Hey guys, today I will be making a brief summary of complex numbers and the concepts surrounding complex numbers.
First, we will start off with the basic concept behind complex numbers, what are they? In order to understand complex numbers we first need to understand Imaginary numbers.
Imaginary Numbers
So what are imaginary numbers? Imaginary numbers can be seen as a tool that allows us to take square roots of negative numbers. Without this special tool, something such as the square root of -1 would not be something that we can evaluate. However, imaginary numbers mean that we can assign the value of i to the square root of -1. This is the foundation of complex numbers in their simplest form. So, sqrt(-1)=i is one of the founding ideals behind imaginary numbers.
Complex Numbers
Complex numbers (which sound more intimidating than they are) are simply numbers that have both a real and an imaginary aspect to them. In other words, any number that can be expressed in the form a+bi with a and b being real numbers is a complex number. In a complex number of form a + bi, a is called the real component while b is called the imaginary component. It is to be noted that any number can be represented as a complex number. For example, the number 5 is simply 5+0i. Complex numbers can also be plotted on a coordinate plane called the complex plane.
When a number has an imaginary component of 0, we can essentially get rid of the vertical axis, this is why numbers that have an imaginary component of 0 can simply be represented on a number line or the horizontal axis.
To plot a complex number on this plane, treat the imaginary and real components of the number as coordinate points with the real component being along the horizontal axis while the imaginary component being along the vertical axis. So, to plot the complex number a+bi, we would go a units along the horizontal (real) axis and b units along the vertical (imaginary) axis.
Operations on Complex Numbers
Addition
To start the discussion of operations on complex numbers, I would like to discuss addition. To add two complex numbers, we can simply add the real components and the imaginary components of the numbers. For example, if a+bi is a complex number and c+di is another complex number, their addition would be (a+c) +(b+d)i. This is very similar to the addition of two vectors.
To subtract simply add the negative of the number being subtracted. So, in the above example if we wanted to subtract c+di from a+bi we would simply add -c-di to a+bi.
Multiplication
Multiplication is a little more confusing as there is no clear analog to multiplication in vectors, however the actual process and formula are fairly straight forward. The formula for the multiplication of two complex numbers is the same as the multiplication of any two binomials.
(a+bi)(c+di)=ac+i(ad) +i(bc) + i^2(bd).
Here we can see that there is a slight problem that prevents us from just simplifying the above problem as is. The powers of i is something that we will discuss later, however, we know from before that i=sqrt(-1) so i^2 is just -1.
Knowing that, we can finish simplifying the above expression.
(a+bi)(c+di)=ac+i(ad)+i(bc) -bd
(a+bi)(c+di)=(ac-bd) + i(ad+bc)
If we want to multiply a complex number by a scalar, say a+bi by c, all the terms with d would simply become 0 in the above equation and we would be left with-
(a+bi)(c)=ac+i(bc)
This is just the basic distributive property we know and love and to take the negative of a complex number we are simply distributing -1 across the complex number.
Division
To divide two complex numbers, say a +bi divided by c+di, represent the division as a fraction and simply multiply the top and the bottom of the fraction by the conjugate of the bottom. The conjugate of a complex number a+bi is a-bi. So, using this information, the above division would look something like this.
(a+bi)(c-di)/((c+di)(c-di))
Simplifying the above expression will result in the division of the two complex numbers expressed in the a+bi form.
This is done to express the division in the form most widely accepted for complex numbers.
Powers of Imaginary Numbers
We have kind of discussed this already, however, there is a pattern with i and its powers.
i=sqrt(-1)
i^2=-1
i^3=i^2*i=-1*i=-i=-sqrt(-1)
i^4=i^3*i=-i*i=-sqrt(-1)*sqrt(-1)=-(-1)=1
i^5= i^4*i=1*i=sqrt(-1)
i^6=i^4*i^2=1*i^2=i^2=-1
i^7=i^4*i^3=1*i^3=i^3=-sqrt(-1)
The first 4 powers of i (1,2,3,4) seem to show a pattern, however, after the 4th power this pattern repeats. Suppose we have i^n, where n is a number greater than 4. We know that all numbers i^a (where a is a power of 4) will be 1. So in order to evaluate i^n we just have to divide n by 4 and use the remainder r from that calculation and evaluate i^r.