Question 1

Wait, are _*all*_ numbers complex?

Accepted Answer

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?

Question 2

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...

Accepted Answer

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!

Question 3

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?

Accepted Answer

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.

Question 4

I have 0 pet cats
I have a real number of cats
I have an imaginary number of cats
I have a complex number of cats
ALL TRUE!

Accepted Answer

Whenever someone ask how many cats you have, you can just say "It's complex"

Question 5

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.

Accepted Answer

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.

Question 6

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.

Accepted Answer

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.

Question 7

Can you give the usage of imaginal number in reality?

Accepted Answer

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.

Question 8

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

Accepted Answer

Yes, all imaginary numbers are also *Complex Number* as they can always be shown to have both a real and imaginary part.

Question 9

Some people have suggested that there are numbers that are NOT complex. Would anyone mind elaborating on what those are?

Accepted Answer

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.

Question 10

in the complex number system, what is the value of the expression 16i^4- 8i^2 +4? (note: i =  square root -1). i am very confused as to how to solve this problem.

Accepted Answer

16 * i^4 - 8 * i^2 + 4
= 16 * (i^2)^2 - 8 * i^2 + 4
= 16 * (-1)^2 - 8 * (-1) + 4
= 16 * 1 - 8 * (-1) + 4
= 16 + 8 + 4
= 28

Complex Number	Standard Form $a + b i$ ‍	Description of parts
$7 i - 2$ ‍	$- 2 + 7 i$ ‍	The real part is $- 2$ ‍ and the imaginary part is $7$ ‍.
$4 - 3 i$ ‍	$4 + (- 3) i$ ‍	The real part is $4$ ‍ and the imaginary part is $- 3$ ‍
$9 i$ ‍	$0 + 9 i$ ‍	The real part is $0$ ‍ and the imaginary part is $9$ ‍
$- 2$ ‍	$- 2 + 0 i$ ‍	The real part is $- 2$ ‍ and the imaginary part is $0$ ‍

	$\begin{aligned} Real \\ (b = 0) \end{aligned}$ ‍	$\begin{aligned} Imaginary \\ (a = 0) \end{aligned}$ ‍	$\begin{aligned} Complex \\ (a + b i) \end{aligned}$ ‍
$\begin{aligned} 7 + 8 i \\ (7 + 8 i) \end{aligned}$ ‍			X
$\begin{aligned} \sqrt{3} \\ (\sqrt{3} + 0 i) \end{aligned}$ ‍	X		X
$\begin{aligned} 1 \\ (1 + 0 i) \end{aligned}$ ‍	X		X
$\begin{aligned} - 1.3 i \\ (0 + (- 1.3) i) \end{aligned}$ ‍		X	X
$\begin{aligned} 100 i \\ (0 + 100 i) \end{aligned}$ ‍		X	X

Integrated math 2

Course: Integrated math 2 > Unit 5

Intro to complex numbers

Defining complex numbers

Check your understanding

Classifying complex numbers

Reflection question

Examples

Now you try it!

Why are these numbers important?

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