Decomplexifying Complex Numbers

By Leo Lam

(Test Preparation (ACT/SAT/SSAT), Math and Physics tutor at The Edge Learning Center)

Disclaimer: before you read further, the word “decomplexifying” is not a real word. It just sounds better than “simplifying” when put in front of “complex numbers.” So please do not use this word in your English papers.

One of the more advanced topics that are popping up in ACT and the new SAT is complex numbers. While the name suggests that it is a complicated concept, both tests only require students to know the basic operations. However, many students might not have encountered this concept in school before, and tackling this on their own can be a daunting task. Today I would like to give you some tips on how to work with this number system for the exam, and introduce some interesting facts should you find the topic fascinating.

>> Read more on Leo’s previous math blog “Fun! Facts about Factorials!”

The  works like a  

If a question asks you to calculate , we know from basic algebra that we can combine like terms. Thus, we would combine the terms with  and add the two constants together to get . Turns out the  in a complex number works very much like a  in an algebraic expression: you can combine them. Hence adding and subtracting complex numbers are as easy as combining like terms. Multiplication works almost the same: we can apply the FOIL method when we multiply two complex numbers together, like below:

The only thing we have to remember is…

 needs to be changed

A complex number is defined as , where both  and  are real numbers. We are not allowed to put any other terms in there. So when we end up with the term , we need to rewrite it. Because  is defined as, we have:

As a matter of fact, we can apply higher integer powers to  and find a pattern:

As we continue, we can see that when we are given a positive integer power of , we can rewrite it in terms of power of  (and if you remember your rules of exponents, this means we multiply the power with 4), and whatever is left will tell us the final form of the number. In another words, these are the rules you need to remember for evaluating , being a positive integer:

  1. if  is a multiple of 4, then 
  2. if dividing  by 4 and the remainder is 1, then 
  3. if dividing  by 4 and the remainder is 2, then 
  4. if dividing  by 4 and the remainder is 3, then 

Dividing requires rationalization

If you get a question where the imaginary number is being placed in the denominator of a fraction, we need to recall from our rules of radicals that roots are not allowed in the denominator. Because , which makes it a radical, we need to rationalize the denominator to make it into a proper form. For example, if we are given:

We need to multiply both the numerator and the denominator by  to achieve a in the denominator and get rid of the . Thus, we get:

Recalling from the previous part that when we expand,  will become -1, we will end up with the answer:

In complex number, we have a special name for the value we multiply to rationalize: a complex conjugate. The idea is as simple as changing the sign for the number with  attached to it. The proper definition is as follows:

Let  be defined as a complex number such that , where  and  are real numbers.

Then the conjugate of , denoted either  or , is defined as:

With this knowledge in your arsenal, you will be able to tackle any questions from the SAT or ACT that involve complex numbers. However, if you are interested in more advanced idea about this number system, here are some additional facts.

It’s not all “imaginary”…

While the name sounds like it’s “made up”, we can actually visualize complex numbers.

Don’t make the “imaginary friends” pun when you work with complex number, courtesy of xkcd.com

The first thing we should see is that complex numbers are made up of two components. If we think of them as the “” and “” of the coordinate plane, we are able to create a plane that has “real” and “imaginary” as the two axes. This plane is called the “complex plane”, and it is here that we visualize complex numbers. Instead of being a point on a plane, however, complex number is shown as a vector instead. That’s because we would end up operating with these entities, and we simply cannot “add” or “multiply” two points. Once we have this plane in place, there are many ways we can visualize some of the more advanced mathematical concepts. For example, proving the Fundamental Theorem of Algebra becomes a task of drawing when we apply this concept.

It’s more than just 

            Using the “” and “” of the coordinate plane as a foundation of visualizing complex number is really an application of the more general “Cartesian Plane” structure. However, there are other ways to represent a complex number. One common way to write a complex number is using the “Polar Coordinate.”

            Polar coordinate uses the angle formed between a line and the length of that line to represent the end of the point. Since we mentioned earlier that complex numbers are drawn as vector in the complex plane, we can easily convert the “” and “” coordinate into polar coordinates by creating a simple right triangle and follow some trigonometric operations.

graph

Here is a picture of how polar coordinate works. Having trouble understanding this? No worry, let Ron the Cartesian Bear help you out.

So why bother with representing the same value in a different way? One reason is the ease of calculation using this different form. Turns out that finding the product, quotient, power, and roots of complex numbers in this form is as simple as adding, subtracting, multiplying , and dividing the angles, respectively (the latter two, which involves power and roots, follow a very important formula called the De Moirve’s Theorem).

polar bear cartesian bear

Here is a picture to help you remember the difference between Polar and Cartesian: Polar is round, Cartesian is boxy.

And we are only scratching the skin of complex numbers. Come join me and learn more about this amazing number system that is bigger than the real numbers you are used to working with. And before we go, let’s see if you are as smart as Homer and understand the joke here:

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