Sunday, August 12, 2012

Argand Plane and Modulus Argument Form

Let's start with some terminology
If z = x +iy
Re(z) = x (real component of z)
Im(z) = y (imaginary component of z)

The Argand plane (also known as the complex plane) is used to plot complex numbers. Complex numbers 'z' are plotted on the Argand plane with Re(z) on the x axis and Im (z) on the y axis.
Eg.
A represents 2+i, B represents 1, C represents -1 +3i, D represents -1-2i.

Now let's look at absolute values again.

The absolute value of a number, also called the modulus of a number is the distance of the number from the origin.
If the number is a real number, then the distance from the origin is just the number without the sign.
Eg. 1
|4|= 4,
Eg. 2
|-3|=3

For a complex number, the modulus can be found using the distance formula.
Eg. 1
|3+4i| = √(32 +42) = 5

 
Eg. 2
|5-12i|= √(52 + 122) = 13
By definition, if z = x +iy, |x +iy|= √(x2 +y2) , this is known as the modulus of z or sometimes shortened to mod z.
It is not enough to define a complex number by its modulus because there are infinite complex numbers that have the same modulus.
So we also define complex numbers by the angle that is created with the x-axis when that point is joined to the origin (known as the argument, often written as arg).
We can find the argument by using the fact that m(gradient) = tan(θ)
Eg. 1
If z = 1 + i√3
First we should determine which quadrant z is in. In this case z is in the first quadrant.
tanθ = √3/1
θ = π/3 + 2πn or 4π/3 +2πn (where n is an integer)
As we can see, there are infinite solutions for θ.
We know that 1 + i√3 is in the first quadrant and therefore 4π/3 + 2πn will not be solutions for θ, but we still have infinite possible solutions, so we limit the range of the answers to –π ≤ θ ≤ π.
So the final answer is arg z = π/3, this is known as the principle arguments, and the other solutions for θ are known as arguments.

Eg. 2
If z = –1 + i 
 
z is in the second quadrant
tanθ = 1/-1
θ = 3π/4 or -π/4 (for –π ≤ θ ≤ π)
Since z is in the second quadrant, arg z ≠ 7π/4 +2πn
Therefore arg z = 3π/4

Eg. 3
If z = -3 – 3i

z is in the third quadrant
tanθ = -3/-3
θ = π/4 or -3π/4 (for –π ≤ θ ≤ π)
Since z is in the third quadrant, arg z = -3π/4.

Now that we know how to work out the modulus and argument of a complex number, we can write it in modulus argument form.
The point z represents that complex number z = x + iy
Let θ = arg z and |z|= r
Therefore x = r cosθ and y = r sinθ
Therefore x +iy = r (cosθ + isinθ), sometimes written as r cisθ, (cis = cos + isin)
This is known as the modulus argument form. The reason why we write complex numbers in this form is that there are various properties that allow use to easily perform operations in complex numbers (will be shown in the next post).
The modulus and argument are sometimes known as polar coordinates. They are written in the form [r, θ] where r is the modulus and θ is the argument. Please note that square brackets are used instead of parentheses for polar coordinates.