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Imaginary numbers...rotate into the space above and below the line. Now it's two dimensional.
Originally posted by ChaoticOrder
reply to post by Bedlam
Imaginary numbers...rotate into the space above and below the line. Now it's two dimensional.
No in fact they cannot be visualized in that fashion, and I went over why that is the case in my opening post.
Originally posted by Bedlam
Originally posted by ChaoticOrder
reply to post by Bedlam
Imaginary numbers...rotate into the space above and below the line. Now it's two dimensional.
No in fact they cannot be visualized in that fashion, and I went over why that is the case in my opening post.
Oh, but they can, and in fact, that's the way you DO visualize them in EE.
j is on the y axis. And that's how you convert from polar to rectangular coordinates with complex numbers.
At this point you might be thinking "well what about the complex plane, that's a 2D plane with an imaginary axis and a real axis and it gives us a way to visualize imaginary numbers". My response to that thinking would be that you are confused, the complex plane plots complex numbers, and complex numbers have both a real part and an imaginary part. That is not the same thing as plotting out the structure of an imaginary number.
The spacings on the imaginary line are completely arbitrary and mean nothing because we have no way of visualizing an imaginary quantity. The expression (2i < 3i) is not even necessary true. If imaginary numbers really are some type of multi-dimensional numbers with multiple components, like a vector with 2 components, we should be able to apply simple vector math to them and also visualize them in a nice way.
You might say all real numbers would exist on the "event horizon" of the imaginary number space, meaning that all real numbers would have a y component equal to 0. However there is no vector which when multiplied by itself results in a vector where x=-1 and y=0 (aka the real number -1). We run into the same problem, a negative vector multiplied with a negative vector is a positive vector. I wish it were that easy, but if it were it would have been figured out already.
Originally posted by halfmask
Imaginary and complex numbers are covered in the standard engineering program I am enrolled in.
Originally posted by ChaoticOrder
You obviously haven't read the opening post properly:
At this point you might be thinking "well what about the complex plane, that's a 2D plane with an imaginary axis and a real axis and it gives us a way to visualize imaginary numbers". My response to that thinking would be that you are confused, the complex plane plots complex numbers, and complex numbers have both a real part and an imaginary part. That is not the same thing as plotting out the structure of an imaginary number.
The spacings on the imaginary line are completely arbitrary and mean nothing because we have no way of visualizing an imaginary quantity. The expression (2i < 3i) is not even necessary true. If imaginary numbers really are some type of multi-dimensional numbers with multiple components, like a vector with 2 components, we should be able to apply simple vector math to them and also visualize them in a nice way.
You might say all real numbers would exist on the "event horizon" of the imaginary number space, meaning that all real numbers would have a y component equal to 0.
However there is no vector which when multiplied by itself results in a vector where x=-1 and y=0 (aka the real number -1). We run into the same problem, a negative vector multiplied with a negative vector is a positive vector. I wish it were that easy, but if it were it would have been figured out already.
What come to mind when you say "plotting out the structure of an imaginary number"? Is there a structure to it? It's just a value. It has all the structure of "5", that is, none. It has a behavior.
The complex plane is for doing vector math with them.
Originally posted by ChaoticOrder
But we clearly cannot plot out imaginary numbers in that way, and complex numbers are not the same thing as imaginary number.
The complex plane is for doing vector math with complex numbers which contain two components: a real components and an imaginary component. It's not for doing math with imaginary numbers directly, and that's the point you need to understand.
What's up and down that y axis?
Originally posted by ChaoticOrder
You need to understand the fact that the y axis is a completely arbitrary, any mathematician will tell you that.
The complex plane is like any other 2D plane which can be used to plot vectors. You're plotting the real component and imaginary component of the complex number as a way to visualize the complex number, it IS NOT a way of visualizing imaginary numbers themselves.
The fact is we have no way of visualizing imaginary numbers and if you dispute that fact you clearly need to do more research on imaginary numbers.
So is any method of visualizing something in mathematics.
...
i is a value, like 5.
Don't visualize it. That's what math is for. I can't visualize Hamiltonians, doesn't stop me from using them.
Originally posted by ChaoticOrder
The imaginary axis on the complex plane isn't meaningful and doesn't represent the true value of imaginary units.
Sure it does. It's not like the thing's made up.
Imaginary numbers are for children.
There are only four possible division algebras: real numbers(1-dimensional), complex numbers (2-dimensional); quaternions (4-dimensional) and octonions (8-dimensional).
Originally posted by ChaoticOrder
reply to post by micpsi
Imaginary numbers are for children.
There are only four possible division algebras: real numbers(1-dimensional), complex numbers (2-dimensional); quaternions (4-dimensional) and octonions (8-dimensional).
The key point you are overlooking is that imaginary numbers are different from all of those. Tell me, how many dimensions does an imaginary number have?
Use something else. Maybe a stick in the ground, we don't care.
Point is: there is a way to visualize complex numbers (which ARE nothing else besides real-part + j*imaginary part). Thousands if not hundreds of thousands are using this tool every day. There is nothing wrong or even debatable with it. It is a mathematical sound way to work with those numbers.
You don't like it. We got that so far.
Please, for the sake of discussion, give an example of a complex number. Is it written in the usual way? Or is there another way?