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## Algebra 2

### Course: Algebra 2>Unit 2

Lesson 2: Complex numbers introduction

# Intro to complex numbers

Learn what complex numbers are, and about their real and imaginary parts.
In the real number system, there is no solution to the equation x, squared, equals, minus, 1. In this lesson, we will study a new number system in which the equation does have a solution.
The backbone of this new number system is the number i, also known as the imaginary unit.
• i, squared, equals, minus, 1
• square root of, minus, 1, end square root, equals, i
By taking multiples of this imaginary unit, we can create infinitely many more new numbers, like 3, i, i, square root of, 5, end square root, and minus, 12, i. These are examples of imaginary numbers.
However, we can go even further than that and add real numbers and imaginary numbers, for example 2, plus, 7, i and 3, minus, square root of, 2, end square root, i. These combinations are called complex numbers.

## Defining complex numbers

A complex number is any number that can be written as start color #1fab54, a, end color #1fab54, plus, start color #11accd, b, end color #11accd, i, where i is the imaginary unit and start color #1fab54, a, end color #1fab54 and start color #11accd, b, end color #11accd are real numbers.
$\begin{array}{ccc} \Large\greenD a&\Large+&\Large \blueD bi \\ \uparrow&&\uparrow\phantom{i} \\ \text{Real}&&\text{Imaginary} \\ \text{part}&&\text{part} \end{array}$
start color #1fab54, a, end color #1fab54 is called the start color #1fab54, start text, r, e, a, l, end text, end color #1fab54 part of the number, and start color #11accd, b, end color #11accd is called the start color #11accd, start text, i, m, a, g, i, n, a, r, y, end text, end color #11accd part of the number.
The table below shows examples of complex numbers, with the real and imaginary parts identified. Some people find it easier to identify the real and imaginary parts if the number is written in standard form.
Complex NumberStandard Form start color #1fab54, a, end color #1fab54, plus, start color #11accd, b, end color #11accd, iDescription of parts
7, i, minus, 2start color #1fab54, minus, 2, end color #1fab54, plus, start color #11accd, 7, end color #11accd, iThe real part is start color #1fab54, minus, 2, end color #1fab54 and the imaginary part is start color #11accd, 7, end color #11accd.
4, minus, 3, istart color #1fab54, 4, end color #1fab54, plus, left parenthesis, start color #11accd, minus, 3, end color #11accd, right parenthesis, iThe real part is start color #1fab54, 4, end color #1fab54 and the imaginary part is start color #11accd, minus, 3, end color #11accd
9, istart color #1fab54, 0, end color #1fab54, plus, start color #11accd, 9, end color #11accd, iThe real part is start color #1fab54, 0, end color #1fab54 and the imaginary part is start color #11accd, 9, end color #11accd
minus, 2start color #1fab54, minus, 2, end color #1fab54, plus, start color #11accd, 0, end color #11accd, iThe real part is start color #1fab54, minus, 2, end color #1fab54 and the imaginary part is start color #11accd, 0, end color #11accd

Problem 1
What is the real part of 13, point, 2, i, plus, 1?

Problem 2
What is the imaginary part of 21, minus, 14, i?

Problem 3
What is the real part of 17, i?

## Classifying complex numbers

We already know what a real number is, and we just defined what a complex number is. Now let's go back and give a proper definition for an imaginary number.
An imaginary number is a complex number start text, a, plus, b, i, end text where start text, a, =, 0, end text.
Similarly, we can say that a real number is a complex number start text, a, plus, b, i, end text where start text, b, =, 0, end text.
From the first definition, we can conclude that any imaginary number is also a complex number. From the second definition, we can conclude that any real number is also a complex number.
In addition, there can be complex numbers that are neither real nor imaginary, like 4, plus, 2, i.
$\purpleD{\boxed{ \begin{array}{c} \text{Complex numbers} \\\\ \begin{array}{lcr} 4+2i&&-3-\sqrt 5 i \end{array} \\\\ \goldD{\boxed{ \begin{array}{c} \text{Real numbers} \\\\ \begin{array}{lcr} \sqrt 5&-12.2&3 \end{array} \end{array}}} \\\\ \maroonD{\boxed{ \begin{array}{c} \text{Imaginary numbers} \\\\ \begin{array}{lcr} \sqrt 5 i&-12.2i&3i \end{array} \end{array}}} \end{array}}}$

### Reflection question

Is the following statement true or false?
Any complex number is either real or imaginary.

### Examples

In the table below, we have classified several numbers as real, pure imaginary, and/or complex.
\begin{aligned}&\text{Real}\\&(b=0)\end{aligned}\begin{aligned}&\text{Imaginary}\\&(a=0)\end{aligned}\begin{aligned}&\text{Complex}\\&(a+bi)\end{aligned}
\begin{aligned}&7+8i\\&(\greenD{7}+\blueD{8}i)\end{aligned}X
\begin{aligned}&\sqrt{3}\\&(\greenD{\sqrt{3}}+\blueD{0}i)\end{aligned}XX
\begin{aligned}&1\\&(\greenD{1}+\blueD{0}i)\end{aligned}XX
\begin{aligned}&-1.3i\\&(\greenD{0}+(\blueD{-1.3})i)\end{aligned}XX
\begin{aligned}&100i\\&(\greenD{0}+\blueD{100}i)\end{aligned}XX
Notice that in the table, all of the numbers listed are complex numbers! This is true in general!

## Now you try it!

Problem 4
What type of number is minus, 2, plus, 3, i?

Problem 5
What type of number is 10, point, 2?

Problem 6
What type of number is minus, 17, i?

## Why are these numbers important?

So why do we study complex numbers anyway? Believe it or not, complex numbers have many applications—electrical engineering and quantum mechanics to name a few!
From a purely mathematical standpoint, one cool thing that complex numbers allow us to do is to solve any polynomial equation.
For example, the polynomial equation x, squared, minus, 2, x, plus, 5, equals, 0 does not have any real solutions nor any imaginary solutions. However, it does have two complex number solutions. These are 1, plus, 2, i and 1, minus, 2, i.
As we continue our study of mathematics, we will learn more about these numbers and where they are used.

## Want to join the conversation?

• Wait, are all numbers complex?
• No BUT --- ALL REAL numbers ARE COMPLEX numbers.
It just so happens that many complex numbers have 0 as their imaginary part. When 0 is the imaginary part then the number is a real number, and you might think of a real number as a 1-dimensional number.

You need linear algebra or complex analysis to get the bigger picture, but for now, trust Stefan and me - real numbers are a special case for complex numbers just as a square is a special case for a rectangle.

This is an argument over semantics ever since it was decided to call these multi-dimensional entities numbers. You might think of complex numbers as two-dimensional. If that isn't rough enough there are numbers with even more that two dimensions, as Stefan alluded to. Don't bother yourself with the ins and outs of multidimensional numbers until or unless there really is a need to. For now just learn the basics of complex numbers so you can get through the traditional undergrad stuff and have a basis for further learning should you decide to go there.
The multidimensional number stuff is the kind of math that you really have to live in for awhile to understand and if you don't have a good motivation to understand then it can drive you crazy trying to get a quick understanding of what we are talking about. You need calc, trig, and analysis - you really do - to understand the reason why we teach complex numbers. Notice there aren't any complex number word problems in undergrad courses?
• If both real and imaginary parts of a complex number are 0, what kind of number is it? Or: is 0 real, complex or pure imaginary?
• This is an interesting question. The real numbers are a subset of the complex numbers, so zero is by definition a complex number ( and a real number, of course; just as a fraction is a rational number and a real number). If we define a pure real number as a complex number whose imaginary component is 0i, then 0 is a pure real number. If we define a pure imaginary number as a complex number whose real component is 0 (or: where a=0 in the general component form for a complex number: a + bi), then 0 is also a pure imaginary number. This makes sense geometrically in the complex plane: the origin is the intersection of coordinate axes, so (0,0) is on both the real and the imaginary axes.
• I have an issue with Problem 2:
What is the imaginary part of 21 - 14i?

The accepted answer was -14, where I'd expect it to be -14i, because the "imaginary part" is including the 'i', and the magnitude of the img part is -14...
• Hello there, steenbergh,
Though the question refers to the "imaginary part of this complex number" it is really refering to the magnitude of the imaginary part. Think of a complex number as a position vector in the complex plane, since the direction of this "imaginary part" is specified (it is in the imaginary axis or the vertical axis in this case) we only need to give its magnitude in that direction as we already know its direction. When we say -14, we are merely saying "negative fourteen units in the imaginary axis/number plane". Hope that makes sense!
• I understand that imaginary number can be helpful for solving math problems. However I am interested to know more on how it is used in quantum mechanics.
• That is very difficult to answer within the confines of the KA discussion page. If you really want to know, you can do a little searching on the net or you could read Hyperspace by Michio Kaku.
• Some people have suggested that there are numbers that are NOT complex. Would anyone mind elaborating on what those are?
• There are number systems beyond the complex numbers, but you don't see them in high-school math. This includes systems like the quaternions, which are 4-dimensional (like how the complex numbers are 2-dimensional), and the hyperreal numbers and surreal numbers, which include versions of infinite and infinitesimal numbers.
• I don't get the polynomial equation showed above: (x*x) - 2x + 5 = 0 and how its complex number solution is 1 + 2i and 1 - 2i.
• Use the quadratic formula to solve the equation and the answers become x = 1+2i and x=1-2i. If you look at the other questions & answers already posted for this page, you will see one near the top of the list (currently the 2nd one) where the answer shows all the work for solving the equation using the quadratic formula.
Hope this helps.
• Can you give the usage of imaginal number in reality?
• They pop up in physics, and you'll probably first see them when dealing with electromagnetism, or quantum mechanics. There's a good article about the usage of complex numbers here:

https://galileospendulum.org/2012/06/09/imaginary-numbers-are-real/

If you think complex numbers are weird, wait until you find out about quaternions... which is useful for computer science.
• Using the same logic that all real numbers are complex, for example 52 = 52 + 0i, couldn't we say that all imaginary numbers are complex as well? For example 2i = 2i + 0, which would be a complex number, right? Thank you in advance
• Yes, all imaginary numbers are also Complex Number as they can always be shown to have both a real and imaginary part.
• So every number, real, imaginary or both are complex?
• Yep, if you want the full picture of kinds of numbers it's like this.

Start with natural nummbers, which are just 1, 2 3 and so on. Adding 0 to the mix makes it whole numbers.

integers includes the negatives

rational numbers get a bit more complcated, basically any number that can be written as one integer divided by another, so this brings in fractions and infinitely repeating decials like 1/3

Real numbers makes things even more complicated with other infinitely repeating decimals, like square roots of non perfect square positive numbers, like square root of 2, and other infinitely repeating numbers like pi or e

The next step up is complex numbers, which is an addition of some real nuber, and one imaginary number. that being said one or both could be 0, so just a real number could be looked at as 5 + 0i, so the imaginary part has 0 multiplying it