wandelaar

Anyone taking the trouble to actually examine the construction of the complex numbers as arrows as it was done in this topic will see that no advanced stuff as presented in the article of Louis Kauffman is needed. That doesn't mean that the complex numbers cannot be introduced in other ways than I did, in fact there are lots of other ways. And I said so earlier. The interesting approach of Kauffman is just one more. Here I took the most simple approach to the complex numbers I could think of, and that's why we got as far as we did in this topic. To take the approach of Kauffman I would first have had to study it myself as I am not familiar with it, but that approach would doubtlessly have been incomprehensible to those not being mathematicians or physicists themselves. One can see that for oneself by taking a look at the article linked by ViYY. So it would have served no useful purpose to use Kauffman's approach here. I have ViYY on my ignore list, and each time I take a look to see whether his posts have improved in the meantime it's clear that they haven't. But I don't have the time and energy to correct all the rubbish ViYY is writing on this forum, so I can only hope that the other Bums are able to recognise it for what it is.

Don't think 1,000,000 + 0i would be a problem?

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.

@ 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

@ dawei Don't see how that applies...

I have just read the text of Henrik Klindt-Jensen again, and I think it is very good, particularly as concerns the comparison of Taoism and Greek philosophy. I wonder whether Henrik Klindt-Jensen has also written a book about this?

More about this here: http://blog.sina.com.cn/s/blog_68dd00750100na80.html

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

OK - we will leave it at this.

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?

@ 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.

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.....

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.

Any questions left?

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

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.

You probably mean without an imaginary term? That can be done. But in that case we will write the result in the form a + 0i . And that is still a complex number. Those complex numbers z wherefore Im(z) = 0 will behave exactly as the corresponding real numbers under addition and multiplication. And that is the reason why the complex number system can be considered as an extension of the real number system. It's essential to the complex numbers that there is a product z*z wherefore Im(z*z) = 0 , because otherwise there would be no solution to the complex equation z*z + (1+0i) = 0 + 0i . The above two points actually form the end point of my explanation. Here we have only reviewed the very basics of the complex numbers. I propose that we go on with this topic till we are able to add and multiply any two complex numbers, and till we understand why the complex numbers can be considered as an extension of the real numbers that gives a complex solution to the equation z*z + (1+0i) = 0 + 0i . After that the theory will become too advanced. But there are great videos on YouTube for further study, and I will be happy to answer any questions you may have as a result of watching those. (This also hopefully answers the question of LiT about what it all means.)

Yes. Can you first show how you calculate the values of the modulus and argument of the product? As soon as you know the modulus and argument of a complex number z you can use trigonometry to express it in the form z = a + bi .

The sum and product of two complex numbers are themselves complex numbers (that is: arrows), not angles. Just as the sum and product of two real numbers are always real numbers, we also want that the sum and product of two complex numbers are always themselves complex numbers. That gives us a "closed" number system. Than we can again add or multiply the sum or product of two complex numbers to another complex number, etc. All intermediary results will always be complex numbers, and consequently the (nested) sums and products will always be defined. No special cases need to be considered.

The angles are relevant for the product.

I don't see how that relates to the geometrical construction of u + v following the definition of the sum.

There are lots of stories in the Chuang tzu that are equally unreal but nevertheless give one food for thought.

@ OldDog The only application that is possible with the very elementary theory of complex numbers that we have covered until now is a demonstration that there is a complex number z such that z*z + (1+0i) = 0+0i . Try: z = 0 + 1i . The real number version of this equation would be: z2 + 1 = 0, and within the real numbers this equation has no solution. So the complex numbers solve a problem that the real numbers are incapable of. This has many repercussions for the theory of algebraic equations. I cannot speed up this exposition any further because each step has to be fully understood before the next step can be taken. As Euclid said to the ruler Ptolemy I Soter when he asked Euclid if there was a shorter road to learning geometry than through Euclid's Elements: There is no royal road to geometry. Now we don't have to struggle through Euclid's Elements but one cannot reasonably expect to see any applications of the complex numbers before the sum and product of the complex numbers are fully understood. Now I have given the definitions for the sum and product. So if you like go ahead and verify that: (0+1i)*(0+1i) + (1+0i) = 0+0i .

Lets call them u and v. Now what is u + v ?

Uh? Could you explain what you are doing?