CTK Exchange
CTK Wiki Math
Front Page
Movie shortcuts
Personal info
Awards
Terms of use
Privacy Policy

Interactive Activities

Cut The Knot!
MSET99 Talk
Games & Puzzles
Arithmetic/Algebra
Geometry
Probability
Eye Opener
Analog Gadgets
Inventor's Paradox
Did you know?...
Proofs
Math as Language
Things Impossible
My Logo
Math Poll
Other Math sit's
Guest book
News sit's

Recommend this site

Manifesto: what CTK is about Search CTK Buying a book is a commitment to learning Table of content Products to download and subscription Things you can find on CTK Chronology of updates Email to Cut The Knot Recommend this page

CTK Exchange

Subject: "A complex number approach..."     Previous Topic | Next Topic
Printer-friendly copy     Email this topic to a friend    
Conferences The CTK Exchange This and that Topic #942
Reading Topic #942
jmolokach
Member since Jan-11-11
Sep-18-10, 09:55 PM (EST)
Click to EMail jmolokach Click to send private message to jmolokach Click to view user profileClick to add this user to your buddy list  
"A complex number approach..."
 
In a recent discussion during a high school math class, my students learned that a + bi multiplied by it's conjugate gives the real number a^2 + b^2. After reading https://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/ (which I found interesting) I thought how the number a + bi being rotated by a - bi winds up being a number on the positive real axis, which has the same length as either conjugate.

I think this is really cool and would make for a great animation. Not sure if this might qualify as a proof. I have a feeling this might turn into another "How do you know that c is the radius of your circle?"

Any thoughts??

molokach


  Alert | IP Printer-friendly page | Reply | Reply With Quote | Top

  Subject     Author     Message Date     ID  
A complex number approach... jmolokach Sep-18-10 TOP
  RE: A complex number approach... alexb Sep-18-10 1
     RE: A complex number approach... jmolokach Sep-19-10 2
         RE: A complex number approach... alexb Sep-19-10 3
             RE: A complex number approach... jmolokach Sep-23-10 4
                 RE: A complex number approach... jmolokach Sep-23-10 5
                     RE: A complex number approach... alexb Sep-23-10 6
                         RE: A complex number approach... jmolokach Sep-23-10 7
                             RE: A complex number approach... alexb Sep-23-10 9
                                 RE: A complex number approach... jmolokach Sep-23-10 10
                                     RE: A complex number approach... alexb Sep-23-10 11
                                         RE: A complex number approach... jmolokach Sep-24-10 12
                                             RE: A complex number approach... alexb Sep-24-10 14
                                             RE: A complex number approach... jmolokach Sep-24-10 17
                                             RE: A complex number approach... alexb Sep-24-10 18
                                             RE: A complex number approach... jmolokach Sep-24-10 19
                                         RE: A complex number approach... jmolokach Sep-24-10 13
                                             RE: A complex number approach... alexb Sep-24-10 15
                                         RE: A complex number approach... jmolokach Sep-24-10 16
  RE: A complex number approach... gaespes Feb-01-11 20
  RE: A complex number approach... gaespes Feb-09-11 21

Conferences | Forums | Topics | Previous Topic | Next Topic
alexb
Charter Member
2758 posts
Sep-18-10, 10:09 PM (EST)
Click to EMail alexb Click to send private message to alexb Click to view user profileClick to add this user to your buddy list  
1. "RE: A complex number approach..."
In response to message #0
 
   >In a recent discussion during a high school math class, my
>students learned that a + bi multiplied by it's conjugate
>gives the real number a^2 + b^2. After reading
>https://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/
>(which I found interesting) I thought how the number a + bi
>being rotated by a - bi winds up being a number on the
>positive real axis, which has the same length as either
>conjugate.
>
>I think this is really cool and would make for a great
>animation. Not sure if this might qualify as a proof.

You mean "... of the Pythagorean theorem" of course. But what is exactly a proof here?

>I
>have a feeling this might turn into another "How do you know
>that c is the radius of your circle?"

No, much simpler. What is the length of a complex number? And, perhaps, where did it come from?


  Alert | IP Printer-friendly page | Reply | Reply With Quote | Top
jmolokach
Member since Jan-11-11
Sep-19-10, 05:29 PM (EST)
Click to EMail jmolokach Click to send private message to jmolokach Click to view user profileClick to add this user to your buddy list  
2. "RE: A complex number approach..."
In response to message #1
 
Here's what I had in mind.... see the following link:

https://docs.google.com/leaf?id=0BygZeXnaKTslMTg2MGYwNTMtYzBiMC00NmNjLTk4MTctNzA0MWI5YzlhNDQw&hl=en&authkey=CLGBlZQH

(The link in the jpeg is not active but it's the one I stated above.)

molokach


  Alert | IP Printer-friendly page | Reply | Reply With Quote | Top
alexb
Charter Member
2758 posts
Sep-19-10, 05:33 PM (EST)
Click to EMail alexb Click to send private message to alexb Click to view user profileClick to add this user to your buddy list  
3. "RE: A complex number approach..."
In response to message #2
 
   I understood the first time.

You take a complex number x and its conjugate x*, multiply the two and get |x|². What it has to do with a proof of the Pythagorean theorem? Whatever the answer to this may be, note that to define |x| one needs the Pythagorean theorem.


  Alert | IP Printer-friendly page | Reply | Reply With Quote | Top
jmolokach
Member since Jan-11-11
Sep-23-10, 02:02 PM (EST)
Click to EMail jmolokach Click to send private message to jmolokach Click to view user profileClick to add this user to your buddy list  
4. "RE: A complex number approach..."
In response to message #3
 
Well in using the last diagram I seem to have 2 similar right triangles. If we call the original blue length c, then c/a = (a^2 + b^2)/c which implies a^2 + b^2 = c^2 / a

???

Why is there an extra a? Either the complex numbers makes this problem messed up or my triangles aren't similar or I am just not seeing the error. I am siding with the latter...

Can anyone explain this?

molokach


  Alert | IP Printer-friendly page | Reply | Reply With Quote | Top
jmolokach
Member since Jan-11-11
Sep-23-10, 02:39 PM (EST)
Click to EMail jmolokach Click to send private message to jmolokach Click to view user profileClick to add this user to your buddy list  
5. "RE: A complex number approach..."
In response to message #4
 
Never mind. I see the error. The larger triangle is not necessarily a right triangle, much less similar to the original one I drew for a + bi.

Sorry about that... Still I wonder if there is a way to prove that the length along the real axis is the square of the blue length by use of Heron's formula or something...

molokach


  Alert | IP Printer-friendly page | Reply | Reply With Quote | Top
alexb
Charter Member
2758 posts
Sep-23-10, 02:43 PM (EST)
Click to EMail alexb Click to send private message to alexb Click to view user profileClick to add this user to your buddy list  
6. "RE: A complex number approach..."
In response to message #5
 
   I have recently learned that it is quite simple to prove that the 3-4-5 triangle is right without recourse to the Pythagorean theorem or its converse. The approach works for all Pythagorean (integer valued side lengths) triangles.


  Alert | IP Printer-friendly page | Reply | Reply With Quote | Top
jmolokach
Member since Jan-11-11
Sep-23-10, 04:24 PM (EST)
Click to EMail jmolokach Click to send private message to jmolokach Click to view user profileClick to add this user to your buddy list  
7. "RE: A complex number approach..."
In response to message #6
 
Gee I would like to see that. Does it have anything to do with my complex conjugates diagram?

By the way I tried to use Heron to find the length of the green length in the picture. It is rather nasty...

I think the fact that that large triangle is not a right triangle makes the proof dead-in-the-water...

molokach


  Alert | IP Printer-friendly page | Reply | Reply With Quote | Top
alexb
Charter Member
2758 posts
Sep-23-10, 04:30 PM (EST)
Click to EMail alexb Click to send private message to alexb Click to view user profileClick to add this user to your buddy list  
9. "RE: A complex number approach..."
In response to message #7
 
   Here it is, with credits to Douglas Rogers.



  Alert | IP Printer-friendly page | Reply | Reply With Quote | Top
jmolokach
Member since Jan-11-11
Sep-23-10, 10:57 PM (EST)
Click to EMail jmolokach Click to send private message to jmolokach Click to view user profileClick to add this user to your buddy list  
10. "RE: A complex number approach..."
In response to message #9
 
This is really cool. I think I will make a poster out of this and put on my classroom wall... I wonder if this might go under the PWW category? It'seems like one could do this for rational numbers as well as integers... just make the distance between the dots 1/n, where n is a common denominator....

I suppose the irrational side lengths would be done another way, but I really like this... great stuff!

I am still stuck on why that triangle in the complex numbers diagram has no geometric relationship to the right triangle... It seems like there should be a way to show that the length along the real axis is the square of the length of the vector from a + bi without using the PT. Have you heard of anyone doing this? I cannot find it on google anywhere...

molokach


  Alert | IP Printer-friendly page | Reply | Reply With Quote | Top
alexb
Charter Member
2758 posts
Sep-23-10, 10:59 PM (EST)
Click to EMail alexb Click to send private message to alexb Click to view user profileClick to add this user to your buddy list  
11. "RE: A complex number approach..."
In response to message #10
 
   >This is really cool. I think I will make a poster out of
>this and put on my classroom wall... I wonder if this might
>go under the PWW category?

Of course.

>It'seems like one could do this
>for rational numbers as well as integers... just make the
>distance between the dots 1/n, where n is a common
>denominator....

Yes. Rational and integers are all the same around here.

>I am still stuck on why that triangle in the complex numbers
>diagram has no geometric relationship to the right
>triangle... It seems like there should be a way to show
>that the length along the real axis is the square of the
>length of the vector from a + bi without using the PT. Have
>you heard of anyone doing this?

No, I haven't.


  Alert | IP Printer-friendly page | Reply | Reply With Quote | Top
jmolokach
Member since Jan-11-11
Sep-24-10, 06:17 AM (EST)
Click to EMail jmolokach Click to send private message to jmolokach Click to view user profileClick to add this user to your buddy list  
12. "RE: A complex number approach..."
In response to message #11
 
In reply to "No I haven't" :

apparently someone has already tried this? The argument may be too weak for a real proof since polar coordinate conversions come directly from the PT.

https://whyslopes.com/etc/ComplexNumbers/ch09A.html

molokach


  Alert | IP Printer-friendly page | Reply | Reply With Quote | Top
alexb
Charter Member
2758 posts
Sep-24-10, 06:19 AM (EST)
Click to EMail alexb Click to send private message to alexb Click to view user profileClick to add this user to your buddy list  
14. "RE: A complex number approach..."
In response to message #12
 
   It's not a question of weakness but that of a vicious circle. Length in R² is defined by means of PT.


  Alert | IP Printer-friendly page | Reply | Reply With Quote | Top
jmolokach
Member since Jan-11-11
Sep-24-10, 08:16 AM (EST)
Click to EMail jmolokach Click to send private message to jmolokach Click to view user profileClick to add this user to your buddy list  
17. "RE: A complex number approach..."
In response to message #14
 
>It's not a question of weakness but that of a vicious
>circle. Length in R² is defined by means of PT.

even if the length is one-dimensional...say using the ruler postulate? Or do you consider the ruler postulate as a special case of the PT?

molokach


  Alert | IP Printer-friendly page | Reply | Reply With Quote | Top
alexb
Charter Member
2758 posts
Sep-24-10, 08:19 AM (EST)
Click to EMail alexb Click to send private message to alexb Click to view user profileClick to add this user to your buddy list  
18. "RE: A complex number approach..."
In response to message #17
 
   >>It's not a question of weakness but that of a vicious
>>circle. Length in R² is defined by means of PT.
>
>even if the length is one-dimensional...say using the ruler
>postulate? Or do you consider the ruler postulate as a
>special case of the PT?

No. And I do not know if anyone does. But why do you ask? Isn't that clear that lenth is defined differently in R and in R²?


  Alert | IP Printer-friendly page | Reply | Reply With Quote | Top
jmolokach
Member since Jan-11-11
Sep-24-10, 12:02 PM (EST)
Click to EMail jmolokach Click to send private message to jmolokach Click to view user profileClick to add this user to your buddy list  
19. "RE: A complex number approach..."
In response to message #18
 
>>>It's not a question of weakness but that of a vicious
>>>circle. Length in R² is defined by means of PT.
>>
>>even if the length is one-dimensional...say using the ruler
>>postulate? Or do you consider the ruler postulate as a
>>special case of the PT?
>
>No. And I do not know if anyone does. But why do you ask?
>Isn't that clear that lenth is defined differently in R and
>in R²?

yes I suppose it is, but it isn't clear in the context of the calculus proof why I can't use the ruler postulate as evidence that as evidence that c is the radius of the circle in the calculus proof, so I suppose I'll continue this discussion over there...

molokach


  Alert | IP Printer-friendly page | Reply | Reply With Quote | Top
jmolokach
Member since Jan-11-11
Sep-24-10, 06:17 AM (EST)
Click to EMail jmolokach Click to send private message to jmolokach Click to view user profileClick to add this user to your buddy list  
13. "RE: A complex number approach..."
In response to message #11
 
the aforementioned site mentons as part of the proof:

"Multiplying a vector a + i b with angle A and length r by its complex conjugate   a - ib gives a complex number with angle 0 = A + (-A) and length r2 units according to the add the angles, multiply the lengths polar coordinate, multiplication rule..."

is this "rule" based in the PT?

if not then this has major ramifications...even to the point of validating my calculus proof... This is why I mentioned originally "Is this another 'How do you know that c is the radius of your circle?'" It's just that this time it's r instead of c...

molokach


  Alert | IP Printer-friendly page | Reply | Reply With Quote | Top
alexb
Charter Member
2758 posts
Sep-24-10, 06:23 AM (EST)
Click to EMail alexb Click to send private message to alexb Click to view user profileClick to add this user to your buddy list  
15. "RE: A complex number approach..."
In response to message #13
 
   >is this "rule" based in the PT?

This rule is based on the definition of multiplication of complex numbers.

>if not then this has major ramifications...even to the point
>of validating my calculus proof...

You may try discussing this on a larger forum, say, at one of mathforum.org groups. In my view, as long as the definition of length requires PT, whatever comes afterwards can't be used to prove the theorem.


  Alert | IP Printer-friendly page | Reply | Reply With Quote | Top
jmolokach
Member since Jan-11-11
Sep-24-10, 06:36 AM (EST)
Click to EMail jmolokach Click to send private message to jmolokach Click to view user profileClick to add this user to your buddy list  
16. "RE: A complex number approach..."
In response to message #11
 
here is a related site which lists that rule as an 'axiom' for complex numbers...

https://whyslopes.com/etc/ComplexNumbers/apCmplx30.html

molokach


  Alert | IP Printer-friendly page | Reply | Reply With Quote | Top
gaespes
Member since Feb-1-11
Feb-01-11, 04:06 PM (EST)
Click to EMail gaespes Click to send private message to gaespes Click to view user profileClick to add this user to your buddy list  
20. "RE: A complex number approach..."
In response to message #0
 
   >In a recent discussion during a high school math class, my
>students learned that a + bi multiplied by it's conjugate
>gives the real number a^2 + b^2. After reading
>https://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/
>(which I found interesting) I thought how the number a + bi
>being rotated by a - bi winds up being a number on the
>positive real axis, which has the same length as either
>conjugate.
>
>I think this is really cool and would make for a great
>animation. Not sure if this might qualify as a proof. I
>have a feeling this might turn into another "How do you know
>that c is the radius of your circle?"
>
>Any thoughts??

***************************************************
https://w3.romascuola.net/gspes/c/lo/5d.html
***************************************************

gaespes


  Alert | IP Printer-friendly page | Reply | Reply With Quote | Top
gaespes
Member since Feb-1-11
Feb-09-11, 07:01 PM (EST)
Click to EMail gaespes Click to send private message to gaespes Click to view user profileClick to add this user to your buddy list  
21. "RE: A complex number approach..."
In response to message #0
 
   I hope this can be a useful summary of my thought about the question:
https://mathematedita.splinder.com/post/24011052

gaespes


  Alert | IP Printer-friendly page | Reply | Reply With Quote | Top

Conferences | Forums | Topics | Previous Topic | Next Topic

You may be curious to have a look at the old CTK Exchange archive.
Please do not post there.

Copyright © 1996-2018 Alexander Bogomolny

Search:
Keywords:

Google
Web CTK