I really displike how too many thngs in mathematics are not explained clearly and simply. Too often really cool things are kept out of the reach of normal people - because the explanations are written in some alien inaccessible language.
What the mathematics community need to realise is - writing stuff in a language only a few of them understand is not big and it's not clever. I'm not impressed.
What is clever - what is hard - is being able to explain those ideas so that the maximum number of people can understand and appreciate them. That should be the measure of success.
If research grants and teaching salaries were based on this measure I wonder how quickly the profession, and the texts it publishes, would change?
Anyway, I'm going to try to demystify complex numbers.
Complex Numbers Are Not Complex
Complex numbers are not complex. That's a really unfortunate name which scares people. It's an accident of history. They should have been called composite or 2-part numbers, or anything that doesn't put barriers up straight away.
Normal numbers, like 2, 3, 4.5, 9.332 are 1-dimensional. That is they describe one thing and only one thing. They could be the length of a rope, or the time it took to finish a race. They can't describe two things at the same time. That is, you can't use a single 1-dimensional number to describe both the width and length of a swimming pool - you have have to have two numbers, one for the width and one for the length.
Complex numbers are 2-dimensional. They have 2 parts. These two parts are independent of each other. One part can do what it likes and is not in any way influenced or constrained by the other. The two parts of a complex numebr are like apples and oranges. You can't mix them up, an apple will always be an apple and not an orange. This is an important thing to keep im mind when we combine complex numbes later.
The following shows the two parts of a complex number. They just happen to be called the real part and the imaginary part. Again, don't let these names put you off, they have been settled on without anyone really considering the best names for these parts.
We could have written these two parts in a different way. For example, (2,3) or (2,3i) or (3i,2) or even (3i+2) ... but the convention is (2+3i) and sometimes without the brackets when it is clear enough what we mean 2+3i. Again, this is just a convention that we arrived at through history becuase it worked well for lots of people. If you are an engineer, you may have seen the symbol j used instead of i. That's ok becuase those engineers often use i to mean other things - they use j to avoid ambiguity. So they might write (2+3j), but most people use i.
What are these numbers good for? Well they are in fact really useful for lots of things in science and engineering, not just for making fractal images!
For now let's just notice that these 2-part complex numbers are very similar to coordinates locating points on a flat 2-dimensional surface. Just like a map grid reference or the familiar (x,y) coordinates when we work with graphs. We can in fact think of comlex numbers as points on a flat surface becuase they have two independent parts. Look at the following diagram and you'll see they aren't that different to the normal (x,y) coordinates for points on a graph.
The first diagram shows the complex number (3+2i). You can see the horizonal distance is 3 and th evertical distance is 2. This is just like (3,2) on the more familiar (x,y) graphs. The second digram shows (-2-2i) which is just like (-2,-2) on (x,y) graphs.
Ok so complex numbers can very easily represent points on a flat surface, just like (x,y) graph coordinates.
That was easy enough!
Add, Subtract, Multiply with Complex Numbers
What about the normal rules of arithmetic? Adding, subtracting, multiplying? Do they make sense for these strange objects we seem to have invented called complex numbers? Let's deal with adding and subtracting first because they are really easy.
Adding complex numbers is really really really easy! Remember when we said the real and imaginary parts of a complex number are like apples and oranges. We said that these were independent because and you can't change an apple into an orange, and the number of apples doesn't influence the number of oranges at all. Well that's really handy. Because if we want to add two complex numbers we simply add up all the apples from the two numbers to tell us how many apples should be in the answer. The same for oranges. Another way os saying this is that we combine the real and imaginary parts separately.So (2+3i) added to (4+5i) means we combine the 2 and 4 (real parts) and also combine the 3 and 5 (imaginary parts). The answer is (6+8i).
If you're still struggling, think of adding the 2 apples to the 4 apples to make 6 apples. Then adding 3 oranges to the 5 organges to make 8 oranges. These are the two parts of the answer:
(2 apples, 3 oranges) + (4 apples, 5 oranges) = (6 applies, 8 oranges)
(2 + 3i) + (4 +5i) = (6 + 8i)
TIP: Did you know that Google can do maths with complex numbers for you. Try it! Type (2+3i) + (4+5i) into the Google search box and you'll get the answer!
What about subtraction? It's the same. You combine the independent real and imagines parts by subtracting them instead of adding them. Or if you prefer, you subtract the apples, then you subtract the oranges. So (2+3i) - (4+5i) = (-2 -2i). Yes, these have negative parts because 2-4 is -2, and 3-5 is also -2. Again Google can do this too.
Actually this is just like adding and subtracting the familiar (x,y) coordinate vectors if you know about those already.
What about multiplication? This is again just like the normal maths that we know, exceot for one tiny special difference. In fact, to make sure no-one is put off by special differences, we'll tell you know that this single tiny difference is the ONLY difference you need to know about. Once you get it, there are no more hidden suprises later on. That single difference is the only thing you need to remember when working wih complex numbers
So how do we multiply complex numbers? Well remember the familiar matrix multiplication? Multiplying (a + b) by (c + d) was done my multiply each combination of things inside the brackets. So to make the answer you'd have to work out a*c, a*d, b*c and b*d ... and once you'd done all t he laborious work you'd collect terms that were similar.
Let's illustrate this with an example (2x + 3y) * (4x + 5y). We've chosen to include x and y bits because of we didn't the example would be too easy and not illustrate collecting like terms: (2+3) * (4+5) is just 5*9 which is just 45. So let's expand those brackets out. The bits we will collect are 2x*4x, 2x*5y, 3y*4x and 3y*5y. Nothing new here, just what we would normally do. Do the multiplications and we get 8x2 + 10xy + 12xy + 15y2. We have collected all the similar terms yet, there are "xy" bits. Once we've tidied it all up we get 8x2 + 22xy + 15y2. Notice there are 3 kinds of animal here, the x2 bit, the y2 bit and the xy bit. We can't simplify any further because the xy bit can't become x2 as they are indepdnent different animals - just like apples and oranges are independent.
Noe let's do this familiar multiplication of brackets with complex numbers. Let's try (2+3i) * (4+5i). The answer will have to have the combinations 2*4, 2*5i, 3i*4 and 3i*5i. Lets do the multiplications to get 8 + 10i + 12i + 15i2. Let's carry on tidying up by collecting the similar terms to get 8 + 22i + 15i2. This just like the example above which used x and y instead of real and imaginary parts - that's why we get the same numbers 8, 22, 15.
So (2+3i) * (4+5i) gives 8 + 22i + 15i2. But we don't stop there. That one single special rule we mentioned for complex numbers allows us to replace the i2 with something simpler.
The rule is i2= -1.
So i*i an always be replaced by -1. You don't have to but it helps simplify the expressions. This is a bit magical - combining two imaginary bits creates a real bit. The oranges multiplied together became a negative apple!
Before you think complex numbers will have all sort os such crazy exceptions and rules - stop worrying now. That's the only rule you have to know - and it's really easy to remember.
So if we apply it to the expression we replace 15i2 by -15 .... to get the final answer: (2+3i) * (4+5i) = 8 - 15 + 22i or more simply (-7 + 22i). Again you can check with Google.
That's it! Easy peasy!
We didn't do division because it's just a tinsy bit more complex in terms of messing about with algebra and doens't introduce new ideas, so we left it out to avoid distraction.
You can find a fuller explanation, slower in pace, and with more examples in the ebook.