Stuff About Complex Number I Wish Somebody Told Me
by Rito Ghosh
Introduction
There are many places that teach you complex number like High School, college classes, YouTube channels, and so on.
But they earlier ones often heavy on solving problems by-hand, and almost every time, they do only that.
The latter ones focus mainly on shiny visuals or advanced properties that are not intended for the uninitiated.
This is a piece written for both people who are getting introduced to Complex Numbers for the first time, and for the people who have studied it in college, and are hazy on the topic. (Why won’t they be? They had to focus on solving boring problems by hand.)
When I saw Jeremy talk about Complex Numbers on the APL study group, I knew I had to write something- I did my Junior year Math paper on Complex Number in High School!
The idea is to:
- quickly introduce Complex Numbers, and how to work with them
- talk about their origins: why need complex numbers in the first place?
- show how they extend the number line
- show how \(i\) works as an operator, and briefly what operators are
- talk about a real world example, and probable future usage
I will keep it short and crisp.
Comples Numbers
Complex Numbers, as you might have seen are written like this:
\[x = a + ib\]
Here, the whole thing in the right-hand side is the number. a is the Real part, and b is the imaginary part.
Do not get misguided by the names. Imaginary numbers aren’t imaginary at all. They make appearance, or rather, they can represent stuff in real life really well! We will soon see.
You can add and subtract complex numbers. You add (or subtract) the real and imaginary parts separately.
Let me show you.
\[(4 + 5i) + (3 + 6i) = (4 + 3) + (5 + 6)i\]
\[ (6i) - (3 + 7i) = (0-3) - (7-6)i = -3 - i\]
They can be multiplied and divided with real numbers and other complex numbers. They work just as you expect them to.
\[(3 + 2i)(1 + 4i) = 3 + 12i + 2i + 8i^2\]
Let’s not spend too much time here.
As how they behave, and more related information can quickly be found- and good ones at that- on internet.
I suggest the Khan Academy course on Complex Numbers.
Oh, I haven’t told you the value of \(i\) yet.
\(i\), is defined such that,
\[i = \sqrt(-1), \text{or}, i^2 = -1\]
That’s it!
Why do we Need i, and the Whole Complex Number Scenery
The story goes, when the Greeks were ridden with famines and other troubles, the Oracle of Delphi told them to double the height of the cube made in honor of Apollo.
The Greeks could not do it! How could they?
To double the size of a cube, you need to find the cube root of 2. The Greeks didn’t know how to that!
Too bad.
You are probably familiar with the story that Pythagoras, frustrated with the fact that the the size of the diagonal of a square with the side of 1 could not be measured, ordered to off someone! Just like that!
Because the square root of 2 was not measurable then. It is not a number that you write down like the others.
Human beings needed something that could be tamed- numbers that always had roots- no matter what the number was!
This is how it came to be. Complex numbers are numbers that always has roots. No matter what.
And with the advent of imaginary numbers, it was possible to calculate the square root of negative numbers, too. But that came later.
Extending the Number Line
The number line is a simple animal.
It is one-dimensional. It extends both ways of zero.
And thus, it was limiting.
Complex numbers was defined in such a way that they added one extra dimension to the number line.
It wasn’t so one-dimensional anymore.
Picture: Wikipedia
Now, numbers weren’t limited to a line, but it was a plane.
And many more operations became possible that simply weren’t before.
i as an Operator
i is not simple. Adding it or multiplying it has consequences.
If you don’t know yet, very simply, operators are stuff that act on other stuff.
(No mathematical rigor whatsoever can be expected from the previous line)
You know operators. We are talking about it in APL. We use them whenever we code.
Functions are also operators.
Think, for example, about the - operator.
This is our friendly negative sign. But add it before a positive number, and the result will be a negative number. The number will no longer remain positive.
Think about the + operator in math. It takes two numbers, not one. And the result is the sum of both of them.
Operator takes stuff in it, and spits out results.
i is also an operator.
Multiplying a number with i rotates the number by 90 degrees counter-clockwise in the complex plain.
Rotate it twice, and you get the negative of a number. Because, you know, i is the square of -1. And multiplying a number with square of i negates it, i.e. rotates the number by 180 degrees, or throws it to the dark, ehm, negative side.
This has important consequences.
Example of Use of Complex Number in Real World: Electronics
The scope of this piece is very limited.
But I will show you one example.
Alternating Current is not steady, or fixed like Direct Current or DC. AC current, the one powers our home is the only practical form of electricity that can be practically transported from one point to another over a long distance.
This form of electricity acts like a wave.
And whenever waves are involved, we have to deal with angles, frequencies, and amplitudes.
When angles become involved, we can use trigonometric functions like cosine and sine.
And complex numbers are related to trigonometric functions via the Euler’s Formula:
\[e^{i \theta} = \cos{\theta} + i \cdot \sin{\theta}\]
And, when we can use the special number, e, we can solve many problems easily because the function returns itself after differentiation and integration.
That makes a lot of the things easier.
While this is the reason for using complex numbers in electronics in the first place, they also make life easier for easier calculation and quantity manipulation.
Why Bother: Isn’t One More Dimension Enough?
You might think that complex numbers are such nuisance, and we could have done the same clever things with adding one more dimension to the real number like we do in geometry. Would not a ‘y’ suffice instead of this dimension riddled with ‘imaginary numbers’?
The answer is no.
You will not have the same behavior then.
You won’t be able to get the niceties that is afforded to you via Euler’s formula and the access it provides you to the exponential function.
I hope I was able to give you some intuition about complex numbers, and their uses.
Please let me know what you think.
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