wandelaar

Added the modulus of a complex number to the picture:

I just now had an idea that might give us some pictures of functions (such as Re( ), Im( ), | | ) from the complex numbers to the real numbers. You know: a picture tells more ... What we need for that is a 3D grapher. We already introduced the Cartesian xy-plane for our complex numbers (= arrows), and adding one more perpendicular axis gives us the possibility to represent the values of the function above (or below) the arrowheads of our complex numbers.

Well - after all we are dealing with imaginary numbers. So...

In a trivial way: yes. Because even for complex numbers (= arrows) z that have Im(z) = 0 we can still write z = a + 0i .

Yes. Yes.

Later today I will answer your questions. First I have some work to do...

Functions are more abstract objects than numbers or arrows. That's why I didn't want to discuss them. But LiT is right that functions are essential to higher mathematics. However giving a formal definition of functions in terms of set theory is out of the question, that would take a huge topic of its own. We will have to make do with a rough idea of what a function is. The best I can think of now as a concrete example is a calculator that always performs the same operation on whatever is fed into it. For instance it could always add 1 to the number that is fed into it. Or it could always multiply the number that is fed into it by 44. But the operation or sequence of operations could be endlessly more complicated. The only restriction is that the calculator must always perform the same operation or sequence of operations on whatever number is fed into it. Such a calculator would be the physical realisation of a mathematical function. And the mathematical function itself could then be considered as the program that describes what operation or sequence of operations the calculator has to perform. I have no more time today, so see you all again tomorrow.

The modulus is a special type of function. You throw a complex number in and you get a non-negative real number out. What you get out depends on what you throw in. If you throw a complex number z in then what you get out in case of the modulus is written as |z| . The modulus is the complex version of the absolute value. Maybe LiT can help some more with explaining the function concept.

Yes - but that is not the answer to my question. The "|"-signs are not ordinary brackets. According to the definition |1 + 0i| is the modulus of 1 + 0i. And the modulus of 1 + 0i is the distance between (0,0) and (1,0).

How about |1 + 0i| ?

DEFINITION: The modulus |z| of a complex number z = a + bi is the distance between the point (0,0) and the endpoint (a,b) of z. Now I have already posted a lot of pictures of complex numbers as arrows, and I like to see whether some of you can calculate the value of the modulus for some of those complex numbers purely on the basis of the above definition.

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.

@ Marblehead We have two more functions to explore. Are you ready?

When (a,b) is the endpoint of the complex number z than: Re(z) = a and Im(z) = b. For a complex number z = a + bi we will get: Re(z) = a and Im(z) = b. So here we have our first "theorem" : z = Re(z) + Im(z) i All this refers to the same geometrical fact, only expressed differently.

LiT is right: both Re( ) and Im( ) are functions. Re(z) gives us the real part of the complex number z Im(z) gives us the imaginary part of the complex number z

Well there is a coordinate system called the complex plane that has a real x-axis and an imaginary y-axis. Most people learn the complex numbers by means of the complex plane. And as a way of representing the complex numbers the complex plane is perfect. But as a way of introducing the complex numbers it isn't. Suppose you know nothing about imaginary numbers and want to learn what they are. Now the teacher draws two axis: a real x-axis and an imaginary y-axes. Along the x-axis there are real numbers, so that's no problem. But along the y-axis there are imaginary numbers. Now what are imaginary numbers? Are you supposed to understand the complex numbers by accepting the imaginary numbers on faith? Apparently you are. And thus there will be two categories of student that are not going to get it. First there are those who are not smart enough to (quickly) understand how to calculate with those mysterious things called complex numbers. And second there are those who are too intelligent to be fooled by a circular explanation. Both will not get it. And so it goes... Actually the magic will happen when we define the multiplication of complex numbers. This will be done in such a sneaky way that the arrows will possess all the properties that we like complex numbers to have.

@ OldDog But I am serious! It would be a good exercise for both of us...

My definitions apply to a real Cartesian coordinate system with (0,0) as starting point for all arrows. When you want to use a coordinate system with an unreal (?) y-axis and a starting point for the arrows that doesn't lie on the intersection of the x- and y-axis than a lot of things will turn out differently. I don't know what would happen. Maybe you could start a topic for the development of such an OldDogian number system and than I could ask critical questions to make sure that your theory is mathematically legitimate?

The point ( Re(z) , Im(z) ) is the endpoint of the complex number z considered as an arrow. This logically follows from the definitions of Re( ) and Im( ). Correct: an arrow needs both a starting point and an endpoint. I don't understand what you are saying here.

Good. It is possible to introduce the complex numbers without an arrow, that could be done by defining the complex numbers as ordered pairs (a,b). Such a construction of the complex numbers would proceed in an algebraic way. We could do that in a new topic sometime in the future. But it will not work to here follow two approaches at the same time. Practical applications in solving algebraic equations will only be possible after we have defined the addition and multiplication of complex numbers. I think that should be doable, but further applications involving Euler's formula are almost certainly too advanced.

You could see it that way, if you leave out all the formal subtleties that make our construction of the complex numbers as arrows mathematically precise. But it is very important to remember that in our approach the "i" is only a sign. The complex numbers themselves are the arrows, and there is nothing undefined about the arrows. Also the imaginary numbers are simply those complex numbers z wherefore Re(z) = 0. So there is also nothing undefined about the imaginary numbers. We can simply draw the imaginary numbers as arrows in our Cartesian coordinate system. To repeat: the "i" is just a sign used in naming the complex numbers. Saying the "i" is undefined is dangerous because that gives the false impression that it somehow is a variable after all or that it should have a value. One thing more: in actual practise the sign "i" is often used as shorthand for "0 + 1i". But it is very confusing to do this while we are still in the process of trying to understand the basics of the complex numbers.

Yes - we are!

Most people would't even have started or have given up long ago. Thus you definitely are motivated. That's: Yes yes. Yes yes. Yes yes. Yes yes. The real part and the imaginary part are defined as the coordinates of the endpoint, and those are real numbers. So now please try again to do this test:

@ Marblehead No no - we are still working in the real Cartesian coordinate system. Both the x-axis and the y-axis contain real numbers. You simply cannot construct a complex plane before you have introduced the complex numbers. That would be meaningless, because then you wouldn't have any imaginary numbers (they are not introduced yet!) to put along the y-axis. Nevertheless in technical education that's the usual way to proceed. And no one will ever notice, because as technicians they are only interested in applications. When you give a technician some new mathematical tools and tricks and show them how to work with it, they will be perfectly happy to start calculating with their new toys. And no technician will ever care about what it all means or why it works. From their practical technical viewpoint it simply doesn't matter what it means or why it works, as long as it does. And so we had all of those Bums jumping in at the start to lance you right away into the complex plane, and present you with Euler's formula before you even knew what was happening. The complex plane where the y-axis contains imaginary numbers is in fact a mathematically useful device, but only after we have put the complex numbers on a solid foundation. And that foundation must naturally not contain the complex or imaginary numbers themselves. Please forget the complex plane for now, because it will only hinder the geometrical approach we are following.