Complex numbers

[wandelaar]

@ OldDog & Jeff

 

At this stage the symbol "i" is to be considered as just a symbol, and nothing more. But 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 done by Marblehead, but besides that there is nothing to be understood about the "i". In fact, trying to understand what the "i" means spoils the whole idea of our geometrical introduction of the complex numbers by means of arrows.

Edited August 27, 2018 by wandelaar

[OldDog]

Soooo ... i is a value (variable?) that we should ignore ... except for its presence. I guess I can suspend my distraction long enogh for the explanation to continue ... but at some point you are going to have to give us a frame of reference for i to go.much further.

[Jeff]

Yes, I am fine with the format and structure to explain the variable.

 

[rideforever]

What do the axes on this graph represent ?
And what do the points on it represent ?
What does i represent ?

[rideforever]

What ?    
Marblehead said in relation to the Red Point that  -3 + 1 = 2 .... and you think the only problem with this is that he didn't include i !!!
Wow that's a good one.
There is no -3 + 1   it is actually -3x + 1y, each axis is fully independent of each other.

Edited August 27, 2018 by rideforever

[wandelaar]

Thank you. But I still have to know whether we now understand how to write the arrows as complex numbers of the form a + bi . I have already given the complex representation of the green arrow myself, but I like to see you guys give the correct representation as a complex number of the red arrow.

[steve]

I would recommend when teaching something like this you first clearly define the axes.

The standard format is to assign the y-axis to the imaginary component.

IMO, the y-axis could should be labeled i for clarity.

The red arrow denotes -3+1i or simply -3+i.

 

[Jeff]

1 minute ago, steve said:

I would recommend when teaching something like this you first clearly define the axes.

The standard format is to assign the y-axis to the imaginary component.

IMO, the y-axis could should be labeled i for clarity.

The red arrow denotes -3+1i or simply -3+i.

 

 

Agreed.  -3 + i

 

[steve]

The other points would be -1.5 - 2.5i and 2 + 3i

[steve]

27 minutes ago, rideforever said:

What do the axes on this graph represent ?

 The x-axis represents the real number component and the y (i) axis represents the imaginary component of the complex number.

 

27 minutes ago, rideforever said:

And what do the points on it represent ?

The points represent complex numbers

 

27 minutes ago, rideforever said:

What does i represent ?

i represents the square root of -1 which is undefined mathematically but can still be very useful in many branches of math and physics.

[steve]

27 minutes ago, rideforever said:

What ?    
Marblehead said in relation to the Red Point that  -3 + 1 = 2 .... and you think the only problem with this is that he didn't include i !!!
Wow that's a good one.
There is no -3 + 1   it is actually -3x + 1y, each axis is fully independent of each other.

 

You don't need to include the x and y as along the x-axis each real number is well defined, there are no variables. Along the y-axis, each number is a multiple of i (the square root of -1). You can't simply add -3 + 1 because the 1 is not 1, it is 1 times the square root of -1 which is not defined mathematically.

[joeblast]

much of what's trying to be said is well detailed in a nice way by the numberphile videos "Imaginary numbers are real"

 

 

 

numberphile and mathologer are excellent channels

[rideforever]

All this makes me wonder if mankind goes deeper into his imaginations and technologies in proportion to his lack of comprehension of himself.
He cannot make progress with himself, so he causes trouble elsewhere.
i is more i-candy for the mind.
The never ending mind-pain whirling around.

"Scientists" today are so mental and dull,  like crazed monkeys on steroids competing to push their mind even faster,  compared to the brightness and soul-journey-ness of the Ancient Greeks.

In general humans don't value real numbers, they don't value reality, they don't sense reality, life is one long dream-nightmare.

 

[liminal_luke]

2 hours ago, wandelaar said:

 

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.

 

Wandelaar,

 

I loved math as a teenager and the topic of complex/imaginary numbers especially intrigued me.  I once checked out a book on differential calculus from the library just because I loved pouring over all the strange (to me) notations.  Although I did my best to decipher the meaning of the text, my engagement with the material never really went beyond an aesthetic fascination with the mathmatical symbols.  

 

For better or worse, my life path hasn`t required me to do much electrical engineering.  I`m purely a consumer of the electrical engineering efforts of others.  That might seem sad, I guess, but nobody can do everything.  Could you say a little more about why the average non-engineer Bum might be interested in the topic of complex numbers.  Is there a tie-in with spirituality?

[Marblehead]

3 hours ago, wandelaar said:

 

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 ?

Well, "i" has not been defined.  Yes, I did miss the "i".

 

I considered bi to define -3.  Upon consideration. If b is -3 then the equation is (-3 X i).  As "i" is still undefined I could apply any value to "i". with all results different.  That doesn't offer me much security.

 

[Zhongyongdaoist]

35 minutes ago, steve said:

I would recommend when teaching something like this you first clearly define the axes.

The standard format is to assign the y-axis to the imaginary component.

IMO, the y-axis could should be labeled i for clarity.

The red arrow denotes -3+1i or simply -3+i.

 

 

Thanks Steve, I have been doing the posting equivalent of biting my tongue for three or so hours, but I didn't want to post again unless absolutely necessary to get past this obstacle, and I was hoping that someone else would bring up these points, so I wouldn't have to.

 

ZYD

[Marblehead]

1 hour ago, wandelaar said:

@ OldDog & Jeff

 

At this stage the symbol "i" is to be considered as just a symbol, and nothing more. But 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 done by Marblehead, but besides that there is nothing to be understood about the "i". In fact, trying to understand what the "i" means spoils the whole idea of our geometrical introduction of the complex numbers by means of arrows.

Okay.  "i" is just imaginary so for now we don't consider it.  But then, the equation (-3 + 1i) cannot be solved because "i" is undefined.

 

[Marblehead]

1 hour ago, wandelaar said:

Thank you. But I still have to know whether we now understand how to write the arrows as complex numbers of the form a + bi . I have already given the complex representation of the green arrow myself, but I like to see you guys give the correct representation as a complex number of the red arrow.

-3 + 1i

 

But I still don't know anything because "i" is still undefined.

 

[wandelaar]

When you consider the complex numbers as something given then of course there would be nothing more to explain would there? Then you can just draw the complex plane with the real and imaginary axis and so on and so forth. What I am trying to do here is to introduce the complex numbers to Bums who don't already know what they are.

 

But as I was doing some shopping others have jumped in to explain the complex numbers, so I will step back for a while to see what happens. Maybe they do know better than me how to explain this thing. We will see...

 

Edited August 27, 2018 by wandelaar

[Marblehead]

3 minutes ago, wandelaar said:

When you consider the complex numbers as something given then of course there would be nothing more to explain would there? Then you can just draw the complex plane with the real and imaginary axis and so on and so forth. What I am trying to do here is to introduce the complex numbers to Bums who don't already know what they are.

 

But as I was doing some shopping others have jumped in to explain the complex numbers, so I will step back for a while to see what happens. Maybe they do know better than me how to explain this thing. We will see...

 

Well, I'm still with you and I haven't been paying attention to what they have been saying so I would be pleased if you continue on while considering my responses whether they are correct or incorrect and especially why they are incorrect if they are so.

 

You have lots of help here so being as optimistic as possible, perhaps something others add to this discussion will help someone better understand the concepts.

 

Please continue.  Be brave.

 

[whitesilk]

haley's comet  has an orbit of 76 years

 

saturn has an orbit of 28 years

 

76 / 28 = 2.71 = e

[Marblehead]

10 minutes ago, whitesilk said:

haley's comet  has an orbit of 76 years

 

saturn has an orbit of 28 years

 

76 / 28 = 2.71 = e

Okay.  You have defined the ratio of the orbit of Haley's comet to the orbit of Saturn around the sun as "e'.  Those are "real" numbers.  Doesn't help understanding imaginary numbers.

 

[wandelaar]

@ Marblehead

 

OK - lets proceed. The best way to remove your problem (at least in our current approach) is to think of the complex numbers as being the arrows. The arrows can be drawn as soon as a Cartesian coordinate system is given. We don't need a symbol "i" for that. Agreed?

[Marblehead]

8 minutes ago, wandelaar said:

@ Marblehead

 

OK - lets proceed. The best way to remove your problem (at least in our current approach) is to think of the complex numbers as being the arrows. The arrows can be drawn as soon as a Cartesian coordinate system is given. We don't need a symbol "i" for that. Agreed?

Agreed.  Based on what you already said, if we ignore "i" we will have a mathematic equation that is solvable.  And the solution will be a real number specifically defined, ie, 37.625

 

But even the solution wouldn't define a specific value of anything unless we also define "i".

 

(Bare with me while I work with the way my brain works.  Hehehe.)

 

[Marblehead]

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.