As an imaginary brother (that’s right, even imaginary girls have imaginary siblings) I felt I needed to contribute to my sister’s online venture. The problem is that I’m more of a down-the-middle music fan than most site subscribers and really couldn’t hold my own when it comes to cutting edge local music (which is probably too bad since I could have brought some new names from the Bay Area). However, we all have our skills, and since mine are more of a scientific nature I got to thinking…
How about an article on Imaginary Numbers?
“Imaginary numbers?” you say. “You mean like the word ‘Google’? — isn’t that an imaginary number?” Well, yes, that’s true (see appendix below) but in fact the phrase imaginary number refers to a very serious and important branch of mathematics that may seem silly at first but has all sorts of important applications.
It all starts with the familiar observation that when you multiply a (non-zero) number by itself you always receive a positive number. So 2*2=4, and (-3)*(-3)=9, but the right side of the equation is always positive.
Well, what if we define a number i to be the square root of -1. That is:
i*i = -1
This then allows us to find the square root of any negative number. For example
sqrt(-4) = sqrt(4 * -1)
= sqrt(4) * sqrt(-1)
= 2i
Any number of this form (some basic multiple of i) is called an Imaginary Number. As with many concepts in math and science it is not really possible for us to “see” an imaginary number (hey, you can not have 2i apples in your basket) but the concept is quite useful in many real world examples in engineering and physics. The name comes from the fact that the concept was frowned upon until the brilliant German mathematician Gauss showed how useful they could be in solving important equations.
First I should state that imaginary numbers are really a special subclass of numbers known as Complex Numbers. Complex numbers combine both a real and imaginary component and are often written like this:
z = a + bi
where a is called the Real Part of z and b is the Imaginary part of z. The branch of mathematics known as Complex Analysis is dedicated to the study of these numbers. Numbers of the form z = bi are called “pure imaginary numbers” since they are complex numbers where the real component is zero.
As an illustration of the usefulness of complex and imaginary numbers I’ll ask you to remember that good-old quadratic equation from high school. Grumble, grumble, you say? Well, I’ll remind you. The quadratic equation is used to solve problems of the form:
A x2 + B x + C = 0
Well, the solution, you may remember, involves a square root:
x = (-b ± sqrt(b2 – 4ac)) / 2a
But it is possible that the expression inside the square root will be negative! In high school you might have been told that this type of quadratic equation has no answer, since you cannot take the square root of a negative number, right? Well, we know better now. If we introduce imaginary numbers into the picture we have, for example:
4x2 + 2x + 5 = 0
implies:
x = (-2 ± sqrt(4 – 4 * 20)) / 8
x = (-2 ± sqrt(-16)) / 8
x = (-2 ± 4i) / 8
x = -1/4 + i/2 and -1/4 – i/2
So we have found an answer to a previously unsolvable problem. To understand a little bit more about what these answers mean in the real world we would have to go further into Complex Analysis and something really cool called the Polar Coordinate System (where we represent complex numbers by their “size” and their “phase” — which is very useful in signal processing and electronics, for instance), but suffice it to say that these answers have very real interpretations in your daily life.
In fact, one of the most amazing things about complex and imaginary numbers is that they really are the fundamental solution to an important class of algebraic problems. Recall that a polynomial is an expression of the form:
y = anxn + an-1xn-1 + … + a1x + a0
Now it has been shown that for any polynomial the equation has exactly n complex solutions of the form a + bi (where n is the highest power of x in the polynomial). This basic result is known as the Fundamental Theorem of Algebra.
I’ll close with a few amazing results that make me scratch my head and wonder how the world was put together.
Euler’s formula: ei? = cos? + i sin?
[where e is a very special number defined from calculus as lim n¬ ? (1 + 1/n)n]
From which, amazingly: ii = e-?/2
And finally, a question for you: Can you tell me what are the square roots of i?
Appendix:
Search engine name Google is actually derived from the word “googol” that was coined by the mathematician Edward Kasner (apparently listening to his nine year old nephew) and indicates the number 10 to the 100th power which illustrates a concept so large as to be beyond human comprehension – and also, incidentally, far larger than the number of particles in the universe. Another common term is the word “googolplex” which refers to 10 to the googol power (10^(10^100)), which is so large as to be absurd.
On a related note, did you know that there are multiple sizes to Infinity? For instance, the number of natural numbers (1, 2, 3 …) is equal to the number of rational fractions (1/2, 3/17, 4/25…), which is probably surprising, but true. However, the number of real numbers (which include irrational numbers like the square root of 2 and pi) is much, much greater than this first infinity. Cool, huh?