Complex numbers

[wandelaar]

In this topic I will give an explanation of complex numbers. Imaginary numbers are a special type of complex numbers so my explanation will at the same time explain what imaginary numbers are. There are basically two ways of introducing complex numbers: it can be done in a geometrical and in an algebraic way. Which way shall we choose?

[Jeff]

geometrical.  

[Lost in Translation]

You're teaching math now? Bold move! Are you going to link this math back to Taoism somehow? That will be impressive!

 

Imaginary numbers always bothered me in school so I look forward to your treatise.

 

 

[Marblehead]

12 minutes ago, Jeff said:

geometrical.  

Sure, let's try it that way and see what happens.

 

[Marblehead]

5 minutes ago, Lost in Translation said:

Are you going to link this math back to Taoism somehow? That will be impressive!

Well, it arose out of Taoism so it should be able to return.

 

[wandelaar]

We will need a Cartesian coordinate system for that. It looks like this:

 

(Source: https://upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Cartesian-coordinate-system.svg/2000px-Cartesian-coordinate-system.svg.png    )

[wandelaar]

Now do we all understand that each point in the plane can represented by an ordered pair of numbers (x,y) such that x gives the horizontal position of the point and y gives the vertical position of the point?

[Jeff]

1 minute ago, wandelaar said:

Now do we all understand that each point in the plane can represented by an ordered pair of numbers (x,y) such that x gives the horizontal position of the point and y gives the vertical position of the point?

 

yes

 

[Marblehead]

10 minutes ago, Jeff said:

 

yes

 

Me too.

 

[wandelaar]

Now each complex number can be represented by a corresponding arrow in the Cartesian coordinate system. The general notation for a complex number is a + bi . The arrow corresponding to the complex number a + bi is the arrow that starts in the point (0,0) and ends in the point (a,b). Is that understood?

Edited August 27, 2018 by wandelaar

[Marblehead]

a + bi is still talking about X,Y … right?  Except we are now drawing an arrow beginning at 0,0 and ending at X,Y?

 

Then a + bi, is the arrow itself and its value of X,Y?

 

 

[wandelaar]

Here is an example. What complex number is represented by the red arrow? Please write it in the form a + bi ?

 

Edited August 27, 2018 by wandelaar

[Marblehead]

Looks like -3 + 1 to me.  Do the math:  -2

 

 

PS  I'm going out for lunch.  I'll catch up later.

 

[wandelaar]

Just now, Marblehead said:

Looks like -3 + 1 to me.  Do the math:  -2

 

 

PS  I'm going out for lunch.  I'll catch up later.

 

Almost right, but you missed the symbol "i". See you later.

 

Anybody else know the correct notation for the red arrow as a complex number a + bi ?

[Jeff]

7 minutes ago, wandelaar said:

Here is an example. What complex number is represented by the red arrow? Please write it in the form a + bi ?

 

 

I get the logic, but this is only describing a form as you are basically declaring one of the axis as an "imaginary realm".

 

[wandelaar]

8 minutes ago, Jeff said:

I get the logic, but this is only describing a form as you are basically declaring one of the axis as an "imaginary realm".

 

Correct! But we need to represent the complex numbers as geometrical objects (arrows) first to be able to give a geometrical definition of their sum and product later. That's the way we choose to do it. And when we have the definitions of the sum and product we can geometrically prove what rules apply to the sum and product of complex numbers.

Edited August 27, 2018 by wandelaar

[rideforever]

Red is -3,1 .... it's a vector, no need for i .... is there ?

[rideforever]

If i= sqrt( -1) .... and then ?   It's a definition.
Or  pf = sqrt (  gigantic pink flamingo ) .... another definition.
er .... you can make as many definitions as you like.

 

Some "interesting" questions are :

- do humans play with their imagination / complex / irrational / unreal ideas because they have not been able to comprehend how to make progress in reality ?

- why are real numbers good ?

- what is the "spiritual" meaning usefulness purpose behind all these concepts ?

- when is imagination not good, and what is a good use for the imagination ?

 

Edited August 27, 2018 by rideforever

[wandelaar]

3 minutes ago, rideforever said:

Red is -3,1 .... it's a vector, no need for i .... is there ?

 

The complex numbers are used in quantum mechanics, in electrical engineering, in solving algebraic equations, in Fourier analysis, in Laplace transforms, etc. All of them hugely useful applications.

[rideforever]

Just now, wandelaar said:

 

 

If you say so, but anyway .... isn't (-3,1) sufficient where is the need for i ?   Is there something I am missing ?

[Zhongyongdaoist]

With all due respect wandelaar, if I were going to discuss complex numbers I would use a number theory approach, and show how both "negative" and "imaginary" numbers arise from doing such basic operations as addition and subtraction, multiplication and division, on what are called "the Natural Numbers", then the nature and origin of these concepts, such as negative numbers can then be seen as answers to questions which naturally arise, like "I know that 8-7 equals 1, but what does 7-8 equal?", and becomes much clearer.

 

All the best in your endeavor,

 

ZYD

[wandelaar]

Maybe some extra examples will be helpful. These are all complex numbers:

 

1 + (-4) i

3.778 + 1 i

-0.000078787 + 10009897.555... i

 

As you can see it doesn't matter what two numbers you put in as long you have an expression of the form a + bi with a and b some numbers and "i" a mere symbol, than you have a complex number. And each such complex number corresponds to an arrow in the Cartesian coordinate system. Now before I can get any further it is important to know whether we understand what arrows correspond to what complex numbers. So again: what is the complex number corresponding to the red arrow?

Edited August 27, 2018 by wandelaar

[wandelaar]

I have done the green one myself:

 

[OldDog]

I don't think the point is getting across. You have presented two things: a cartesean coordinate system with some arrows drawn and some expressions that (algebraicly) involve a coefficient i. Now, it's easy enough to accept a notation like (a,b) ... but it is failing to relate ... at least in my mind ... to the imaginary number i. Need to get over that hump.

[Jeff]

16 minutes ago, OldDog said:

I don't think the point is getting across. You have presented two things: a cartesean coordinate system with some arrows drawn and some expressions that (algebraicly) involve a coefficient i. Now, it's easy enough to accept a notation like (a,b) ... but it is failing to relate ... at least in my mind ... to the imaginary number i. Need to get over that hump.

 

I would agree with OldDog.