The complex axis of a quadratic equation - Z not y

NilsP360 shared this idea 2 years ago
Declined

I've found a minor bug about the complex axis in Geogebra. I filled in the function:

f(x) = x^2+1.

Its the same for all ax^2+bx+c functions, but this is the most simple one. I wanted to find the roots. Of course with the Root command, you will just get the "undefined" output. But later i found the fantastic command ComplexRoot(). Then i got the complex roots: 0+i and 0-i. When i tried to visualize the ComplexRoots in the 2D graphics, the roots got placed one unit in both directions along the y-axis. Which is of course wrong, the z axis is the complex axis, because of this i tried to find a sloution.

There is nothing on the internet, so i decided to try to fix it myself. After 5 hours of work and research, I finally managed to display the complex roots along the third axis.

I did this by finding the exact X and Y coordinates of the "Complex roots", then i plotted that in as the X and Z coordinates for two actual Complex Roots. I simply called them C_1 and C_2.

If you want to see this amazing masterpiece, the file is below.

I would like to know if you think im wrong, and if so, how. If you dont understand something of it, please do ask me.

Best Answer
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First of all complex numbers belong to the complex plane. Following your statement it would be equally wrong to display them on the z-x-plane or any other plane. They should be displayed on a real-imaginary-plane. As you wrote we can not display this in a common way within two or three dimensional graphs.

I assume the programmers choose to display the complex numbers on the x-y-plane as the common teaching of complex numbers uses the decomposition of z=x+yi (complex number) into x (real part) an y (complex part). This is a convention in naming and displaying while teaching those numbers. Criticizing the displaying of those numbers in this way you sought for a better way to display those numbers because you realize that this kind of displaying could be misleading.

Well and if someone wants to solve a equation within the x-z-plane or y-z-Plane? If so the numbers would confuse him on the same way. Those understanding those numbers know that they are not located on the same plane as they might be displayed on. They would surely use the second graphical view to display them and would label those axes with Re and Im as it is less misleading. You yourself tried to find a better fitting way of visualization. And it was your approach to display it. There is no correct way of visualization there are numerous.

The programmers choose to plot them on their primary used plane (x-y-plane).

Deal with it as there is no significant reason to implement all possible visualizing ways. Find your own way to display it. There are a lot of people out there fiddling their way around on programs build to display spatial data displaying time-position data or other kind of data on them. GGb will not restrict your way of displaying it. The only limitation are the GGb-tools and your imagination.

And if you can not display it in GGb:

Be a real scientific man/women and find an appropriate program or program it yourself. As GGb is primarily made to teach non-scientists/engineers or to be used by non-scientists/engineers.

Comments (6)

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Since i cant edit my post any more, im going to write it here:

At the end of the second paragraph, i'd like to add the following:


The x-axis visualizes the different x values for a function and what y value you wound get for that x value. The y axis visualizes the different values for the entire function. The z-axis (only visible in the 3D graphics window (CTRL+SHIFT+3)) visualizes what X and Y values you get for a Z value (a complex number).

Example: f(x) = x^2+1 ComplexRoots: i and -i. X, Y and Z coordinates for the roots: C_1 = (0,0,1), C_2 = (0,0,-1). Example 2: g(x) = x^2+x+1 ComplexRoots: -5+0.87i and -5-0.87i. X, Y and Z coordinates for the roots: C_1 = (-0.5,0,0.87), C_2 = (-0.5,0,-0.87)

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Which is of course wrong, the z axis is the complex axis,
What's your reference for that?

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Im not citing anyone, its just for consistency. Because, by default, the x axis plots the real part of the input for a function, the y axis plots the real part of the output, if you want to use complex values for a function, you technically need 4 dimensions, 2 for the input and 2 for the output, but of course, since we only live in a 3 dimensional world (as far as we know), we are one dimension short. But the compromise is using our third axis, the z axis to represent the imaginary part of the output or input. (You have to choose one)

And to stay consistent with the fact that the x axis is for real inputs, y axis for real outputs, it would only make sense if an imaginary number were placed along the z axis (or w axis, which we dont have). Geogebra seems to place imaginary numbers on the same axis as the real numbers for the ouput for a graph, overlapping the existing data. This is not optimal, but i can see the logic in that then you dont have to expand the view to the 3D perspective. But there should at least be possible to choose wether you want imaginary numbers to be placed along the z axis, or if you want to let them stay overlapped with the y axis and the real valued outputs.

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2

First of all complex numbers belong to the complex plane. Following your statement it would be equally wrong to display them on the z-x-plane or any other plane. They should be displayed on a real-imaginary-plane. As you wrote we can not display this in a common way within two or three dimensional graphs.

I assume the programmers choose to display the complex numbers on the x-y-plane as the common teaching of complex numbers uses the decomposition of z=x+yi (complex number) into x (real part) an y (complex part). This is a convention in naming and displaying while teaching those numbers. Criticizing the displaying of those numbers in this way you sought for a better way to display those numbers because you realize that this kind of displaying could be misleading.

Well and if someone wants to solve a equation within the x-z-plane or y-z-Plane? If so the numbers would confuse him on the same way. Those understanding those numbers know that they are not located on the same plane as they might be displayed on. They would surely use the second graphical view to display them and would label those axes with Re and Im as it is less misleading. You yourself tried to find a better fitting way of visualization. And it was your approach to display it. There is no correct way of visualization there are numerous.

The programmers choose to plot them on their primary used plane (x-y-plane).

Deal with it as there is no significant reason to implement all possible visualizing ways. Find your own way to display it. There are a lot of people out there fiddling their way around on programs build to display spatial data displaying time-position data or other kind of data on them. GGb will not restrict your way of displaying it. The only limitation are the GGb-tools and your imagination.

And if you can not display it in GGb:

Be a real scientific man/women and find an appropriate program or program it yourself. As GGb is primarily made to teach non-scientists/engineers or to be used by non-scientists/engineers.

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I did not intend to criticize the ggb team, i just wanted a change to make it easier to visualize complex roots of a polynomial utilizing the third axis, namely the z-axis. But as you wrote yourself, there are numerous ways to visualize it, with no best way to do it. After reading your explanation, i realize that the way ggb is currently constructed is for the best, and dont really need such a change after all. But damn, it would have been very cool to be able to see an entire 4 dimensional graph of a complex valued function... too bad we're stuck in our measly 3 dimensions...

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I can only imagine to add the 4th dimension by using color. Sadly GGb is not really strong in displaying 3D graphs (implicit) or displaying colored fields/functions. There are some users trying to fiddle their way around. Two of them are gwengler and Roman Chijner. You can search their forum topics by clicking through their forum profiles and you can see some of their material by looking at their GGb-profiles.

I use GGb mainly as testing, visualizing or prototyping tool (all on short term). I wont use it as a fixed tool to solve all my problems as it is still at beta state (quite buggy in my view) and is sometimes changing his functionality (some features get removed changed and so on). So i do not bother complaining on some features or missing tools. I create them by myself or just do it on other programs. I still do report some bugs which are annoying me.

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