Complex numbers

[whitesilk]

sorry, my imagination is cyclical. If you were to evaluate e^i what would you have?

 

I thought i = square root ( -1 )

Edited August 27, 2018 by whitesilk

[whitesilk]

16 minutes ago, Marblehead said:

Also, my brain tells me that, depending on the value we could have an arrow pointing at an infinite positions around the 360 circumference of the graph.

 

 

enter euler's formula e^ix = cos(x) + isin(x)

 

since cos(x) + sin(x) defines a circle of radius 1, e is a special number?

 

[Marblehead]

4 minutes ago, whitesilk said:

sorry, my imagination is cyclical. If you were to evaluate e^i what would you have?

 

I thought i = square root ( -1 )

Please don't be sorry.  I know nothing about "i"s square root.

 

Yeah, my brain is linear as is my thinking.  Some things take a long time for me to understand.

 

[Marblehead]

1 minute ago, whitesilk said:

 

 

enter euler's formula e^ix = cos(x) + isin(x)

 

since cos(x) + sin(x) defines a circle of radius 1, e is a special number?

 

Now you are WAAAAAY ahead of me.  I have no idea what you just said.

 

[wandelaar]

@ Marblehead

 

As I said we are considering arrows that start in the point (0,0) and end in a point of the plane (a,b), where a and b can be any (real) numbers. We don't consider curly arrows. And points at infinity are used in certain forms of geometry but we will not do that here. It is already difficult enough. Now in all this there is no symbol "i". So the complex numbers as arrows don't depend on a symbol "i" for their existence. Will you please start here, and say whether you understand this or not?

[whitesilk]

Marblehead, back in the day, 3-4 centuries ago, I think, people contrived solutions to problems that were unsolvable. One problem that was currently impossible, and still is, to take the square root of a negative number, so it is labeled i in math, and j in engineering

 

Edited August 27, 2018 by whitesilk
typo

[whitesilk]

5 minutes ago, Marblehead said:

Now you are WAAAAAY ahead of me.  I have no idea what you just said.

 

 

think of it like this: cos(x) + y * sin(x) [left that y out above] on the x,y plane

[Marblehead]

1 minute ago, wandelaar said:

@ Marblehead

 

As I said we are considering arrows that start in the point (0,0) and end in a point of the plane (a,b), where a and b can be any (real) numbers.

Okay.  "a" + b" is an arrow that begins at 0,0 and ends wherever the values of a + b indicate such as X,Y indicated.  Am I okay so far?

 

1 minute ago, wandelaar said:

 

We don't consider curly arrows.

Good.  I don't do curly arrows.

 

1 minute ago, wandelaar said:

 

And points at infinity are used in certain forms of geometry but we will not do that here.

Good.  Finding the end of infinity is always a problem.

 

1 minute ago, wandelaar said:

 

It is already difficult enough. Now in all this there is no symbol "i".

Great.  Real numbers. I can deal with that and any needed math.  (Well, in the most part.)

 

1 minute ago, wandelaar said:

 

So the complex numbers as arrows don't depend on a symbol "i" for their existence.

Are you saying that a complex number is the solution to an equation?  Or is there more to it?

 

1 minute ago, wandelaar said:

 

Will you please start here, and say whether you understand this or not?

I did my best.  Did I fall into the deep end?

 

[Marblehead]

4 minutes ago, whitesilk said:

 

 

think of it like this: cos(x) + y * sin(x) [left that y out above] on the x,y plane

Okay.  Hang in there with me.  I've never done this before.

 

[whitesilk]

grab a piece of paper, draw one horizontal line (the x axis), intersect one vertical line (this y axis).

 

the (0,0) point is the intersection between the vertical line and the horizontal line.

 

draw a random dot.

 

connect a line from the random dot to the (0,0) point.

 

draw a second horizontal line through the random dot.

draw a second vertical line through the random dot.

 

measure the real distance as the distance between the (0,0) point and the second vertical line intersecting with the x axis.

measure the imaginary distance as the distance between the (0,0) point and the second horizontal line intersecting with the y axis.

 

 

[wandelaar]

@ Marblehead

 

OK - the complex numbers are now considered as arrows that start in the point (0,0) and end in the point (a,b), where a and b can be any (real) numbers. This is the basic supposition, and we can clearly picture those complex numbers as arrows in our mind. 

 

Now the only role of a notation is to designate something. The expressions of the form a + bi designate arrows that start in the point (0,0) and end in the point (a,b). Now what does the "i" mean , what is its value? At this moment it has no meaning or value but is just a symbol to make sure we will not mindlessly add a to b to get a real sum. Nothing more is involved, at this point. The arrows are the complex numbers, and the expressions a + bi are notations of those complex numbers (= arrows).

[joeblast]

26 minutes ago, Marblehead said:

Okay.  Hang in there with me.  I've never done this before.

 

watch the video series - it'll wind up taking you less time, and you'll have a much better understanding of why i is a very natural extension of numbers.  basically, if you apply algebra to its logical conclusions, you cannot avoid i.  multiplying by i is what gives rotation, so if you think of a 2d sine wave but instead 3d now you can start doing things with a function like a corkscrew

[Marblehead]

7 minutes ago, joeblast said:

watch the video series - it'll wind up taking you less time, and you'll have a much better understanding of why i is a very natural extension of numbers.  basically, if you apply algebra to its logical conclusions, you cannot avoid i.  multiplying by i is what gives rotation, so if you think of a 2d sine wave but instead 3d now you can start doing things with a function like a corkscrew

Thanks but I would rather stay with Wandelaar with this project.  It's not like it is something I need to know because I am planning to send a rocket to Pluto.  It's something I mentioned that I wasn't able to grasp years ago.  It is more of a test to see if I am able to grasp it this time.  I get immediate feedback here so maybe that will help.

 

[Marblehead]

16 minutes ago, wandelaar said:

@ Marblehead

 

OK - the complex numbers are now considered as arrows that start in the point (0,0) and end in the point (a,b), where a and b can be any (real) numbers. This is the basic supposition, and we can clearly picture those complex numbers as arrows in our mind. 

 

Now the only role of a notation is to designate something. The expressions of the form a + bi designate arrows that start in the point (0,0) and end in the point (a,b). Now what does the "i" mean , what is its value? At this moment it has no meaning or value but is just a symbol to make sure we will not mindlessly add a to b to get a real sum. Nothing more is involved, at this point. The arrows are the complex numbers, and the expressions a + bi are notations of those complex numbers (= arrows).

Okay. 

 

So 0,0 and a,b is the complex number (arrow) containing both 0,0 and a,b?  How would that be presented on paper?  0,0 -> a,b?

 

So "a + bi" presently is undefined but it will at some point become a real number?

 

[Lost in Translation]

4 hours ago, wandelaar said:

when we would leave out the symbol one could just add the two numbers together (as Marblehead did) , and than we wouldn't have anything new. So the symbol "i" does serve a purpose as part of the expression "a + bi" namely in blocking the addition

 

This is so important! 'the symbol "i" does serve a purpose as part of the expression "a + bi" namely in blocking the addition'. I wish my teachers had stated this one fact. This alone was worth the entire thread. Thank you!

[wandelaar]

The explanatory posts of the other Bums are correct, but much too advanced to start with. Here we will need a very elementary introduction that gives a clear picture of complex numbers and of the way we can add and multiply them. When we succeed in that then I will consider this topic as a success. And then from there on there are indeed lots of books and video's one could use for further study.

 

But it is getting very late where I live, so I will stop for now.

[voidisyinyang]

11 hours ago, liminal_luke said:

 

 

 

 Is there a tie-in with spirituality?

Quote

The simplest discrete system corresponds directly to
the square root of minus one, when the square root of
minus one is seen as an oscillation between plus and
minus one. This way thinking about the square root
of minus one as an iterant is explained below. More
generally, by starting with a discrete time series of
positions, one has immediately a non-commutativity
of observations and this non-commutativi ty can be
encapsulated in an iterant algebra as defined in Sec-
tion 2 of the present paper.

https://www.worldscientific.com/doi/pdf/10.1142/9789813232044_0001

I corresponded with Math professor Louis Kauffman recently about complex numbers. There was an electrical engineering paper out of China that tried to dismiss Kauffman by claiming you do not need noncommutative math to explain complex numbers. http://ecoechoinvasives.blogspot.com/2018/01/rukhsan-ul-haq-wani-and-time-of-reality.html

Quote

My claim, and original idea, has been that this is circumnavigating a T'ai Chi (Yin/Yang) symbol! More recently (Oshins, 1993b) I have suggested that this proximate technique can be used to realize Wing Chun kung-fu's "bong sau/tan sau" movement [youtube] out of the Kauffman/Oshins "quaternionic arm" discussed and referenced below in end note 5.

 

He said he would respond. This is his latest paper. Kauffman was a colleague of Eddie Oshins at SLAC - Stanford Linear Accelerator Center - where Oshins realized that noncommutative quantum logic is the secret of Daoist Neigong training. http://ecoechoinvasives.blogspot.com/2017/05/the-physics-of-tao-eddie-oshins-cracked.html

Quote

These Majorana Fermions can be symbolized by Clif-
ford algebra generators a and b such that a [squared]
= b[squared]= 1
and ab = −ba. One can take a as the iterant corre-
sponding to a period two oscillation, and b as the
time shifting operator. Then their product ab is a
square root of minus one in a non-commutative con-
text.

Kauffman 2011

Quote

In this sense the square root of minus one is a clock and/or a clock/observer. 

http://ecoechoinvasives.blogspot.com/2018/01/lawrnence-domash-former-professor-where.html

Quote

...superconductivity within one neuron could become phase coherent with that in an adjoining cell by virtue of quantum tunnelling, and this could be stimulated by the macroscopic analog of stimulated emission (alluded to before in connection with the mantra), that is an AC Josephson effect. ...At a more interesting level, the quantum vacuum state may be said to be empty (of excitation) and yet full in the sense of pure potentiality; it contains "virtual" (unphysical) representatives of all possible modes of matter and excitation in the form of vacuum fluctuations or "virtual particles" (zero-point excitations of each field mode, assigned one-half quanta of energy, due directly to the non-commutative property of the field operators).

and then a collaborator of Kauffman gives us more details:

Quote

imaginary number x Planck's Constant as h-bar [ h divided by 2π,]....a comparable commutator between energy [frequency] and time... [change in]energy x [change in]time = 1/2 h-bar [half of the quantum phase space is quantum spin, which cannot be put in a Poisson bracket].

A Proof for Poisson Bracket in Non-commutative Algebra of Quantum Mechanics by Sina Khorasani, University of Vienna, 2014 pdf

 

 

noncommutative math as the T'ai Chi secret of Neigon

Edited August 28, 2018 by voidisyinyang

[wandelaar]

7 hours ago, Marblehead said:

So 0,0 and a,b is the complex number (arrow) containing both 0,0 and a,b?  How would that be presented on paper?  0,0 -> a,b?

 

An arrow pointing from (0,0) to a point (a,b) can be represented in several ways. One often writes it in the form of a 'row vector', and that would look thus: ( a  b ), however when we want to interpret the arrow as a complex number we usually write a + bi. Just as the brackets "(" and ")" in the notation as a row vector have no meaning of their own, so also the "+" and the "i" in the complex number notation have no meaning of their own. The "+" and "i" are just components of a notation for an arrow (= complex number) in the Cartesian coordinate system, and nothing more is involved (at this point). 

 

7 hours ago, Marblehead said:

So "a + bi" presently is undefined but it will at some point become a real number?

 

No and no. The expression "a + bi" is just a way to represent the arrows (= complex numbers) in a Cartesian coordinate system in a way that somewhat looks like numbers. This is done on purpose because later on when we have defined how to add and multiply arrows (= complex numbers) we will see that the arrows also behave just like numbers. And that is the fundamental idea behind all this. There are no real numbers z with the property that z*z = -1, but as we will see there is an arrow (= complex number) z wherefore we will have z*z = -1. So the equation z² + 1 = 0 has a solution for arrows (= complex numbers) where the same equation z² + 1 = 0 has no solution for ordinary real numbers. But we are getting ahead of ourselves. The important thing for now is to remember that "a + bi" is just a notation for the arrow pointing from (0,0) to the point (a,b). And a notation has no meaning besides what it designated. Thus "a + bi" designates an arrow pointing from (0,0) to the point (a,b), and the actual complex number that is involved is the designated arrow. And as the arrows are concrete geometrical objects so the same is true for the complex numbers, because in our chosen geometrical approach the complex numbers are considered as being the arrows.

[rideforever]

10 hours ago, Lost in Translation said:

This is so important! 'the symbol "i" does serve a purpose as part of the expression "a + bi" namely in blocking the addition'. I wish my teachers had stated this one fact. This alone was worth the entire thread. Thank you!

 

It doesn't "block the addition" !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
If you have a normal graph with x and y,   and you have a point (3,5).    You STILL can't add 3 + 5 = 8 !!!!
The i has got nothing to do with it.
That's like saying my friend lives 20 miles north and 20 miles east.   Where does he live ?   40 miles away !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

[Marblehead]

11 hours ago, wandelaar said:

The explanatory posts of the other Bums are correct, but much too advanced to start with. Here we will need a very elementary introduction that gives a clear picture of complex numbers and of the way we can add and multiply them. When we succeed in that then I will consider this topic as a success. And then from there on there are indeed lots of books and video's one could use for further study.

 

But it is getting very late where I live, so I will stop for now.

Yeah, I need to stay with the basics until I am sure my brain has a grasp of them.  No need to go further with me until that has happened.

 

See you in the morning.

 

[Marblehead]

4 hours ago, wandelaar said:

 

An arrow pointing from (0,0) to a point (a,b) can be represented in several ways. One often writes it in the form of a 'row vector', and that would look thus: ( a  b ), however when we want to interpret the arrow as a complex number we usually write a + bi. Just as the brackets "(" and ")" in the notation as a row vector have no meaning of their own, so also the "+" and the "i" in the complex number notation have no meaning of their own. The "+" and "i" are just components of a notation for an arrow (= complex number) in the Cartesian coordinate system, and nothing more is involved (at this point).

Is the complex number always assumed to begin at 0,0?

 

4 hours ago, wandelaar said:

No and no. The expression "a + bi" is just a way to represent the arrows (= complex numbers) in a Cartesian coordinate system in a way that somewhat looks like numbers. This is done on purpose because later on when we have defined how to add and multiply arrows (= complex numbers) we will see that the arrows also behave just like numbers. And that is the fundamental idea behind all this. There are no real numbers z with the property that z*z = -1, but as we will see there is an arrow (= complex number) z wherefore we will have z*z = -1. So the equation z² + 1 = 0 has a solution for arrows (= complex numbers) where the same equation z² + 1 = 0 has no solution for ordinary real numbers. But we are getting ahead of ourselves. The important thing for now is to remember that "a + bi" is just a notation for the arrow pointing from (0,0) to the point (a,b). And a notation has no meaning besides what it designated. Thus "a + bi" designates an arrow pointing from (0,0) to the point (a,b), and the actual complex number that is involved is the designated arrow. And as the arrows are concrete geometrical objects so the same is true for the complex numbers, because in our chosen geometrical approach the complex numbers are considered as being the arrows.

A double "No".  I did good.  Hehehe.

 

Let's don't start adding arrows yet, please.  My arrow is still crooked as hell.

 

So in that first graph with the arrow, "a + bi" is just pointing to an point on the graph from 0,0 to (a + bi).  Nothing else should be assumed at this point?

 

[joeblast]

1 hour ago, Marblehead said:

Yeah, I need to stay with the basics until I am sure my brain has a grasp of them.  No need to go further with me until that has happened.

 

See you in the morning.

 

like my boss keeps telling me "you've got to put this into normal people terms or they wont understand it"

[Marblehead]

Okay.  My brain asked me a question that I couldn't answer.  I need to answer the question before I can go any further.

 

The question is:  What are we doing?

 

So I will ask:  What are we doing?  Are we establishing a formula that is to be used in the future in order to solve equations using complex numbers?

 

I have to satisfy my brain with this before I can go any further.

 

[wandelaar]

43 minutes ago, Marblehead said:

So I will ask:  What are we doing?  Are we establishing a formula that is to be used in the future in order to solve equations using complex numbers?

 

I have to satisfy my brain with this before I can go any further.

 

We are not establishing a formula but we are trying to understand what kind of things complex numbers are. Now there are several ways to introduce complex numbers, but here we have  chosen the geometrical way. And I think that is the easiest to understand. When you use the geometrical way the complex numbers are considered to be arrows in a Cartesian coordinate system. As soon as you understand that the complex numbers are actually arrows you can then geometrically manipulate them to form a sum  and a product of those arrows. That is all we need to do. As  soon as you can add and multiply the arrows you can see that they behave like numbers. And that is why they are usually called "complex numbers" (and not "complex arrows").

 

What we are doing now is trying to understand what arrows are designated by what expressions a + bi. And that is just a convention, there is no calculation or equation involved. The convention is this:

 

The expression a + bi with a and b real numbers designates the straight arrow starting at the point (0,0) and ending in the point (a,b).

 

There is nothing more to it. Example: the expression 5 + 8i designates the arrow starting at the point (0,0) and ending in the point (5,8).

 

Thus these arrows are called complex numbers because - as we will see later - they behave like numbers when we form sums and products of them.

Edited August 28, 2018 by wandelaar

[Lost in Translation]

5 hours ago, rideforever said:

It doesn't "block the addition" !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
If you have a normal graph with x and y,   and you have a point (3,5).    You STILL can't add 3 + 5 = 8 !!!!
The i has got nothing to do with it.
That's like saying my friend lives 20 miles north and 20 miles east.   Where does he live ?   40 miles away !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

 

You are correct. Perhaps I should have been more precise. Although the topic centers on manipulating X,Y coordinates on a graph I was speaking in general, algebraic terms.