wandelaar

Here's the connection: https://en.wikipedia.org/wiki/Ronnie_James_Dio#Rainbow

It is not so much that I expect z1 to be displayed, but that I expect the real part Re(z1) and the imaginary part Im(z1) of z1 to be displayed. And that would look like this: Re(z1) = 2 Im(z1) = 3

A later one of Black Sabbath with Ronnie James Dio as singer:

No - the idea is good, but your notation is weird. When a mathematician wants to say that the real part of z1 is 2 they will use the concise notation: Re(z1) = 2 . And when a mathematician wants to say that the imaginary part of z1 is 3 they will use the concise notation: Im(z1) = 3 . We better get this right. Can you now give the answers to my previous post using the concise notation with the symbols Re( ) and Im( )?

Another test: Can you now give the following values? Re(z1) = Im(z1) = Re(z2) = Im(z2) = Re(z3) = Im(z3) =

Still looking for a general key (or set of keys) to open up those kind of paradoxes. I have the impression that there must be a general principle (or a few general principles) behind them all. I have read lots of explanations for individual paradoxes, but I can't seem to remember them later on because I can't fit them into some kind of general scheme. I know what most of you are thinking now: there is no general scheme, you have to go beyond logic, the paradoxes are all expressions of non-duality. Maybe that is so. But I am not prepared to give up my search for a general rational interpretation yet.

We have been discussing this chapter not so long ago, but I don't remember where?

I must be getting really old! Jimi Hendrix was god playing electric guitar. Everybody knew Jimi Hendrix! Sad to realize that nowadays House, Rap, and r&b (no not Rhythm-and-Blues - that would have been great) are setting the scene, and that the superb pop and rock music from the sixties and early seventies is (almost) forgotten. But modern pop music is awful. No not awesome - awful ! Listen here why this grumpy old man is right:

That's almost correct. The usual mistake for beginners (and you are also falling into it) is to give the imaginary part complete with the "i", but that's not how it is defined. Our definition is: And the y-coordinate of a point in our Cartesian coordinate system is always a real number. That's why in the example of my previous post (regarding z1) I left out the "i" and wrote: Im(z1) = 3 .

Let me do z1 as an example: Re(z1) = 2 Im(z1) = 3

Two further definitions: The x-coordinate of the endpoint of a complex number (= arrow) z is called the real part Re(z) of the complex number z . The y-coordinate of the endpoint of a complex number (= arrow) z is called the imaginary part Im(z) of the complex number z . Below I have drawn four complex numbers (= arrows) z1, z2, z3, z4 and I like you to give me the real and imaginary parts of those complex numbers.

One minor correction: the arrow is written as -1 + 3i , not as -1,3i . But apart from that I think you understand all there is to understand at this point. All we have done is naming arrows in a Cartesian coordinate system in a certain way. The meaning of all this is that we can now consider complex expressions of the form a + bi as designating arrows in a Cartesian coordinate system. So in our approach the complex numbers are the arrows. Complex numbers considered as being arrows thereby loose their aura of mysticism. Statements about those numbers become statements of geometrical facts about arrows. So we will not need any of the usual heavy algebraic machinery to introduce the complex numbers. We only need the understanding that in our approach the complex numbers are the arrows.

It's up to you. Do you want some more training before moving on to the next phase? Or do you have some questions left?

The "i" is only a sign (not a variable) used in the complex notation (a + bi) of arrows. However there is an arrow with the complex notation 0 + 1i that will later on play the role of the "imaginary unit". And in shorthand that arrow will often be written as "1i" or even simply as "i". Now there is no real number x such that x*x = -1. But as we will see later on there are complex numbers z (and in our geometric approach arrows z) such that z*z = -1 + 0i . That is the way mathematicians solve this issue. They create a larger number system (namely that of the complex numbers) in which there are solutions z such that z*z = -1 + 0i. And those solutions aren't figments of the imagination but concrete objects (arrows in our case) from the set of complex numbers.

Why can't I add whitesilk to my ignore list?

Both axes of my Cartesian coordinate system are real! When you introduce the complex numbers by taking the horizontal axis as real and the vertical axis as imaginary then you are already presupposing the existence of the imaginary numbers. That's a vicious circle. New mathematical objects must - whenever possible - be constructed from the already trusted old ones. As soon as the existence of the complex numbers is proven to be non-problematic then one can suppose the vertical axis to be imaginary.

@ OldDog You can't apply a theory before it is developed. You are asking the impossible. Complex numbers have many applications. I already named some of them in a reaction to rideforever. And the arrows are meant to give a solid foundation to the complex numbers. But most practical applications will only become possible after we have defined ez where z is a complex number. But that's advanced stuff. I doubt whether we will come that far. The complex numbers as arrows are perfectly defined objects. Nothing mysterious about that. And that is the reason I am building this theory on the arrows. Please ignore the misinformed rantings of rideforever.

That's only shows that you don't know what you are talking about. You are making a fool of yourself. Lao tzu considered it very important to know the boundaries of ones knowledge. Nobody knows everything, and personally realising that fact is a great form of (self)knowledge.

OK - let me know when you are ready for the next phase. That will then be studying the real and imaginary part and the argument and modulus of a complex number. (Sounds more difficult than it is.)

Just when it seemed you understood it all, and now this..... Please look again?

One more very special case is the "arrow" that starts and ends in (0,0). How would you write that arrow in the form a + bi ?

You could of course formally give that expression a meaning by defining it to mean b + ai. But lets not do that until the basics of the theory are well understood.

@ OldDog I am not giving the one and only explanation of the complex numbers, but only the one that I consider the easiest to understand. There are many other ways of introducing the complex numbers besides that. Maybe it can also be done with a curly arrow, or with an arrow that starts form somewhere else than (0,0). Maybe there is an alternative to using an arrowhead at the end to give it a direction. Who knows? I am just giving one possible way to do it, no more and no less. So basically your questions of "why this?" and "why that?" cannot really be answered. It's just a choice to do it this way or that way...

That is completely correct! Now when actually calculating with those objects it quickly becomes a nuisance to write the zeros when they occur. So in practice one often writes 2i as a shorthand for 0 + 2i, and 3 as a shorthand for 3 + 0i. But we have to remember that in the context of the complex numbers when we see "2i" or "3" that it is just a lazy way of writing "0 + 2i" and "3 + 0i". So when we are dealing with complex numbers the expressions of the form "bi" and "a" are shorthand for "0 + bi" and "a + 0i". Thus 4i , -1i , and 3.6784i are shorthand for 0 + 4i , 0 + -1i , and 0 + 3.6784i . And again in the context of complex numbers 2 , 7.899 , and 101 are shorthand for 2 + 0i , 7.899 + 0i , and 101 + 0i .

Two special cases: How do we represent the green arrow and the red arrow as a complex number of the form a + bi ?