(This post was cut and pasted from one of my other blogs.)
So... I have this Maths Specialist investigation on complex numbers in polar form, and they say that the best way to learn something is to teach other people, which is what I'm going to do!
These posts really need diagrams... but I'm pretty lazy, so I'll just add diagrams when I feel like it/ when you pressure me to :)
ANYWAY, I'll get to the point.
Polar Coordinates
Most of the time, when you are given coordinates for a graph, you are given the coordinates in rectangular form. Rectangular form is basically (x, y) where x is the number of units to the right and y is the number of units up. Polar coordinates differ quite drastically: they give coordinates in the form (r,θ) where r is the distance from the origin, and θ is the angle anticlockwise from the positive x-axis.
So, how do you go about converting rectangular coordinates to polar form? It's simple...
To find r, the direct distance from the origin, all you have to do is use Pythagoras' theorem. According to Pythagoras' theorem, r^2 = x^2 + y^2, so r is the square root of x^2 + y^2. (This would all make a lot more sense if I put a graph here, but as I said, I'm pretty lazy.)
Finding the angle between r and the x-axis takes a bit more work. Using SOHCAHTOA (sin = opp/hyp, cos = adj/hyp, tan = opp/adj), tan θ is equal to y/x. (Once again, it would be a lot easier to see if I could be bothered drawing a graph.) Therefore, θ is simply the reverse tangent of y/x.
The above was for if the point (x,y) is in the first quadrant of the graph (i.e. x and y are positive). But what happens if the point (x,y) is in a different section of the graph? No worries- you may simply have to add or subtract 180 or 360 to give the anticlockwise angle from the point (x,y) and the positive x-axis. If the graph lies in the 2nd quadrant (i.e. x is negative, y is positive), you need to subtract the angle from 180 degrees. If the graph lies in the 3rd quadrant (i.e. both x and y are negative) then you need to add 180 degrees. If the graph lies in the 4th quadrant (i.e. x is positive, y is negative), then you need to subtract the angle from 360 degrees. If you ever get confused, draw a graph and draw an angle anticlockwise starting from the positive x-axis. Diagrams are a very useful tool for helping your understanding.
By the way, there are two acceptable formats for giving an angle: the angle anticlockwise from the positive x-axis is given as a positive angle, while the angle clockwise from the positive x-axis is given as a negative angle. Therefore, the angle 270 degrees can also be written as -90 degrees.
To convert from polar form to standard Cartesian coordinates, it helps to draw a quick sketch of the triangle with its hypotenuse and angle marked. You can then easily use SOHCAHTOA to find x and y.
Complex Numbers
Complex numbers are made up of two parts: a real part and an imaginary part. They take on the form z = x + yj (i is sometimes substituted for j, but they both mean the same thing). x is the real part, and yj is the imaginary part. j (or i) is equal to the square root of negative one (which is a real number), but x and y themselves are real numbers- one example of a complex number is 5 + 3j.
Let's take any real number and call it a. If you have the square root of a, and you raise it to the power of 2, you end up with a. Similarly, if you take the square root of -1, and you raise it to the power of 2, you should theoretically also end up with -1. As j = -1, j^2 must also equal -1. This is important for complex number arithmetic.
Adding, subtracting and multiplying complex numbers is easy. When adding or subtracting, just add or subtract the real parts together and then add or subtract the imaginary parts. When multiplying, just multiply as you would normally do in algebra. However, if you end up with j^2, make sure to substitute -1 in its place. (Additional substitution may be required for higher powers of j.)
Dividing requires a new concept: the conjugate of a complex number.
Basically, to get the conjugate of a complex number, just change the addition sign to a subtraction sign and vice versa.
When dividing complex numbers, multiply by the conjugate of the bottom number over itself. For example, if you want to do (2 + 3j)/(4 - 6j), multiply by (4 + 6j)/(4 + 6j). This works for two reasons:
Reason 1: Difference of Two Squares.
When you multiply the bottom number by its conjugate, you get a difference of two squares expression. (x + yj)(x - yj) = x^2 + xyj - xyj - (yj)^2. xyj and -xyj then cancel each other out and the resulting expression can be expanded to x^2-(y^2)(j^2). As I have said earlier, j^2 = -1, so (x + yj)(x - yj) = x^2 + y^2 which is a real number! Therefore, after multiplying a fraction by the conjugate of the bottom number over itself, you really end up simply dividing by a real number at the end.
Reason 2: The Conjugate Over Itself is Equal to 1.
When you multiply the fraction by the conjugate of the bottom number over itself, you simplify it so that you are dividing by a real number, but you are not changing the value of the expression. This is because any number over itself is equal to 1. Therefore, when you are multiplying by the conjugate over itself, the value of the expression remains unchanged.
Complex numbers are able to be graphed. The name for the graph is called the Argand diagram but it's actually very similar to any other graph. The main difference is that the x-axis and y-axis are given different names: the x-axis is the real axis and the y-axis is the imaginary axis. To find the location complex number x + yj on the Argand diagram, simply move x units to the right and y units up.
Now on to the next part...
Complex Numbers in Polar Form
Unfortunately, converting complex numbers to polar form isn't as easy as converting normal numbers to polar form. I think that this is because a complex number has two components (real and imaginary), while a normal number only has one real component. Nevertheless, it's not that hard.
First, we have to use a little trigonometry (again, I need a diagram here). Let r be the distance from the origin to the point x + yj on the Argand diagram and θ be the angle counterclockwise between the positive real axis and r.
cos θ = adj/hyp = x / r. Therefore x = r cos θ.
sin θ = opp/hyp = y / r. Therefore y = r sin θ.
Substituting these values into the general complex number form z = x + yj you get:
z = r cos θ + (r sin θ)j = r (cos θ + j sin θ)
Now, using Pythagoras, r = sqrt(x^2 + y^2). Now |z| = sqrt(z multiplied by the conjugate of z). As I stated in the above section about complex numbers, z multiplied by its conjugate is x^2 + y^2. Therefore, |z| = r = sqrt(x^2 + y^2).
Therefore, z = |z| (cos θ + j sin θ).
Some terminology for you: |z| is the modulus of the complex number. θ is the argument (or arg).
Now let's move on to... the powers of complex numbers in polar form.
No, I didn't mean that they had powers, I meant raising them to a power. Don't get too excited.
When you raise z to a power p, the modulus also gets raised to a power p. The argument, on the other hand, simply gets multiplied by the power. It's like the log laws!
log (z^p) = p log z
arg (z^p) = p arg z
Multiplying and Dividing Complex Numbers
Multiplying and dividing complex numbers in polar form are like raising complex numbers in polar form to powers. The modulus gets treated like a normal number while you have to treat the argument like a logarithm.
When multiplying two complex numbers, say z and w, the modulus is simply |z|*|w| = |zw|. The argument is simply arg (z) + arg (w) = arg (zw).
When dividing two complex numbers, say z and w, the modulus is simply |z|/|w| = |z/w|. The argument is simply arg (z) - arg (w) = arg (z/w).
I think that the laws about multiplying, dividing and raising the modulus to a power are the same as those around normal numbers because the modulus is an ordinary real number as well. But why are the above laws surrounding arguments the same as the laws surrounding logs? Let's take a brief glance at two trigonometry laws and see how we can use them to explain multiplying (and possibly dividing too). These laws are:
cos (θ + φ) = cos θ cos φ - sin θ sin φ
sin (θ + φ) = sin θ cos φ + cos θ sin φ
Let z = |z|(cos θ + j sin θ)
Let w = |w|(cos φ + j sin φ)
zw = |z|(cos θ + j sin θ)|w|(cos φ + j sin φ)
= |zw|(cos θcos φ + i sin θcos φ + cos θ i sin φ + i sin θ i sin φ)
= |zw|((cos θcos φ - sin θsin φ)+ i(sin θcos φ + cos θsin φ))
Using the laws stated previously,
zw = |zw|(cos (θ + φ) + sin (θ + φ))
The modulus of z and w have been multiplied together and the arguments have been added together.
There's a website on how to explain why arg(z^p) = p arg (z), but it looks too confusing for me. If you're interested, though, you can find it at http://math.tutorvista.com/trigonometry/de-moivre-s-theorem.html .
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