Defining Complex Numbers

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Algebra Two $\longrightarrow$
Complex Numbers $\longrightarrow$
 
Objectives
  • Define what complex numbers are
  • Understand that each real numbers and imaginary numbers are subsets of the set of complex numbers
  • Know the concepts of norm and conjugate as applicable to complex numbers
Lesson Description

Real numbers and imaginary numbers are not like terms and cannot be combined, but their sum comprises a type of number we call a "Complex Number". These are very useful for polynomial analysis and other situations where imaginary numbers are meaningful, such as advanced Physics applications.

 
Practice Problems

Practice problems and worksheet coming soon!

 

Complex, Not Complicated

When we learn about number sets, such as the set of integers, the set of rational numbers, or the set of real numbers, we tend to learn a lot more minutia than we really need for practical purposes. For example, it's nice to understand that the set of rational numbers is a subset of the set of real numbers, but it doesn't have a lot of applicability for skills that help us "do math". The same is true for complex numbers, so keep that in mind as we look at their definition and use. Most uses of complex numbers are for topics related to the study of polynomials.
Define: Complex NumberA complex number has the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary number (defined as either $i=\sqrt{-1}$ or the number such that $i^2 = -1$ depending on your teacher's preference).
The reason we define complex numbers as a major "set" of math is that, along side the real numbers, we have a complete set of answers to work with in any application that requires imaginary answers. Since pure imaginary numbers take the form $bi$, we can see that any imaginary number can be called a complex number, where the "a value" is zero. e.g. $3i$ is really $0 + 3i$.
You Should Know
There isn't a lot of new concept to see here. It's more about understanding what structure a complex number has, and that any imaginary number answer is a complex number.
It is customary that we write complex numbers with the real number followed by the imaginary number, but it's not a requirement.

Complex Number Characteristics

In advanced math, there's a way to do almost anything with complex numbers that you can do with "regular" numbers. However, we will probably never care about some of these things (e.g. taking the square root of a complex number, doing calculus with imaginary numbers, etc.) unless we're bored. There are, of course, a number of basic things we will want to do.ArithmeticOf course, we want to be able to do some basic arithmetic operations with complex numbers. While much of the process and results of complex number arithmetic are intuitive enough that you could guess about how they work, our next few lessons » carve out the specifics.MagnitudeWhen working with real numbers, we know how to order them based on their direction (positive or negative) and magnitude (size). Picture the number line - there is a way to take any handful of real numbers and order them. Number order is something we learn very young and feels natural and obvious.For two complex numbers, how do you determine which is "bigger" than the other? Is $2+5i$ bigger or smaller than $6 + i$?Using the real and imaginary coefficients of a complex number, $a$ and $b$ respectively, we use the norm of a complex number as the measure of its magnitude.
Define: NormThe norm of a complex number $z = a+bi$ is defined as$$|z| = \sqrt{a^2 + b^2}$$
You Should Know
The norm of a complex number is used to compare the size of two or more complex numbers, regardless of direction. Think of this as analogous to absolute value for real numbers.
In my not-so-humble opinion, norm is best understood visually. We'll look at it when we discuss plotting complex numbers ».ConjugateThere is no "real number equivalent" of conjugates - rather, this is a convenience name we use to reference a certain thing.
Define: Complex ConjugateFor a complex number, the complex conjugate is the complex number that has the same real number coefficient but the negative of its imaginary number coefficient. The notation is that the conjugate has a bar over the number.For example, for $z = a+bi$:$$\bar{z} = a - bi$$You can also put a bar over the number itself instead of its name:$$\bar{a + bi} = a - bi$$e.g.$$\bar{4 - 5i} = 4 + 5i$$
It's worth mentioning that the conjugate of a conjugate is the original number. That is, if $w = \bar{z}$ and $v = \bar{w}$, then $z = v$. For a real example, if we start with $-3 + 5i$, the conjugate is $-3 - 5i$, but the conjugate of $-3 - 5i$ is $-3 + 5i$, which is exactly what we started with.We will use conjugates very soon for rationalizing complex expressions ». It has a few minor uses in later random topics, but it's biggest algebra use is for rationalizing.

The Master Set

It is worth quickly pointing out and understanding that the set of complex numbers is actually the most complete set of numbers we've ever worked with in Algebra - any real number including integers, fractions, and even irrational numbers like $\pi$ and $\sqrt{7}$ are all also part of the set of complex numbers, where the "b value" is zero. For example, although it's a waste of ink, we could say the following equivalent statements:$$x=3 \longrightarrow x = 3 + 0i$$$$x=\pi \long rightarrow x = \pi + 0i$$$$x=\sqrt{7} \longrightarrow x = sqrt{7} + 0i$$So in short, every real and imaginary number can be written as $a+bi$ if we wanted to, and therefore everything can be a complex number! It's worth noting that we often don't put "plain" numbers in the context of complex numbers, but if you can follow along with the idea that you could write any number this way, that will get you through any concept quiz question about how complex numbers fit into the world of number sets.
 
Lesson Takeaways
  • Know what the structure of a complex number consists of
  • Understand the concept of norm as a measure of size, analogous to absolute value for pure real numbers
  • Know what complex conjugates are and how to get them
  • At a basic level, understand how the set of Complex Numbers encompasses the set of Real Numbers

Lesson Metrics

At the top of the lesson, you can see the lesson title, as well as the DNA of Math path to see how it fits into the big picture. You can also see the description of what will be covered in the lesson, the specific objectives that the lesson will cover, and links to the section's practice problems (if available).

Key Lesson Sections

Headlines - Every lesson is subdivided into mini sections that help you go from "no clue" to "pro, dude". If you keep getting lost skimming the lesson, start from scratch, read through, and you'll be set straight super fast.

Examples - there is no better way to learn than by doing. Some examples are instructional, while others are elective (such examples have their solutions hidden).

Perils and Pitfalls - common mistakes to avoid.

Mister Math Makes it Mean - Here's where I show you how teachers like to pour salt in your exam wounds, when applicable. Certain concepts have common ways in which teachers seek to sink your ship, and I've seen it all!

Put It To The Test - Shows you examples of the most common ways in which the concept is tested. Just knowing the concept is a great first step, but understanding the variation in how a concept can be tested will help you maximize your grades!

Lesson Takeaways - A laundry list of things you should be able to do at lesson end. Make sure you have a lock on the whole list!

Special Notes

Definitions and Theorems: These are important rules that govern how a particular topic works. Some of the more important ones will be used again in future lessons, implicitly or explicitly.

Pro-Tip: Knowing these will make your life easier.

Remember! - Remember notes need to be in your head at the peril of losing points on tests.

You Should Know - Somewhat elective information that may give you a broader understanding.

Warning! - Something you should be careful about.

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