Complex numbers

[wandelaar]

 

Yes - we are!

[Marblehead]

5 minutes ago, wandelaar said:

Yes - we are!

Well, yes, for hanging in there with me all this time you deserve a medal.  How about one of my Good Conduct medals from the Army?

 

So let me ask you this question:

 

Whatever it is attached to, X or Y, the "i" is what makes it an imaginary number because Yi is undefined due to "i" being undefined and therefore it is impossible to multiply Y by "i"?  (At least at this point in time.)

 

 

 

[OldDog]

Ok. So I think where I am at, at this point, I can follow the rules of notation and label a point correctly as Re(z), Im(z). Neatly, this alternate notation avoids the use of i and all the attendant questions about it.

 

Still do not understand the significance/importance of why we are putting an arrow there. Maybe its time to suggest what wecan do with an arrow object. Or, is there more to be discovered about the object?

[wandelaar]

1 hour ago, Marblehead said:

Whatever it is attached to, X or Y, the "i" is what makes it an imaginary number because Yi is undefined due to "i" being undefined and therefore it is impossible to multiply Y by "i"?  (At least at this point in time.)

 

You could see it that way, if you leave out all the formal subtleties that make our construction of the complex numbers as arrows  mathematically precise.

 

But it is very important to remember that in our approach the "i" is only a sign. The complex numbers themselves are the arrows, and there is nothing undefined about the arrows. Also the imaginary numbers are simply those complex numbers z wherefore Re(z) = 0. So there is also nothing undefined about the imaginary numbers. We can simply draw the imaginary numbers as arrows in our Cartesian coordinate system. To repeat: the "i" is just a sign used in naming the complex numbers. Saying the "i" is undefined is dangerous because that gives the false impression that it somehow is a variable after all or that it should have a value.

 

One thing more: in actual practise the sign "i" is often used as shorthand for "0 + 1i". But it is very confusing to do this while we are still in the process of trying to understand the basics of the complex numbers.

 

Edited August 30, 2018 by wandelaar

[wandelaar]

1 hour ago, OldDog said:

Ok. So I think where I am at, at this point, I can follow the rules of notation and label a point correctly as Re(z), Im(z). Neatly, this alternate notation avoids the use of i and all the attendant questions about it.

 

Good.

 

Quote

Still do not understand the significance/importance of why we are putting an arrow there. Maybe its time to suggest what wecan do with an arrow object. Or, is there more to be discovered about the object?

 

It is possible to introduce the complex numbers without an arrow, that could be done by defining the complex numbers as ordered pairs (a,b). Such a construction of the complex numbers would proceed in an algebraic way. We could do that in a new topic sometime in the future. But it will not work to here follow two approaches at the same time.

 

Practical applications in solving algebraic equations will only be possible after we have defined the addition and multiplication of complex numbers. I think that should be doable, but further applications involving Euler's formula are almost certainly too advanced. 

 

Edited August 30, 2018 by wandelaar

[OldDog]

19 minutes ago, wandelaar said:

But it will not work to here follow two approaches at the same time.

 

Agreed.

 

Couple of more question before we proceed. Sorry, but this has to do with arrows, again.

 

Is it incorrect to think of Re(z), Im(z) as a point on a plane?

 

It seems that (0,0) as the origin is important to defining an arrow. While it may be a trivial case, it seems without an origin the arrow pointing to z would have no direction or dimension.

 

OMG ... suddenly hit me that to consider some other origin other than (0,0) would mean that such an origin would have to be a point that described by another point, another z with an ordered pair that contained an imaginary component. That is if the plane is consistent.

 

 

[Lost in Translation]

53 minutes ago, wandelaar said:

To repeat: the "i" is just a sign used in naming the complex numbers. Saying the "i" is undefined is dangerous because that gives the false impression that it somehow is a variable after all or that it should have a value.

 

The symbol "i" is a notation symbol. Check! Good to know. Not to be confused with an algebraic variable. Got it.

 

[wandelaar]

10 minutes ago, OldDog said:

Is it incorrect to think of Re(z), Im(z) as a point on a plane?

 

The point ( Re(z) , Im(z) ) is the endpoint of the complex number z considered as an arrow.

This logically follows from the definitions of Re( ) and Im( ). 

 

Quote

It seems that (0,0) as the origin is important to defining an arrow. While it may be a trivial case, it seems without an origin the arrow pointing to z would have no direction or dimension.

 

Correct: an arrow needs both a starting point and an endpoint.

 

Quote

OMG ... suddenly hit me that to consider some other origin other than (0,0) would mean that such an origin would have to be a point that described by another point, another z with an ordered pair that contained an imaginary component. That is if the plane is consistent.

 

I don't understand what you are saying here.

[OldDog]

8 minutes ago, Lost in Translation said:

The symbol "i" is a notation symbol.

 

Yeah, thanks LT for stating that. I had always considered it like and undefined-variable. Maybe that is what has contributed to my confusion. 

 

Gonna take a bit of time for that to sink in.

[OldDog]

11 minutes ago, wandelaar said:

I don't understand what you are saying here.

 

Well, you have not ruled out that there can be another starting point  other than (0,0) ... you seem to have deferred discussion.

 

If there can be some other starting point on a coordinate system described by X being real and Y not being real, then that point must be of the form Re(z),Im(z), right ... consistency, right?

 

[wandelaar]

46 minutes ago, OldDog said:

If there can be some other starting point on a coordinate system described by X being real and Y not being real, then that point must be of the form Re(z),Im(z), right ... consistency, right?

 

My definitions apply to a real Cartesian coordinate system with (0,0) as starting point for all arrows. When you want to use a coordinate system with an unreal (?) y-axis and a starting point for the arrows that doesn't lie on the intersection of the x- and y-axis than a lot of things will turn out differently. I don't know what would happen. Maybe you could start a topic for the development of such an OldDogian number system and than I could ask critical questions to make sure that your theory is mathematically legitimate?

Edited August 30, 2018 by wandelaar

[wandelaar]

@ OldDog

 

But I am serious! It would be a good exercise for both of us...

[OldDog]

I'm sorry, Wandelaar. I am not really trying to be obtuse. Just trying to develop a frame of reference for thinking about this. I honestly thought we had established that the Y axis was not real. Seems I leapt to a wrong conclusion.

 

So, X and Y axis are both real. So you are saying that on this plane defined by real X and Y coordinates there exist points that can be located by real (x,y) positions but they have some other (imaginary) attribute other than their real position. Did I get that right?

 

Asking questions is really the only way to test my understanding.

 

 

Edited August 30, 2018 by OldDog
Spelling

[OldDog]

7 minutes ago, wandelaar said:

But I am serious!

 

I know you are. My laughter was at the turn of the phrase " OldDogian number system"  Not laughing at you ... more with you. 

 

Apologies! Still chuckling though.

[Lost in Translation]

18 hours ago, wandelaar said:

The x-coordinate of the endpoint of a complex number (= arrow) z is called the real part Re(z) of the complex number z .

The y-coordinate of the endpoint of a complex number (= arrow) z is called the imaginary part Im(z) of the complex number z .

 

I don't want to put words in anyone's mouth but it seems we may need to back up slightly. In the above quote, Wandelaar is introducing something called a function, a very basic calculus concept. When he writes "Re(z)" what he is saying is that there is a function called "Re" that relates to a variable "z". I assume "Re" is short-hand for "Real." Similarly "Im" is probably short-hand for "Imaginary".

 

What is a function, you ask? Well, it's basically an algorithm, a set of ordered steps that perform an action on an argument expressed as a variable. For example, let's make a function called "Christmas" that takes as an argument a variable called "child." This would look like this: Christmas(child). The definition of Christmas(child) could be something like follows:

 

Christmas(child)

   if child is good

      then

         bring lots of presents

   otherwise

         bring a lump of coal

 

There's nothing magical about functions. They are just a convenient way to encapsulate logic into a reusable component. I am sure that at some point Wandelaar will give a definition of the functions he has introduced.

[OldDog]

5 minutes ago, Lost in Translation said:

Wandelaar is introducing something called a function

 

Is he?  I did not get that. Pretty familiar with the notation f(x) where f is a placeholder for whatever kind of function we might be talking about. The use of f(x) simply reminds us that we are talking about a function.

 

I don't think that is what Wandelaar is saying. I think he adopted the notation Im(z) because I/we seemed to be struggling with the notstion i.

 

Now, if I am wrong about this ... then I am still out in the hinterland.

[Marblehead]

I used to have a function.  I'm not sure that I still do.

 

[wandelaar]

1 hour ago, OldDog said:

I'm sorry, Wandelaar. I am not really trying to be obtuse. Just trying to develop a frame of reference for thinking about this. I honestly thought we had established that the Y axis was not real. Seems I leapt to a wrong conclusion.

 

Well there is a coordinate system called the complex plane that has a real x-axis and an imaginary y-axis. Most people learn the complex numbers by means of the complex plane. And as a way of representing the complex numbers the complex plane is perfect. But as a way of introducing the complex numbers it isn't. Suppose you know nothing about imaginary numbers and want to learn what they are. Now the teacher draws two axis: a real x-axis and an imaginary y-axes. Along the x-axis there are real numbers, so that's no problem. But along the y-axis there are imaginary numbers. Now what are imaginary numbers? Are you supposed to understand the complex numbers by accepting the imaginary numbers on faith? Apparently you are. And thus there will be two categories of student that are not going to get it. First there are those who are not smart enough to (quickly) understand how to calculate with those mysterious things called complex numbers. And second there are those who are too intelligent to be fooled by a circular explanation. Both will not get it. And so it goes...

 

Quote

So, X and Y axis are both real. So you are saying that on this plane defined by real X and Y coordinates there exist points that can be located by real (x,y) positions but they have some other (imaginary) attribute other than their real position. Did I get that right?

 

Actually the magic will happen when we define the multiplication of complex numbers. This will be done in such a sneaky way that the arrows will possess all the properties that we like complex numbers to have.

Edited August 30, 2018 by wandelaar

[wandelaar]

LiT is right: both Re( ) and Im( ) are functions.

 

Re(z) gives us the real part of the complex number z

Im(z) gives us the imaginary part of the complex number z

[Lost in Translation]

9 minutes ago, wandelaar said:

LiT is right: both Re( ) and Im( ) are functions.

 

Re(z) gives us the real part of the complex number z

Im(z) gives us the imaginary part of the complex number z

 

Ah, ha!

 

So if we have a point (3,2) where X=3 and Y=2 then

 

Given z = (3,2)

Re(z) returns the 3 part and

Im(z) returns the 2 part.

 

It's syntactically the same as 3 +2i, just expressed in a different notation.

 

 

Edited August 30, 2018 by Lost in Translation
Added z for clarity

[wandelaar]

When (a,b) is the endpoint of the complex number z than: Re(z) = a and Im(z) = b.

 

For a complex number z = a + bi we will get: Re(z) = a and Im(z) = b.

 

So here we have our first "theorem" :

 

z = Re(z) + Im(z) i

 

All this refers to the same geometrical fact, only expressed differently.

 

[wandelaar]

[Marblehead]

1 hour ago, wandelaar said:

First there are those who are not smart enough to (quickly) understand how to calculate with those mysterious things called complex numbers. And second there are those who are too intelligent to be fooled by a circular explanation. Both will not get it. And so it goes...

Well, I almost didn't get it.  I won't put myself in a basket though.

 

 

[Lost in Translation]

2 minutes ago, Marblehead said:

Well, I almost didn't get it.  I won't put myself in a basket though.

 

Remember basic training? Brain off. Follow orders, quickly!

 

[Marblehead]

Just now, Lost in Translation said:

 

Remember basic training? Brain off. Follow orders, quickly!

 

I was never good at following orders until after I had asked "Why?".  Yes, that caused a lot of problems.