wandelaar

As to a possible connection between the complex numbers and Taoism I can mention the application of the complex numbers in the analysis and study of periodic (and other dynamic) phenomena.

@ Marblehead The complex expression a + bi has only two places where numbers may be inserted, and that is on the places of a and b. The brackets in 2 + (-3i) give the false impression that -3i is itself a number. But as yet -3i has no meaning, because only complete expressions of the form a + bi have a meaning as the arrow pointing from (0,0) to (a,b). But we will give a meaning to expressions such as -3i as we go on.

Yes. When you define a thing to be such and so, then it is such and so by definition. Otherwise mathematics would become impossible. No - the expression a + bi is chosen because the complex numbers will be seen later on (after we have given definitions for the sum and product of two complex numbers) to behave like we would expect for numbers consisting of the sum of a real and an imaginary part. The expression a + bi is suggestive of this. No - it isn't. Yes! The angle between the positive x-axis and the arrow will play an important role in the eventual theory of the complex numbers. Yes - the term complex is chosen to refer to the real and imaginary components of a complex number. Taking the point (0,0) as the starting point of the arrow is by far the easiest thing to do. Why make it more complicated than it already is?

Formally correct would be 2 + -3i or 2 + (-3)i. But in actual practice one uses 2 - 3i. The last expression can be made mathematically legitimate by considering the as jet undefined expression a - bi as a shorthand for a + (-b)i.

Well - that's almost correct but not quite. The correct answer is: 1 + 2i . Only then is it in the form a + bi. We have to be rigorously correct now, for otherwise the problems will reappear later on. So another test to get it right: How do you write the blue arrow in the form a + bi ?

Can you now represent the red arrow (= complex number) by means of an expression of the form a + bi ?

The definitions of the sum and product will involve geometrical constructions with the arrows. But that's no problem for now. First I want to make sure that you now know what a complex number considered as an arrow is and how it can be designated by an expression of the form a + bi. When that is done, then the rest will cause no further problems.

@ Marblehead Yes!!! It's party time. I think you now got it. But first I will post some more tests to see whether you can now handle and recognize all complex numbers as arrows.

Black Sabbath is one of my favorites! Particularly: The Wizard Black Sabbath War Pigs Hand of Doom Wheels of Confusion Changes Snowblind Children of the Sea Lonely Is the Word Falling Off The Edge Of The World The Sign of the Southern Cross

Now what are a and b in the complex number representation a + bi for the red arrow (= complex number) in the picture below? Remember this: Thus how does it work? First we know that all arrows that are complex numbers start in (0,0), so we only have to know the endpoint of the arrow to be able to draw it. Finding the endpoint goes as follows: given an expression like 5 + 8i than the first real number (5) in the expression gives us the x-coordinate of the endpoint of the arrow, and the second real number (8) in the expression gives us the y-coordinate of the endpoint of the arrow. And thus the endpoint of the arrow is known (in our example it will be (5,8) ). The expression 5 + 8i thus designates the arrow starting at the point (0,0) and ending in the point (5,8).

I have cleaned up the Cartesian coordinate system to work with some concrete arrows (= complex numbers). We will use this:

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.

Both Lao tse and Chuang tse refer to meditation as an important element of their way of life. So even philosophical Taoism has to take meditation seriously. Now legends and speculations abound on this subject. But what I'm looking for in this topic is a scientific investigation (book or article) of the question to what exact form(s) of meditation Lao tse and Chuang tse were referring.

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). 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.

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.

@ 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).

The problem with fundamental science is that it concerns itself with things we don't yet understand, and that's why we can't predict whether or not a certain form of fundamental research will be of benefit to humanity.

@ 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?

The study of electricity and magnetism was not done in the interest of humanity but out of pure curiosity and its practical utility at the time was as good as absent. One simply cannot foresee what kind of fundamental research will or will not prove useful in the future.

@ 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?

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...

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.

@ 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.

I have done the green one myself:

Maybe some extra examples will be helpful. These are all complex numbers: 1 + (-4) i 3.778 + 1 i -0.000078787 + 10009897.555... i As you can see it doesn't matter what two numbers you put in as long you have an expression of the form a + bi with a and b some numbers and "i" a mere symbol, than you have a complex number. And each such complex number corresponds to an arrow in the Cartesian coordinate system. Now before I can get any further it is important to know whether we understand what arrows correspond to what complex numbers. So again: what is the complex number corresponding to the red arrow?