Okay, those are all - well, I think I can kind of see some relation to complex numbers in there, but it's very vague.

So, let me describe how I understand complex numbers. To do that, we'll have to go right back to the very basics of mathematics; numbers.

Imagine, for a moment, an infinite piece of paper. (Or you can get a piece of paper and draw this, if you like; you won't need to draw the whole, infinite thing, just enough to get the idea)

Take a point, nice and central. Mark it "zero".

Select a second point (traditionally, this point is chosen to the right of zero, but the location doesn't matter). Mark it "one".

Now, let us call the distance between zero and one a "jump". You start from zero, you move a jump in a particular direction, you get to "one". You move another jump in the same direction, you get to "two". Another jump, "three". Another jump, "four". And so on, to infinity. These are the positive integers.

Now, consider an operation; addition. If I apply addition to any pair of positive integers, I get another positive integer. Any of these numbers that I add gives me a number I already have; I can add no new numbers with addition.

However, I can also invert the addition operation, to get subtraction. If I want to find X+Y, I hop X jumps from the zero point,then Y more jumps. But if I want to find X-Y, I must jump X jumps to the right, then Y jumps to the left; and this gives me the negative integers. Add them to the mental numberline.

At this point, multiplication gives us no new numbers. Division , however, does.

You will now notice, there are still gaps between the numbers. To fill these gaps, we turn to division; X/Y gives us a plethora of new numbers (1/2, 2/3, 3/4, 4/5, so on and so forth), hundreds and millions and billions of little dots between each point on the numberline. These are the rational numbers.

Is the numberline full yet? Hardly; it turns out that the rational numbers are so small a proportion of the numberline that it's still more empty space than marked point. I could say that there's billions of irrational numbers for every rational number, but that severely underestimates the number of irrational numbers that there are.

But let's add all the irrational numbers as well. (If you're actually drawing this, just take a ruler and draw a line across the page, such that all your integers fall on the line).

This line, then, is the famous numberline. I'm sure you've seen it before, on classroom walls and similar. It contains all the real numbers and, now that we've added the irrational numbers, it is full; there is no space on the line where another number can be added.

Now, let's consider squaring. The square of one is one. The square of any positive number greater than one is an even greater positive number (for example, two squared is four). The square of any positive number between zero and one is a positive number closer to zero (0.5 squared is 0.25).

The square of zero is zero.

The square of any negative number is equal to the square of the corresponding positive number; thus the square of negative two is four.

Therefore, four has two square roots; 2 and -2. Similarly, one's square roots are one and minus one.

So, a question then emerges; where are the square roots of minus one?

They cannot be on the numberline. There is no space for new numbers on the line, and the square of every number on the line is a positive number (or zero).

Let us call the square roots of minus one i and -i (somwhat arbitrary notation that was used once and stuck) Where do we put them on the line?

Since the line is full, we cannot put them on the line. If you place the line such that the zero is in front of you, the positive numbers head off to the right, and the negative numbers go to the left, then i is found one jump directly up from zero. Similarly, -i is one jump directly down from zero.

So, they are numbers, but they are not on the number line.

And now that we have placed i and -i, we can apply the same operation as we used earlier.

Addition: adding 1 is a jump to the right. Similarly, adding i is a jump upwards. There is a 2i two jumps above zero; a 3i three jumps above zero, and so on.

In fact, by following the same steps as were used to construct the original, real number line, we can create an imaginary number line at right angles to it; so that we can point to, say, 2.5i, or even pi i.

Then, if we want t find the point where (say) 3+4i is, we first jump three jumps to the right, then we jump four jumps up; adding the numbers 3 and 4i. 3+4i is thus a clearly defined point on the numberplane (since it's no longer one-dimensional, "numberline" is not exactly accurate anymore).

Adding and subtracting complex numbers on this plane is perfectly straightforward (though actually describing what i apples look like is beyond me). Multiplication follows the rules for multiplying additive expressions; that is, (a+b)*(c+d) = ac+ad+bc+bd. So, therefore:

(3+4i)*(2+5i) = (3*2)+(3*5i) + (4i*2) + (4i*5i) = 6 + 15i + 8i + 20*i*i

But since i is defined such that i*i=-1, that means:

(3+4i)*(2+5i) = 6 + 15i + 8i + 20(-1) = 23i-14

Voila, multiplication.