The base idea for understanding how to use complex numbers is understanding i. i is a symbol representing √-1. Thus, i^2 = -1. In the simplest form, i is the solution to the equation x^2=-1. As alluded to in the history and background section, before imaginary numbers were invented, this problem was unsolvable. One of the basic properties in mathematics is the idea that any number that is squared will always be positive. If this is true, and we know that it is, then, how can there be a solution to the equation x^2=-1? This dilemma was solved with the creation of the imaginary number i. While the idea can be difficult to grasp, the creation of i was essential since there are processes in our modern world that can only be modeled with equations similar to x^2=-1. Understanding the basic knowledge that i=√-1 is the core to everything that is used in the realm of imaginary and complex numbers. If you want an extensive introduction to imaginary numbers, watch the following video.
As mentioned above, once one comprehends the variable i, complex numbers are rather simple to understand. A complex number has a real and an imaginary part. For example, in the complex number (a+bi), a is the real part, and b is the imaginary part. One key thing to remember when working with complex numbers is the need to keep the real and imaginary parts separate since it makes no sense to combine a real number with an imaginary number. Mostly for your amusement, here is a video that infers the reason for keeping both parts separate.
If you want to further review the concept of complex numbers, this following video from Khan Academy provides a helpful summary.
To understand some of the basic mathematical properties of complex numbers, this page is split up into two sections. In the first section below labeled Complex Numbers Algebra, you will learn how complex numbers can be manipulated algebraically. The second section will give an overview of how to graph complex numbers. Both sections will include video help and links where you can learn about and practice the mathematics.
Complex Numbers Algebra
As mentioned, complex numbers are numbers represented with a real and an imaginary part such as (a + bi). Just like any other numbers in mathematics, complex numbers contain many of the same properties and can be manipulated by addition, subtraction, multiplication and division. The following section shows some examples of how complex numbers can be manipulated algebraically. The conclusion of this section will include worksheets that you can access to practice algebraically manipulating complex numbers.
Addition
When adding complex numbers, one simply combines like terms. That is, if adding (a+bi) to (c+di), one combines the real parts (a+c) and the imaginary parts (c+d)i. The following illustration demonstrates this property with an example. If you want more examples of adding complex numbers watch the video from Khan Academy.
Subtracting complex numbers is similar to that of adding complex numbers where one simply combines the like terms, such as (a+bi) - (c+di) = (a-c) +(b-d)i. Here is an illustration and another video of this practice.
The process of multiplying complex numbers is similar to that of expanding polynomials. Thus,(a+bi)*(c+di) = (ac + (ad)*i +(bc)*i +(bd)*i^2). The interesting aspect to this expansion is that (bd)*i^2 is a constant since i^2=-1. Thus the result will be of the form (bd+ac)+(ad+bc)*i. The following picture displays a problem of this type. Also available is two video links on this practice; one from Khan Academy and one personal video.
When dividing complex numbers, it is most helpful to multiply both terms by the conjugate of the bottom term. A conjugate is simply the negation of the second term of the complex number. For example, the conjugate of (a+bi) is (a-bi). This action of multiplying the top and bottom terms by its conjugate is helpful because if the bottom term is (7+5i), you multiply by (7-5i) and the imaginary part disappears. (49 - 35i +35i - 25*(-1)) = 74. This makes the division much easier since one is now only dividing by a constant. The following picture provides a demonstration of this process.
Complex numbers can be graphed on a two dimensional plane by graphing the imaginary part on the Y axis, and the real part on the X axis. For example, (a-bi) would correlate to the point (a,-b). With this representation, they can be represented as vectors where a+bi = <a,b>. Being able to graph these numbers helps clarify what each specific complex number means in context. The following picture shows examples of a several complex numbers graphed on the complex plane.
The following applet will give a demonstration of how each complex number can be graphed as a vector. Input any complex number to see the corresponding vector. In addition, you can see what happens to graph of each complex number when it is manipulated algebraically such as by addition, subtraction, multiplication, or division.
Below is a video that portrays how graphing with complex numbers can help solve problems that are unsolvable without the use of complex numbers. It provides a scenario that may have motivated the discovery of imaginary numbers.
As you may be realizing there is a lot more to complex numbers than most think. Complex numbers are a unique entity in mathematics that contain many unique properties in their use and application. While it isn't expected for one to completely understand the concept of complex numbers from these explanations, the previous sections provide a clear summary of the concept of complex numbers. Below is one more helpful video in understanding imaginary numbers and some application in mathematics.
