Complex numbers

[wandelaar]

7 hours ago, OldDog said:

Does this mean that i with a coefficient of 0 is a non-zero value thus requiring the term to re retained.

 

No! The expressions of the form "a+bi" are to be considered as 'unbreakable wholes' . Not only the "i" but also the "+" in this expression is only a sign or component of the whole notation. The actual complex numbers are the arrows designated by the expressions. Again the "i" has no value, it's just a sign. Our theory of the complex numbers as arrows would have been exactly the same when we would have used another notation without any "i" or "+" like this silly one:

 

 

In case of the above notation we would state that the first real number a in the left red box is the x-coordinate of the endpoint of our arrow in the Cartesian coordinate system and second real number b in the right green box is the y-coordinate of the endpoint of our arrow in the Cartesian coordinate system. And in this way we would have designated our complex numbers as arrows from (0,0) to (a,b) with no "i" in sight! 

 

But what you will see in practice is that "i" and "bi" are written as shorthand for 0+1i and 0+bi. With practical calculations it very quickly becomes a nuisance to use the formally correct expressions like 0+1i and 0+bi, and that is the reason why you will often see the shorter "i" and "bi". But to avoid confusion you can always change the "i" and "bi" back to the correct formal expressions 0+1i and 0+bi and do the calculation in the formally correct way.

Edited September 3, 2018 by wandelaar

[wandelaar]

6 hours ago, Lost in Translation said:

I "cheated" and looked up some videos on YouTube to help me understand what's going on here.

 

The important thing is that you now understand the basics of the complex numbers. We are not in school here.

 

6 hours ago, Lost in Translation said:

Video showing how to add and subtract complex numbers. I found this video easy to understand.

 

  Reveal hidden contents

 

 

A good practical explanation using "j" instead of "i" as is usual in electrical engineering. No foundations or definitions are given, but we already did that here.

 

6 hours ago, Lost in Translation said:

Here's a video showing the multiplication and division of complex numbers using the FOIL method. I also found this easy to understand.

 

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I am not happy with this one. No understanding is involved, it's just "monkey see, monkey do". And i is defined as √(-1) , which is just nonsense. The complex numbers don't just magically spring into existence by declaring i to be √(-1) . One has to show that it is logically possible for mathematical objects with the wished for properties of the complex numbers to exist, and showing that is what I have been doing here by introducing the complex numbers as arrows.

 

6 hours ago, Lost in Translation said:

Lastly here is a video showing how to multiply complex numbers in polar form. This video made no sense to me. The only reason I was able to follow it at all was due to what we had already discussed in this thread.

 

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The cis-notation is new to me. It is explained here: https://en.wikipedia.org/wiki/Cis_(mathematics)

 

This is the kind of stuff you will have to learn when you want to understand the more advanced applications of the complex numbers. 

[wandelaar]

Any questions left?

[OldDog]

2 hours ago, wandelaar said:

The expressions of the form "a+bi" are to be considered as 'unbreakable wholes' .

 

The red-a,green-b analogy is a good one for helping dispel attachment to i.

 

Still, attachment is strong. As a young person stuggling with concepts in mathematics, algebraic rules were hammered in at the expense of geometric concepts. Looking back on the education experience, there seems relatively little that connects geometry with algebra. Indeed, it was not until a class in Analytic Geometry that the two began to come together.

 

So, back to the quote above. It would seem then that statements of the form a+bi ... or perhaps just the bi part ... cannot be manipulated using the rules of algebra. I say that as sort of a hyperbole. This discussion seems to make clear that we should not be thinking of i as anything other than a form of notation to indicate the presence of a component that cannot be resolved. As such, it cannot be manipulated using algebraic rules.

[wandelaar]

In mathematics we have the unhappy phenomenon called "abuse of notation". Human beings normally don't have the patience to consistently write everything out in a ruthlessly rigorous and formal way. As soon as we understand that the complex number "a + 0i" behaves exactly as the real number "a" we lose the motivation to view the complex number "a+ 0i" and the real number "a" as conceptually different mathematical objects. And after that the will power to keep writing the formally correct expression "a + 0i" in stead of the shorthand "a" will quickly go down the drain. Something similar goes for the shorthand's "bi" and "i" that would have to be written as "0 + bi" and "0 + 1i" to be formally correct. Calculating with "i" as if it were a variable will often give the correct result, and as the complex numbers are often introduced in a purely practical way in the school environment there will be nothing to correct the wrong impression that "i" stands for a variable, albeit a mysterious one. 

 

Edited September 3, 2018 by wandelaar

[OldDog]

17 minutes ago, wandelaar said:

Human beings normally don't have the patience to consistently write everything out in a ruthlessly rigorous and formal way.

 

AMEN!

 

18 minutes ago, wandelaar said:

albeit a mysterious one

 

Sooo ... the i that can be told of is not the real i.  

 

... just needed a way to bring this discussion back around to the general topic of the forum. 

[wandelaar]

We need Euler's formula for the more interesting applications of the complex numbers in the context of Taoism. This will go beyond my originally planned program that is finished by now. But I can search for appropriate follow up video's if you wish.....

[Lost in Translation]

5 minutes ago, wandelaar said:

We need Euler's formula for the more interesting applications of the complex numbers in the context of Taoism. This will go beyond my originally planned program that is finished by now. But I can search for appropriate follow up video's if you wish.....

 

I don't think you'll find much support if this thread becomes a full on course in mathematics. Perhaps you can explain in layman's terms how all of this relates back to Taoism?

[Lost in Translation]

n Imaginary Number, when squared, gives a negative result.

Try

Let's try squaring some numbers to see if we can get a negative result:

• 2 × 2 = 4
• (−2) × (−2) = 4 (because a negative times a negative gives a positive)
• 0 × 0 = 0
• 0.1 × 0.1 = 0.01

No luck! Always positive, or zero.

It seems like we cannot multiply a number by itself to get a negative answer ...

... but imagine that there is such a number (call it i for imaginary) that could do this: i × i = −1  Would it be useful, and what could we do with it?

Well, by taking the square root of both sides we get this:

Which means that i is the answer to the square root of −1.

Which is actually very useful because ...

... by simply accepting that i exists we can solve things 
that need the square root of a negative number.

Let us have a go:

Example: What is the square root of −9 ?

√(−9)= √(9 × −1)

 = √(9) × √(−1)

 = 3 × √(−1)

 = 3i

(see how to simplify square roots)

Hey! that was interesting! The square root of −9 is simply the square root of +9, times i.

In general:

√(−x) = i√x

So long as we keep that little "i" there to remind us that we still 
need to multiply by √−1 we are safe to continue with our solution!

Using i we can also come up with new solutions:

Example: Solve x² + 1 = 0

Using Real Numbers there is no solution, but now we can solve it!

Subtract 1 from both sides:

x² = −1

Take the square root of both sides:

x = ± √(−1)

x = ± i

Answer: x = −i or +i

Check:

• (−i)² + 1 = (−i)(−i) + 1 = +i² + 1 = −1 + 1 = 0
• (+i)² +1 = (+i)(+i) +1 = +i² +1 = −1 + 1 = 0

Unit Imaginary Number

The "unit" Imaginary Number (the equivalent of 1 for Real Numbers) is √(−1) (the square root of minus one).

In mathematics we use i (for imaginary) but in electronics they use j (because "i" already means current, and the next letter after i is j).

 

Examples of Imaginary Numbers

i

12.38i

−i

3i/4

0.01i

−i/2

Imaginary Numbers are not "Imaginary"

Imaginary Numbers were once thought to be impossible, and so they were called "Imaginary" (to make fun of them).

But then people researched them more and discovered they were actually useful and important because they filled a gap in mathematics ... but the "imaginary" name has stuck.

And that is also how the name "Real Numbers" came about (real is not imaginary).

Imaginary Numbers are Useful

 

Complex Numbers

Imaginary numbers become most useful when combined with real numbers to make complex numbers like 3+5i or 6−4i

 

Spectrum Analyzer

Those cool displays you see when music is playing? Yep, Complex Numbers are used to calculate them! Using something called "Fourier Transforms".

In fact many clever things can be done with sound using Complex Numbers, like filtering out sounds, hearing whispers in a crowd and so on.

It is part of a subject called "Signal Processing".

 

Electricity

 

AC (Alternating Current) Electricity changes between positive and negative in a sine wave.

When we combine two AC currents they may not match properly, and it can be very hard to figure out the new current.

But using complex numbers makes it a lot easier to do the calculations.

And the result may have "Imaginary" current, but it can still hurt you!

 

Mandelbrot Set

The beautiful Mandelbrot Set (part of it is pictured here) is based on Complex Numbers.

 

Quadratic Equation

The Quadratic Equation, which has many uses, 
can give results that include imaginary numbers

Also Science, Quantum mechanics and Relativity use complex numbers.

Interesting Property

The Unit Imaginary Number, i, has an interesting property. It "cycles" through 4 different values each time we multiply:

1 × i   = i i × i   = −1 −1 × i   = −i −i × i   = 1 Back to 1 again!

 

So we have this:

i = √−1

i2 = −1

i3 = −√−1

i4 = +1

i5 = √−1

...etc

 

Example What is i⁶ ?

i⁶= i⁴ × i²

 = 1 × −1

 = −1

And that leads us into another topic, the complex plane:

Conclusion

The unit imaginary number, i, equals the square root of minus 1

Imaginary Numbers are not "imaginary", they really exist and have many uses.

 

https://www.mathsisfun.com/numbers/imaginary-numbers.html

 

 

[Lost in Translation]

On 8/30/2018 at 12:24 PM, wandelaar said:

Here is the road map:

 

1. Complex numbers as arrows

2. The real and imaginary part of a complex number

3. The modulus and argument of a complex number

4. The addition of complex numbers

5. The multiplication of complex numbers

6. Examples of complex solutions to quadratic equations

 

Beyond 6. is advanced stuff. We are now on the brink of 3.

 

We've covered 1-5. Do you still plan to cover 6 or shall we leave that as self study?

[wandelaar]

@ LiT

 

Thank you. I don't like the hand-waving manner in which MATH is FUN introduces the complex numbers, but they do a good job in illustrating some of their applications in layman's terms.

 

Important in the context of Taoism is the fact that complex numbers greatly simplify the analysis of periodic, cyclical and dynamic phenomena and that complex numbers are essential to fractal geometry which in many ways corresponds to the geometry of natural objects.

 

Edited September 3, 2018 by wandelaar

[OldDog]

Ugh .... think I'll pass. Topic is getting a bit weary since I am having a hard time relating it to anything practical.

[wandelaar]

3 minutes ago, OldDog said:

Ugh .... think I'll pass. Topic is getting a bit weary since I am having a hard time relating it to anything practical.

 

LiT just now posted an article mentioning some of the applications in layman's terms.

 

And I mentioned some more in non-layman's terms.

 

What more can we do?

[Lost in Translation]

I think it's time to put this topic to bed.  I've spent more time working with imaginary numbers in the last week than I have since I was 17 in high school...

[wandelaar]

OK - we will leave it at this.

 

[wandelaar]

Life without complex numbers. Ooooh! Already feel it coming:

 

 

[voidisyinyang]

On 9/3/2018 at 10:57 AM, wandelaar said:

Important in the context of Taoism is the fact that complex numbers greatly simplify the analysis of periodic, cyclical and dynamic phenomena and that complex numbers are essential to fractal geometry which in many ways corresponds to the geometry of natural objects.

 

OOPS! You oops here. I pointed out earlier in the thread that complex numbers are based on noncommutative phase and that is the connection to Daoism. Physicist Charles B. Madden contacted me about my master's thesis, wanting to publish it, but he could not understand my physics claim. So I read his book "Fractals and Music" and I discovered my error. He points out that the yin-yang symbol is NOT a fractal because the yin-yang symbol is not symmetric math logic. https://books.google.com/books?id=88TMcxkJ810C&printsec=frontcover#v=onepage&q&f=false

It's in the google preview of his book - chapter 2.

Figure 2.1

The  Yin-Yang Symbol is not a fractal.

 

Quote

 

Fractals in Music: Introductory Mathematics for Musical Analysis

https://books.google.com/books?isbn=0967172756

Charles Madden, ‎Charles B. Madden - 1999 - ‎Music

In contrast, many things that seem identical in all directions are not fractal. ... Although the yin-yang symbol shown in figure 2.1 is a reflected and inversely colored it too is not a fractal because there is no scaling... A photographic image makes clear that viewing the array at an angle ...

 

 

So fractals use logistic symmetric math while the yin-yang Taiji is actually noncommutative phase which is the secret of complex numbers, as math professor Louis Kauffman points out in his most recent academic research article that I linked in the thread.

As math professor Steve Strogatz points out - fractals are a "Platonic ideal" that does not exist in Nature.

So I know it's a New Age trope to promote fractals as being spiritual but sorry - it's not true.

 

[OldDog]

Complex numbers ... imaginary numbers ... noncommutative phases ... the mind boggles! Although it is beyond my ability to deal with such things, I am fascinated by those who can.  

 

Is it not amazing that there are those who have so trained their minds ... in concepts that so defy practical representation ... that they are able to freely and seemingly consistently be able to discuss them? How much like a sage are such people?

 

Not trying to be facetious here. I find it interesting that there are apparently so many people on this site ... seemingly disproportional to the numbers found in the ordinary world ... that have some level of capability in such discussions. 

 

Interesting group of people here.

[wandelaar]

@ OldDog

 

I wouldn't take the posts of ViYY too serious.  The complex numbers as arrows with the addition and multiplication as defined in this topic form a commutative system. I can prove that if you wish to see it. There is no need whatever to use non-commutative mathematics to understand the complex numbers.

 

Further the Yin-Yang symbol isn't a fractal and I didn't claim it was. But there are lots of fractals that are similar to natural forms. See:

https://www.mnn.com/earth-matters/wilderness-resources/blogs/14-amazing-fractals-found-in-nature

[OldDog]

@voidisyinyang

 

btw ...

 

Speaking of books on strange applications of mathematics ... I am reminded of the book Goedel, Escher and Bach: An Eternal Golden Braid, by Douglas Hofstadter. Have you (or anyone else for that matter) read it? It is an interesting exploration of mathematical principles found in works of art. Kind of a fun read.

[OldDog]

9 minutes ago, wandelaar said:

I can prove that if you wish to see it.

 

No, please, no proof necessary. I am still reeling from our exploration of complex/imaginar numbers! All matters of correctness or non-correctness aside ... I do find these kind of discussions interesting. It's just that my attention span threshold is much lower than it used to be.

[wandelaar]

Yes, I read Gödel, Escher, Bach many years ago. It's mainly about (symbolic) logic, levels of interpretation, paradoxes, artificial intelligence, consciousness, etc. There is some talk about Bach and Escher, but not much about art in general. It's very abstract stuff, much more so than this topic. 

Edited September 7, 2018 by wandelaar

[Lost in Translation]

22 minutes ago, OldDog said:

I am reminded of the book Goedel, Escher and Bach: An Eternal Golden Braid, by Douglas Hofstadter. Have you (or anyone else for that matter) read it?

 

I read that book back in the mid nineties. It was interesting but didn't make a lot of sense to me. My main takeaway from that work was that there is a little cartoon version of Achilles from the Illiad who likes to ask questions. Oh, and there was a cartoon turtle, too.

 

[voidisyinyang]

2 hours ago, OldDog said:

@voidisyinyang

 

btw ...

 

Speaking of books on strange applications of mathematics ... I am reminded of the book Goedel, Escher and Bach: An Eternal Golden Braid, by Douglas Hofstadter. Have you (or anyone else for that matter) read it? It is an interesting exploration of mathematical principles found in works of art. Kind of a fun read.

yes read that book when I took quantum mechanics at Hampshire College in 1990.

[voidisyinyang]

3 hours ago, OldDog said:

Complex numbers ... imaginary numbers ... noncommutative phases ... the mind boggles! Although it is beyond my ability to deal with such things, I am fascinated by those who can.  

 

Is it not amazing that there are those who have so trained their minds ... in concepts that so defy practical representation ... that they are able to freely and seemingly consistently be able to discuss them? How much like a sage are such people?

 

Not trying to be facetious here. I find it interesting that there are apparently so many people on this site ... seemingly disproportional to the numbers found in the ordinary world ... that have some level of capability in such discussions. 

 

Interesting group of people here.

 

The OP continually refers to me in the third person despite my direct engagement with his "complex numbers" attempt to connect to Daoism. He says he can "prove" that complex numbers are commutative and not non-commutative. This is very funny because in the post I made in this thread - the first post - this is now my third post - the first post I made I stated how I had corresponded with math professor Louis Kauffman about how a couple Chinese electrical engineers were claiming in a paper that complex numbers are not based on non-commutative logic. Professor Kauffman has made a career out of emphasizing that noncommutative logic source of complex numbers - so their paper was directed at his claim. He responded to me that he had not seen that paper and so he would respond. Then I posted his latest research paper which is his response - in that paper he demonstrates that complex numbers are derived from noncommutative logic.

 

Now the connection to Daoism is that I had corresponded with math professor Kauffman before about his collaboration with Eddie Oshins, the founder of quantum psychology at Stanford Linear Accelerator Center. Both Kauffman and Oshins worked at SLAC together. Oshins also taught Wing Chun martial arts and he realized that the secret of Daoist neigong was the same noncommutative logic of quantum physics based on complex numbers.

 

So you can study Eddie Oshins articles for more details. For example Oshins also worked with Karl Pribram on Pribram's "holographic mind" model of neuroscience. But Pribram could not understand the noncommutative logic of quantum physics. So Pribram continues to make the mistake of promoting a symmetric math logic based on Fourier analysis. Oshins pointed out Pribram's error but instead Pribram just had his ideas promoted in the New Age scene - especially Michael Talbot's book The Holographic Universe. So qigong master Chunyi Lin said he read that book, "The Holographic Universe" and says yes it is an accurate portrayal of what reality is for qigong masters.

 

So then the other scientist featured in that book the Holographic Universe is David Bohm. I have also corresponded with quantum physics professor Basil J. Hiley who collaborated with David Bohm. Bohm had expanded on de Broglie's first critique of relativity, in the foundation of quantum physics. Schroedinger's wave equation is based on de Broglie but Schroedinger took out the relativity foundation and so the noncommutative logic was lost. Instead it became the "measurement problem" as Schroedinger's Cat. But the new "weak measurements" analysis of the double slit experiment has proven once again that the noncommutative logic is the foundation of reality, and so corroborates Daoism as well.

 

Eddie Oshins was pissed that the woo-woo New Age science scene promoted the wrong logic for the Holographic Model as quantum consciousness and he died early - but his legacy lives on.  Here is math professor Louis Kauffman newly demonstrated a math move of complex numbers as noncommutative logic - based on his work with Eddie Oshins.

 

Quote

The "Quaternion Handshake" illustrates the fundamental orientation-entanglement relation that interlocks the structure of the quaternions with the geometry and topology of an object connected to a background in three dimensional space. In this case the objects are human hands, the background is the body and the connection is the arm that links hand to body. ...

Starring:

Martial Arts By: Louis Kauffman
John Hart
Eddie Oshins

Starring:

Martial Arts By: Louis Kauffman
John Hart
Eddie Oshins

 

https://www.evl.uic.edu/hypercomplex/movies/handshake.mpg

 

http://www.quantumpsychology.com/

 

So if you study SLAC - these are real quantum physicists who acknowledge that quantum physics enables precognition and other paranormal abilities as well. They are high level scientists and yet the "general science" promoters as "skeptics" know nothing about noncommutative logic. It is quite funny.

 

I had to really dig to discover Eddie Oshins! I figured - I just read books on my own, despite my master's degree and so SOMEONE must have discovered this same secret besides me. Sure enough! Thank you Eddie Oshins.

 

 

Here is Oshins demonstrated the Complex Number noncommutative logic hand shake with another scientist at SLAC.

 

http://ecoechoinvasives.blogspot.com/2017/05/the-physics-of-tao-eddie-oshins-cracked.html

 

So I blogged on this - 1 year and 4 months ago.

 

Quote

 

 Soon after creating the binary number system (used in modern day computers), Leibniz found out about this ancient symbolism and attributed the origination of the binary numbers to the Chinese. In addition to their usage as oracles or as representations of patterns of Nature, these symbols were precursors to a school of health systems, mind exercises and martial arts known collectively as neigong/noi kung (the "inner" or "internal" school of "shaddow boxing"). The most well known of these esoteric skills are taijiquan/t'ai chi chuan ("grand pinnacle" boxing), hsing-i chuan ("form of mind/will/intent" boxing), and baquazhang/pa kua chang ("8 trigrams palm" [boxing]).

In this talk, Eddie will give a short history of the above concepts and, in light of some work he has been developing in his Quantum Psychology Project®, he will propose a new reinterpretation of these symbols. He will demonstrate mathematical aspects, such as the consequent "orientation-entanglement relation" and the Kauffman-Oshins "quanternionic arm." Eddie will use these concepts to illustrate his notion of "self-referential motion," and relate such understanding to both gongfu (kung-fu) and psychology.

 

So it is originally called the Dirac Dance. I learned this in my first year of college in my quantum mechanics class taught by Professor Herbert J. Bernstein. He is using noncommutative logic for NASA to test out his quantum teleportation signal system with satellites.