Complex Numbers Definition
All the numbers we know are called real numbers and can be set on a straight line called the line of the real numbers. So numbers like 1, 2, -7, 4.78, -3.14, 2/5, the square root of positive numbers, are real numbers. They are set on the real numbers straight line and we can apply to them all the operations, like addition, multiplication, etc.
What if we want to calculate the square root of a negative number? There are no real numbers that could be the solution of the square root of a negative number. In this case the imaginary numbers are used.
We define the number i as follows:
An imaginary number is the compound of a number with i: 5i, 8i, 5.34i,…
The square root of a negative number is:
In Electrical and Electronic Engineering, instead of using the symbol i, often is used the symbol j, to avoid confusion with the symbol of the current. From this point, this is the symbol we will be using.
Imaginary numbers are also set on a straight line called the imaginary numbers line and we can apply to them all the operations.
The compound of a real number with an imaginary number is called complex number. They have the form z=a+bj. For example four complex numbers could be:
The Complex Plane
The combination of the line of the real numbers with the line of the imaginary numbers constitutes the complex plane. The horizontal axis is the axis of the real numbers and the vertical is the axis of the imaginary numbers. In image 1 we can see the previous complex numbers, set on the complex plane.
Image 1: The Complex Plane
Forms of Complex Numbers
Consider the complex number of image 2:
Image 2: z=x+yj
From the trigonometry we know that for z=x+jy it is:
x=rcosθ and y=rsinθ.
So the complex number can be written as:
Also, from Euler’s formula we have:
The complex numbers can be written in the following four forms:
Consider the complex number:
To get the conjugate of this complex number we toggle the sign of the imaginary part. If the imaginary part is positive, we make it negative and if it is negative, we make it positive. So the conjugate of z is:
Image 3: Complex Number and its Conjugate
In image 3 we can see the complex plane with number z and its conjugate z*. As we can see from this image, the conjugate has the opposite angle from the complex number. The relation between them can be written in the four forms: