Imaginary Numbers

Don't let "imaginary" throw you as a term, any more than "irrational" or "transcendent", it's just qualifiers that mathematicians attach to concepts.

A way to sort of visualize imaginary numbers is this - take your grade school number line. Positive numbers move to the right, negative numbers move to the left. Imaginary numbers...rotate into the space above and below the line. Now it's two dimensional.

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 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.

eta:
Hint: it's called an Argand diagram, and it's something you pick up in your freshman year in EE. Seriously. Pure imaginary numbers lie on the y axis. Pure real numbers on the x axis. It's the way you work with them.

After you use complex numbers for a while, the "what's inside this 'j' term?" thing sort of becomes commonplace. You don't sit around and debate the whichness of what of the y axis. It's sort of like transcendentals and irrationals, I don't worry that pi has no exact solution, and don't sit around worrying about what the best approximation formula is. If you're into number theory, I guess it might be your job to contemplate. To me it's a tool I use every day.

etaa: If you multiply a negative real number by a negative real number, you get a positive number. It puzzles me why this bothers you for a pair of purely real negative numbers in Argand space...it's the same result.

Imaginary and complex numbers are covered in the standard engineering program I am enrolled in. We need them for analyzing AC circuits and, from what I am told, used to calculated quantum mechanical states. Other then the core introduction of this stuff, I don't know the rest, I will have the AC course starting in September, or February, and the quantum mechanics course will be in 2 or 3 years from (School takes a while).

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.

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.

Originally posted by halfmask
Imaginary and complex numbers are covered in the standard engineering program I am enrolled in.

Learn them well, you'll never be rid of them for the rest of your engineering career. Welcome to all the trig you never wanted to know.

There are a few paths you can follow to avoid them but all the really interesting stuff invokes complex numbers.

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.

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 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.

The complex plane is for doing vector math with them. In fact, it's often what you use vector math for. I don't think i is "multi dimensional", it's a value. Other than, of course, they lie on the y dimension of an Argand plane. There are many things you can't directly visualize. That's not a big deal...your system for visualizing wasn't evolved for higher math and physics. It doesn't mean the things you can't visualize are non-existent or useless. That's why you have to resort to math to describe a lot of things in physics. I literally can't verbally describe some things well in EE and physics. They're just not amenable to verbal description with accuracy. It's one of the beauties of math.

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.

On an Argand space, that's true.

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.

On the real axis, you can't do that - it's one of the defining rules of the real number line. A negative real multiplied by a negative real is a positive real number.

It's ok...it gets worse. When you flip AC onto the imaginary axis, it doesn't generate heat either. e.g. I use a capacitor as a dropping resistor all the time - it acts like a perfectly good impedance element and since the current's at right angles to the voltage, there's no heat dissipation except in the esr of the cap. As long as you don't mind the lag between the current and voltage. Vars are pretty oddball that way, nothing like imaginary power.

reply to post by Bedlam

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.

I was saying if imaginary numbers really are multi-component vectors then we should be able to plot out the structure on a 2D grid. 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 represents something completely different and it's important not to think that it's a way for visualizing imaginary numbers. It's not a way for visualizing imaginary numbers.

The complex plane is for doing vector math with them.

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.

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.

What's up and down that y axis?

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.

Is the number line for doing math with real numbers?

reply to post by Bedlam

What's up and down that y axis?

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 like we do real numbers, and if you dispute that fact you clearly need to do more research on imaginary numbers.

Originally posted by ChaoticOrder
You need to understand the fact that the y axis is a completely arbitrary, any mathematician will tell you that.

So is any method of visualizing something in mathematics.

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.

And again, when you say "visualizing imaginary numbers" what does that mean to you? i is a value, like 5. The only difference is, the value of i is the square root of -1. Yes, that's tough to swallow. It's no worse than imaginary power, though.

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.

Oddly, I've been using them for about 23 years. Seems to work for me. If you say we have no way to visualize them, fine. Don't visualize it. That's what math is for. I can't visualize Hamiltonians, doesn't stop me from using them. Can't visualize a var, either, but they work all the live long day.

reply to post by Bedlam

So is any method of visualizing something in mathematics.
...
i is a value, like 5.

Except the real number 5 can be expressed on the real number line and the distance between each unit on the line is meaningful. It's a clear way of visualizing real numbers and plotting them out on a 1D line. The same thing cannot be done, or has not been done, with imaginary numbers. The imaginary axis on the complex plane isn't meaningful and doesn't represent the true value of imaginary units.

Don't visualize it. That's what math is for. I can't visualize Hamiltonians, doesn't stop me from using them.

I didn't say we shouldn't use them, I'm simply saying we should look for a good way to visualize them because I believe it is possible.

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. You may not like it in that form, but to call it meaningless sort of flies in the face of a myriad of correct answers and complex number math theory that has gone before. You've got a long row to hoe if you want to prove the imaginary axis in Argand space is meaningless and/or arbitrary. Phasors are your friend. Embrace the suck.

reply to post by Bedlam

Sure it does. It's not like the thing's made up.

I give up. You clearly aren't going to grasp what I'm talking about.

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).
Octonions are for adults. This is because they turn out to be nature's numbers. Electromagnetic theory can be written in the form of complex numbers. M-theory (including superstring theories and supergravity), which includes electromagneticism, can be expressed in terms of octonions.

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?

imaginary numbers have a curious effect on things, but i'm not sure they're as mysterious as they're made out to be.

what is positive and what is negative is based on perspective, it's all relative. if two equal but opposite wave functions were graphed, which would be the negative, and which the positive?

i think it's a way to designate something as being opposite while working around the impossibility of taking the square root of a negative (or perhaps we haven't discovered a sufficient mathematical way to describe the action, so imaginary numbers act as placeholders).

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?

Okay, either you have a very unique view on numbers or we are simply not speaking about numbers at all but about something much more esoterical (with "dimensions", "structures" and complex numbers which are obviously not the usual kind of combined real-part + imaginary-part)..

Because, the way many posters before tried to explain, in EE you work with complex numbers on a daily basis. Complex apparent power and other parts of our daily calculations (at the university, nowadays I could use a 8-digit calculator and am using Excel for its simplicity) are always visualized in the complex plane.

You don't want to use something mathematicians developed to show that real numbers are orthogonal different to imaginary numbers? Fine.

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?

reply to post by ManFromEurope

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.

You seem to be completely misunderstanding me. I never once said I don't like complex numbers or the idea of a complex plane. I simply said the complex plane is a way of visualizing complex numbers, not imaginary numbers. And that if we want to truly have a legitimate way of visualizing imaginary numbers then we must come up with a new system which can do so. Is that so hard to understand?

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?

A complex number is typically of the form (z = a+bi) where a is the real part and bi is the imaginary part and b represents the magnitude of i. Together these two components form z, which is the complex number and can be manipulated as if it were a 2D vector. The y axis of the complex plane is really plotting the magnitude of i (aka b), it's not plotting the true value of any given imaginary number at any point on the y axis.

And I think what I should also make extremely clear is that I don't believe imaginary numbers are multi-dimensional numbers as many mathematicians tend to claim. If imaginary numbers did have multiple components themselves, they could be a 2-dimensional number like a complex number, which would actually make a complex number a 3-dimensional number, which only further illustrate how the y axis on the complex plane doesn't truly represent the meaning of the imaginary numbers plotted on that axis.

However as I mentioned in the opening post I don't believe that to be the case, I simply think some of our mathematical operations apply differently to imaginary numbers, there is an interesting inversion process which occurs when you use imaginary numbers. And if you think about why that occurs it's quite clear, in order to arrive at the square root of -1 you need some sort of numbers which behave in the opposite way to normal numbers. They're almost like anti-numbers or something.