This guide already assumes you have a basic understanding of what a complex number is!
We already know that there are two parts to a complex number, the "Real" part and the "Imaginary" part. On CASIO Calculators, this is denoted by the commands ReP and ImP respectively. This shorthand will be used in this guide, as it is a nice shorthand.
Let's take the following Complex Numbers:
z1=1+1i
z2=1+3i
z3=−3+2i
z4=−3+−2i
z5=−2i
But, what is the easiest way to represent this data?
What if I were to rewrite z5 as:
z5=0+−2i
Can you see, how every single complex number has two data points? The ReP and the ImP!
And, what is another way of storing 2-dimensional data?
A coordinate! i.e: (3,4), (−1,4), (1.31,√2)
+-- --+ +-- --+ +-- --+
| 01 | | -14 | | -3 |
| | | | | |
| 02 | | 3 | | 06 |
+-- --+ +-- --+ +-- --+
Or even in ij notation. (NOTE: I am hesitant to put this example in, because some people may confuse the term i with the imaginary part i. I have left this here anyway): x = i - j [WARNING: IN THIS NOTATION i IS THE X-AXIS, THIS IS THE OPPOSITE FOR ARGAND DIAGRAMS!]
But, how do we go about drawing these? A quick definition:
Cartesian coordinates are coordinates in the form (x,y)
Cartesian equations are equations in the form y=mx+c or x^2+y^2=r^2
Firstly:
- The ReP corresponds to the x-axis in a cartesian coordinate.
- The ImP corresponds to the y-axis in a cartesian coordinate.
Im |
|
|
| Re
-------+-------
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|
Now, let's look at the different ways that we can represent complex numbers on a graph!
The first way is to convert the complex number into a cartesian
coordinate!
This is done in the following way:
Let z be any complex number, as a cartesian coordinate this would be
(ReP(z),ImP(z)) .
This creates a new general way of writing complex numbers:
z=x+iy
This form is extremely important later on, so you had better get used to it!
A common way for an exam question to tell you to draw as a coordinate:
> A complex number z is represented by the point P in the Argand Diagram
Im |
| 2
|
3 |
|
| 1
|
------------x-----------------
| Re
|
|
4 5
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|
The other way of representing a Complex Number is by drawing it as a
vector!
This can be done using the form:
+-- --+
| ReP(z) |
| |
| ImP(z) |
+-- --+
It is actually rather useful to show complex numbers in complex form, because they follow a lot of the same rules as vectors!
Im |
| ^
|/
-----|-----
| Re
|
| z1 = 1 + 1i
So, you may be wondering, what is so good about representing this as a
vector?
Well, what if I wanted to add two complex numbers?
Let's create two new complex numbers:
zA= 1 + 2i
zB=-2 + 1i
Now, I warn you, this doesn't look very good in ASCII. I recommend checking out the image-based version of this site, if you want to understand it more easily!
|
| ^
| |
<- |/
\-||
--------|-----
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|
Adding these two complex numbers gives:
zA+zB=(1+2i)+(-2+1i)=(-1+3i)
Let's define this as zL
I warn you, the ascii art is about to get even worse!
|
^ |
| |
| | ^
\| |
<- ||/
\|||
--------|--------
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|
Can you see how this is just simple vector addition?
No?
Well, you probably can't tell this, because you are looking at ASCII art,
but I will ask you to do something. Go get a piece of squared paper, and
draw out these vectors!
I still really recommend looking at the images for this area of maths,
because it will make your life 4000x easier (don't sue me if it doesn't)
Now that you've drawn the vectors, draw lines from the tip of your zA and
the tip of your zB to the tip of your zL vector!
Can you see how the line you draw from zA to zL is the same as zB?
And how the line from zB to zL is the same as zA?
Hopefully you can see that it doesn't matter which way you add the complex
numbers, you get an end result that's the same!
__
z = 1 + 1i : z = 1 - 1i
1 1
__
z = 1 + 3i : z = 1 - 3i
2 2
__
z =-3 + 2i : z =-3 - 2i
3 3
__
z =-3 - 2i : z =-3 + 2i
4 4
__
z = 0 - 2i : z = 0 + 2i
5 5
Can you see what happened there?
That's right, the conjugate of a complex number, is a reflection in the
x-axis!
Also, notice that z4 is the conjugate of z3
Therefore, given we put z5 into a vector c->:
+-- --+
| 00 |
c = | |
| -2 |
+-- --+
+-- --+ +-- --+ +-- --+
| 1 0 | | 00 | | 00 |
| |*| | = | |
| 0 -1 | | -2 | | +2 |
+-- --+ +-- --+ +-- --+
Therefore the conjugate of z5 is 2i.
The Modulus Function |z| is quite simply a fancy way to write the pythagorean theorem. Let's have a look at z2 (and |z2| respectively)
It is exactly the same as vector mathematics!
^
/|
/ |
√(3^2+1^2) / |
/ |3
/ |
/ |
/-------
1
As you can see, this is simply pythagoras completed on the right-angled
triangle of height 3, width 1.
Knowing this, we can do this calculation for any complex number, without
needing to use a graph.
|z1| = √(1^2+1^2) = √2
|z2| = √(1^2+3^2) = √10
|z3| = √[(-3)^2+2^2] = √13
|z4| = √[(-3)^2+(-2)^2] = √13
|z5| = √[0^2+(-2)^2] = 2
It is important to notice that both |z3| and |z4| are equal, even though their complex numbers are different!
Therefore, |z| = |conj z|
The argument of a complex number, often written as Arg or arg.
This can be understood as the angle of the vector. Let's take the example
of our previous z2 complex!
And our question is: What is Arg(z2)?
^
/|
/ |
√(3^2+1^2) / |
/ |3
/ |
/\? |
/--\----
1
The argument is the calculation of what angle is at "?"
Well, if you have a look at it, it's just a right angled triangle.
After seeing this, you may've been tempted to do this on your calculator:
NOTE: tan^-1 = arctan
tan x = 3/1
arctan 3/1 = x
71.565... x
But, you'd be very slightly wrong. The actual answer is:
1.249... = x
At this point, you've probably started to cry tears of pain. "WHY IS THE
NUMBER SO SMALL?" I hear you cry.
Welcome to the world of radians!
You are probably used to the following facts:
1 circle=360 deg
1 square=360 deg
1 triangle=180 deg
The "conversion rate" for radians is the following:
1 circle=pi rad
1 square=pi rad
1 triangle=pi/2 rad
The symbol we use is ^c, don't ask me why we don't use ^r. NOTE: CASIO calculators use ^r.
The most important conversion rate here is π rad = 180 deg. You will most likely use this conversion every time you draw an Argand Diagram, so try and memorise it!
Now for something slightly confusing, we need to look at the way angles are measured on an argand diagram!
> All the angles are measured from the POSITIVE X-AXIS.
But what does this actually mean???
Firstly, draw one of your complex numbers as a vector!
Then draw the shortest angle you can, from the point (1,0) on the diagram!
When the angle goes underneath the x-axis (the complex number has a
negative y-value), the radian will be -.
When it is above, the radian is positive!
A lot of teachers have different ways to teach this next section, but I will use my own method, because it's what I find easiest!
(x * x )*(y * y )
a b a b
cos x = --------------------
|a|*|b|
+- -+
| (x * x )*(y * y ) |
-1 | a b a b |
x = cos |-------------------- |
| |a|*|b| |
+- -+
Let's try this with some actual numbers!
+-- --+ +-- --+
| 01 | | -3 |
a = | |, b = | |
| 02 | | 3 |
+-- --+ +-- --+
Now, we know that |a|=√[x^2+y^2] and |b|=√[x^2+y^2] (REMEMBER: it's just pythagoras)
+- -+
| (x * x )*(y * y ) |
-1 | a b a b |
x = cos |-------------------- |
|√[x^2+y^2]*√[x^2+y^2]|
| a a b b |
+- -+
Now let's subsititute the numbers we have from the vertex
+- -+
-1 |(1 * -3 )*(2 * 3 ) |
x = cos |--------------------- |
|√[1^2+2^2]*√[-3^2+2^2]|
+- -+
x = arccos(3/10)
x = ~75.242... deg = ~1.266... rad (~ is shorthand for roughly)
"But James!!! This is longer and harder than what my teacher has taught
me???" I hear you again!
Do you remember how I said earlier that all the calculations are done
based off the angle between the positive x-axis, and the complex?
Well, what vector represents the x-axis?
+-- --+
| 01 |
x = | |
| 00 |
+-- --+
Now let's let a=x
+- -+
| (1 * x )*(0 * y ) |
-1 | b b |
x = cos |-------------------- |
|√[1^2+0^2]*√[x^2+y^2]|
| b b |
+- -+
This can then be simplified
+- -+
-1 | x |
x = cos |----------|
|√[x^2+y^2]|
| b b |
+- -+
Voila! James' fancy formula!
For example, the angle for z3 was found using the following formula:
+- -+
-1 | -3 |
x = cos |-----------|
|√[-3^2+2^2]|
+- -+
x = 2.55359... rad