Complex numbers

[wandelaar]

Two special cases:

 

 

How do we represent the green arrow and the red arrow as a complex number of the form a + bi ?

[Marblehead]

Well, let's see:

 

Green:  0 + 2i

 

Red:  3 + 0i

 

 

 

How badly did I mess that up?

 

[Marblehead]

A question that popped into my mind:

 

Will it ever be the case where it is ai + b?

 

 

 

 

[wandelaar]

12 minutes ago, Marblehead said:

Well, let's see:

 

Green:  0 + 2i

 

Red:  3 + 0i

 

 

 

How badly did I mess that up?

 

That is completely correct!

 

Now when actually calculating with those objects it quickly becomes a nuisance to write the zeros when they occur. So in practice one often writes 2i as a shorthand for 0 + 2i, and 3 as a shorthand for 3 + 0i. But we have to remember that in the context of the complex numbers when we see "2i" or "3" that it is just a lazy way of writing "0 + 2i" and "3 + 0i".

 

So when we are dealing with complex numbers the expressions of the form "bi" and "a" are shorthand for "0 + bi" and "a + 0i". Thus 4i , -1i , and 3.6784i are shorthand for 0 + 4i , 0 + -1i , and 0 + 3.6784i . And again in the context of complex numbers 2 , 7.899 , and 101 are shorthand for 2 + 0i , 7.899 + 0i , and 101 + 0i .

[Marblehead]

5 minutes ago, wandelaar said:

 

That is completely correct!

 

Now when actually calculating with those objects it quickly becomes a nuisance to write the zeros when they occur. So in practice one often writes 2i as a shorthand for 0 + 2i, and 3 as a shorthand for 3 + 0i. But we have to remember that in the context of the complex numbers when we see "2i" or "3" that it is just a lazy way of writing "0 + 2i" and "3 + 0i".

I would rather stick with the "0 + 2i" for the time being so that my brain doesn't become confused.  Simplify later.

 

5 minutes ago, wandelaar said:

 

So when we are dealing with complex numbers the expressions of the form "bi" and "a" are shorthand for "0 + bi" and "a + 0i". Thus 4i , -1i , and 3.6784i are shorthand for 0 + 4i , 0 + -1i , and 0 + 3.6784i . And again in the context of complex numbers 2 , 7.899 , and 101 are shorthand for 2 + 0i , 7.899 + 0i , and 101 + 0i .

Okay.  Let's keep it longhand for now if you please.

 

[Marblehead]

Others have become silent.  I wonder if they are way ahead of me or if they just got tired.

 

 

 

[OldDog]

No, I'm stil here .... but struggling still with the idea of arrowness. What is it that give the geometric thing direction ... and for that matter straightness?  Why not a curved line ... with or without an arrow on the end?

[Marblehead]

6 minutes ago, OldDog said:

No, I'm stil here .... but struggling still with the idea of arrowness. What is it that give the geometric thing direction ... and for that matter straightness?  Why not a curved line ... with or without an arrow on the end?

I think you might be wanting to get ahead of Wandelaar.  I think we are still with Algebra working with real numbers.  

 

But I'm glad you are still with us.  I don't want to be the only one making mistakes here.

 

[wandelaar]

@ OldDog

 

I am not giving the one and only explanation of the complex numbers, but only the one that I consider the easiest to understand. There are many other ways of introducing the complex numbers besides that. Maybe it can also be done with a curly arrow, or with an arrow that starts form somewhere else than (0,0). Maybe there is an alternative to using an arrowhead at the end to give it a direction. Who knows? I am just giving one possible way to do it, no more and no less. So basically your questions of "why this?" and "why that?" cannot really be answered. It's just a choice to do it this way or that way...

 

 

 

[wandelaar]

1 hour ago, Marblehead said:

A question that popped into my mind:

 

Will it ever be the case where it is ai + b?

 

You could of course formally give that expression a meaning by defining it to mean b + ai. But lets not do that until the basics of the theory are well understood.

[wandelaar]

One more very special case is the "arrow" that starts and ends in (0,0). How would you write that arrow in the form a + bi ?

[Marblehead]

Hehehe.

 

i

 

 

 

[wandelaar]

Just when it seemed you understood it all, and now this.....

 

Please look again?

Edited August 29, 2018 by wandelaar

[Marblehead]

How about 0 + 0i

 

Now, understand, this is messing with my brain.  I need to be given slack here.

 

 

[Marblehead]

It's time for me to get away from the computer for a while.  I'll be back.

 

[wandelaar]

OK - let me know when you are ready for the next phase.

 

That will then be studying the real and imaginary part and the argument and modulus of a complex number. (Sounds more difficult than it is.)

[OldDog]

a+bi or ai+b ... it doesn't really matter does it?  You are specifying a point on a Cartesian coordinate system (to continue the geometric model) where one dimension id real and the other is not ... contains an imaginary term. The notation is arbitrarty, no?

 

Sorry about the fixation on arrowness. One of the reasons I struggle with math in general ... and imaginary numbers in particular ... is that if I cannot relate it to a real world situation ... a frame of reference for how to apply it ... it seems devoid of meaning. I took a fair amount of math along the way. The only way I could maintain my sanity was to think of it as an arbitrary game with objects and rules for manipulating them. The rules didn't have to make sense, just learned. Kinda like life. 

[rideforever]

Mathematicians cannot solve :   sqrt ( -1 ).   

They don't know how.   

So in order that they may still make some progress with equations what they do is call     i =  sqrt( -1 ).    

So they define i to equal the square root of minus one.

Why ?
Well .... because they want to do some work with their equations and they sort of package up this unknown thing into the term   i.   and that allows them to proceed and at least some work with their equations.

You might call   i a white elephant.

So a normal number is     17.   Seventeen.    A complex number is    17  + 5i, seventeen plus five white elephants.
What is a white elephant it is  i   or the sqrt ( -1 ) that we can't solve right now, so we call it white elephant.

 

So if you have 2 complex numbers :

 

17 + 5i ..... seventeen plus five white elephants

20 + 2i  ..... twenty plus 2 white elephants

 

... then if you add them together what do you get ?

 

You get

 

37 + 7i .... thirty seven plus seven white elephants.

 

or

 

(17+5i)   + (20+2i) = 37 + 7i

 

[OldDog]

26 minutes ago, rideforever said:

.... because they want to do some work with their equations ...

 

That brings up a good point. If it is undefined ... how can we assume it will behave the same way at real numbers? If its undefined I would think we cannot assume anything about it.

 

@wandelaar Hope you will touch on this. You have already suggested in the arrow example that we would be performing some operations on those arrows.

[wandelaar]

20 minutes ago, rideforever said:

Mathematicians cannot solve :   sqrt ( -1 ).   

They don't know how.  

 

That's only shows that you don't know what you are talking about. You are making a fool of yourself.

 

Lao tzu considered it very important to know the boundaries of ones knowledge. Nobody knows everything, and personally realising that fact is a great form of (self)knowledge.

[rideforever]

21 minutes ago, wandelaar said:

Lao tzu considered it very important to know the boundaries of ones knowledge.

He also left the city after failing to reconcile his situation there.

But anyway I am not saying I do know.  I just wanted to talk about elephants.
This thread needs a bit of levity, or gravity, or ... stuff.

My understanding is that mathematicians still cannot solve sqrt(-1).   Has that changed ?

[wandelaar]

@ OldDog

 

You can't apply a theory before it is developed. You are asking the impossible. Complex numbers have many applications. I already named some of them in a reaction to rideforever. And the arrows are meant to give a solid foundation to the complex numbers. But most practical applications will only become possible after we have defined e^z  where z is a complex number. But that's advanced stuff. I doubt whether we will come that far.

 

The complex numbers as arrows are perfectly defined objects. Nothing mysterious about that. And that is the reason I am building this theory on the arrows. Please ignore the misinformed rantings of rideforever.

[whitesilk]

24 minutes ago, rideforever said:

This thread needs a bit of levity

 

Eye Aye or i?

[whitesilk]

ayeaye blog post 1

07/03/2018

I am currently taking steps toward internet independence, hence this blog. For some time, I've been expressing an 'aye aye captain' additude. Being that I live in a Free Nation, and raised as such, I have also lived as if the certain freedoms that I enjoy are equally available to every other member of this free nation. On the world wide web, however, this is not so. There are a vast array of Servers and Clients all interconnected to bring services to the world. Having been a client of the web for my entire life, this is the first time that I've taken steps to serve.

About four months ago now, I've sent out a distress signal. A simple ---___--- at the end of a job application claiming that I've been on a researching island. In my research, I have found something that I would like to share with the world. Why light a lamp and put it under a bushel basket? However, I feel the need to share discretely, as if I can answer the door, or choose to keep it locked.

So back to the aye aye mentality that I've been living in. While taking Ghandi's quote into account that an eye for an eye would cause the world to go blind, I belive that temptation ought best be resisted. I once had a book. I gave this book to an oriental indian because the book was a series of speeches by swami vivikananda. In on of his speeches, this swami took the stance that the only way to get rid of temptation is to give into it until no longer desire that which you are tempted. The internal conflict of resisting temptation causes the temptation itself. tat vam asi.

So, another viewpoint is that of guatama; resist mara (the tempter) though insight; rein your mind and body like a good horse. For a well thached roof keeps the floor dry. How can one serve if the there is flooding in the basement? The actions I take I cannot forsake, spoken from a point of view that takes minimal action in life. Perhaps the undo shortcut is there to remind us all of that idea.

So, Jesus would say, "Make a tree good and its fruit will be good, or make a tree bad and its fruit will be bad, for a tree is recognized by its fruit... For out of the overflow fo the heart the mouth speaks. The good man brings good things out of the good stored up in him." MATTHEW 12:33-35 NIV. I translate aye aye as yes yes or oui oui. So if I am always saying yes to every question, am I really doing myself and others good? To live in a reactionary state of mind is troublesome regarless if your react positvely or negatively. So because I think this, I more often than not do nothing, think nothing, and say nothing.

Lao-Tzu would say that to speak rarely is natural. Aye aye can also be an expression of exasperation. A solemn mind at peace with him/her self is able to reflect clearly the nature of others. So, the seeking of understanding oneself is necessary fore the wise person may see clearly. Even so, the wise may only be a sounding board for others in distress. Every individual ought have the ability to find their solution; a core ideal termed freedom.

 

[wandelaar]

2 hours ago, OldDog said:

You are specifying a point on a Cartesian coordinate system (to continue the geometric model) where one dimension id real and the other is not ... contains an imaginary term.

 

Both axes of my Cartesian coordinate system are real! When you introduce the complex numbers by taking the horizontal axis as real and the vertical axis as imaginary then you are already presupposing the existence of the imaginary numbers. That's a vicious circle. New mathematical objects must - whenever possible - be constructed from the already trusted old ones.

 

As soon as the existence of the complex numbers is proven to be non-problematic then one can suppose the vertical axis to be imaginary.