Imaginary numbers are simply an expansion of the system of real numbers, invented by mathematicians to provide solutions to equations requiring an even root of a negative number. The imaginary number is defined by taking the square root of a negative one. Thus, and . By using , we can simplify and perform operations with expressions such as . For example,
The realm of complex numbers includes all numbers that possess a real and imaginary part and can be written in the form where a is the real part and b is the imaginary part. This means that real numbers and imaginary numbers can both be considered complex numbers. Real numbers are simply ...view middle of the document...
When graphing complex numbers we use a real and imaginary axis like the ones shown below. The real number axis is the horizontal axis and the imaginary number axis is the vertical axis. Thus, the graphical representation of the complex number is plotted by moving 3 units right and 2 units down from the origin.
We can perform operations on complex numbers graphically as well as algebraically. Given the complex numbers and we can use the complex number plane to find the complex number . For example, let’s say that and . You’ll see both points plotted on the complex number plane below.
We can find their sum by using a few different methods. First, we can use the same “head to tail” method we would use to add vectors by graphing one point as though the other were the origin. This method is illustrated below.
As you can see in the diagram above, the points , , and create three vertices of a parallelogram with the fourth being at the origin. Constructing this parallelogram would be a second method of using the complex number plane to add complex numbers.
We can also use a graphical representation to help us understand division of complex numbers. To do this, we first have to understand the distance to any point on the complex plane from the origin is the same as the absolute value of that complex number. Given the complex number , then . For this example, we are going to...