem of algebra) was first proven by the great mathematician Gauss.

Posted by: srandby | December 15, 2008 2:14 PM | Report abuse

I don't think my previous post addressed your concern about the existence of imaginary numbers.

You ask: "If nothing times itself can equal a negative number, than how can these numbers exist at all?" This is a common question for students new to imaginary numbers. But this question contains a misconception.

Compare the following two statements:

1. A number times itself cannot equal a negative number.
2. A real number times itself cannot equal a negative number.

The first statement is false while the second statement is true. Yet, the only difference between the two statements is the term "real." The common misconception this illustrates is the idea that the real numbers are the only numbers that exist. In fact, the real number system is just one of many other types of number systems. What makes the real numbers seem special is that these are the kinds of numbers used in everyday life.

So, just think of i as a new kind of number, one whose square is -1. Whenever you see i*i or i^2, just replace it with -1. Nothing to it.

Posted by: srandby | December 15, 2008 2:40 PM | Report abuse

Picture the real number line as a horizontal line, running from -inf to +inf.

The number line for imaginary numbers intersects the real number line perpendicularly at the origin.

Multiplying a number by i (which you refer to as the square root of negative one) rotates that number by +90 degrees. So if you take 1 on the real number line, multiply by i, your number is now on the imaginary axis. Multiply i by i and that's another 90-degree rotation, now falling on -1.

Imaginary numbers are extremely useful in all areas of physics, because you can use the natural exponent and imaginary powers (exp(a+bi)) to represent sinusoids that grow or decay over time. That's everything from a weight on a spring to current through a wire (which is my field) in a single general representation.

You're actually just reaching the edge of one of the most beautiful parts of mathematics.

See here:
http://en.wikipedia.org/wiki/Euler%27s_formula

http://en.wikipedia.org/wiki/Euler%27s_identity

Posted by: PseudoNoise | December 15, 2008 10:16 PM | Report abuse

Amongst the prose, a little poetry.

e^i*pi = -1

Think about it. The identity element for multiplication (1) combined with the two most important transcendental numbers in a single equation. 2.318... to the 3.14159 something equals 1??? How is that possible?

I'm reminded of a moment from basic analysis in college. Take as an assumption that a function is its own derivative. All the properties of exponentials follow. Math is cool.

BB

Posted by: FairlingtonBlade | December 15, 2008 11:07 PM | Report abuse

"I'm reminded of a moment from basic analysis in college. Take as an assumption that a function is its own derivative. All the properties of exponentials follow. Math is cool."

No arguments with the last bit, but you also need to assume that your function is not identically zero. Granted, the zero function is important, but it's not nearly as interesting.

Posted by: one2three4five | December 16, 2008 1:26 AM | Report abuse

"why ... use them?"
for *solving equations*, of course!

the "natural" numbers (not the reals!)
are the ones we use in "everyday life":
*counting* is among the most basic
of those intellectual activities we count
as "mathematical". it may even be
*so* basic as to count as "pre-mathematics".

but *solving equations* presumably
counts as true math on any model.
and check it out: the natural number
equation x + 5 = 3 has *no solution*!
and so mote it be, from pre-history
even unto just a short handful
of centuries ago, when some advanced
avant-garde community (lone geniuses
had long since invented the idea
*many times*) decided to say,
"hey, why not just *say* there's a number
[call it "-2"] that solves this equation?"

in my imagination, this caught on because
it makes financial accounts easier to keep:
credits and debits count plus and minus
(respectively or not; accounts differ).
anyhow, it caught on.

there's a sense in which negative numbers
are "more imaginary" than positive:
they *just don't occur* in certain
common contexts. but we should never
imagine that even counting numbers
are any less a figment of our imaginations.

one could go on to point out that
equations like 7x = 11 also call
for a "new" kind of number: fractions
(we call 'em "rationals" in the trade).
these arise in my private epistemology
as parts-of-wholes rather than as
solutions-of-equations, so one could also
go on to omit this example.

point is, even after we "complete"
the rational numbers (positive and
negative "fractions", together with 0)
to the so-called reals, certain very
simple equations admit of no solution.
specifically, x^2 + 1 = 0 famously
has no "real" solution. but by extending
the concept of number *one more time*
we get the "algebraically closed field"
called The Complex Numbers (or just
plain C [i call it ${\Bbb C}$ ...]).

this means that at long last
every polynomial equation
(whose variable can't be eliminated
to get an identity or contradiction
like "x+x = 2x" or "x = x+1")
can be considered to *have* a solution
and we'll know [in principle] how
to *calculate* the doggone thing
and we can get on with whatever it was
that made us want to solve such-and-such
an equation in the first place.

that's if you like that kind of thing.
some people believe math is the servant
of science but opinions differ.
for *me*, the complex domain is there
so we can do *algebraic number theory*.

exercise: which natural numbers
can be written as a^2 + b^2,
where a and b are themselves naturals?

note that there are no square roots
of negatives in this *question*.
you can also easily state its *answer*
in terms of natural numbers alone.
but if you set out to *prove* it
(or, better still, to *understand* it),
you'll probably find C to be useful.

the editor clobbers links around here;
here's the wikipedia entry on the
sums-of-squares "exercise":
http://en.wikipedia.org/wiki/Fermat%27s_theorem_on_sums_of_two_squares


Posted by: vlorbik | December 16, 2008 1:06 PM | Report abuse

It has been so interesting to read the daily blogs and comments. My niece is Tricia Winning, the teacher, and I am so proud of her!! Jan Lynch, Garden City, New York

Posted by: jlynch3 | December 16, 2008 1:54 PM | Report abuse

Imaginary numbers pop up all the time in electronics and in quantum mechanics, and I'm sure in other systems as well. So long as you require that the result of your calculation is real (which can be done), like if you're calculating a current or a resistance, then you can use imaginary numbers to solve the problems. And in quantum mechanics, where results are probabilistic, while the probabilities are real, the original wavefunctions are actually in general complex (real and imaginary parts). It's cool stuff!

Posted by: UberJason | December 16, 2008 9:34 PM | Report abuse

Hopefully, I'm not too late to the party. I thought I would throw in my two cents.

The first thing to remember is mathematics isn't handed down to us by some deity, people are making it up as they go along. Imaginary numbers are no exception.

If I recall correctly, imaginary numbers first come up in the solution of equations. There were certain equations (called cubic equations) that had procedures for solving them that involved taking square roots. When these methods were invented, it was always assumed that the numbers would be the regular numbers(which I shall call 'real numbers') that you and I know so well. The problem is that every so often, you would find that you needed to take the square root of a negative number.

Naturally, this didn't make any sense to anybody. So many did the reasonable thing of just stopping the calculation. Others, argued that if you just pretended these square roots of negative numbers were 'real' numbers and continued the calculations, you always got the answers to the equation you were trying to solve. You knew these were the solutions because they were real numbers and you could put them into the original equation and see that they solved it. So, there was a powerful incentive to use these 'imaginary' numbers because it meant you could solve problems that the people who wouldn't use them could not.

Thus one major application of complex numbers is increase the number of problems that you can't solve using just real numbers.

Posted by: mathlete | December 17, 2008 1:22 PM | Report abuse

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