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Is there's a real purpose to imaginary and complex numbers?

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I guess its really just a by product of how our math works, but If I swing on the swing and I graphed the position off the ground, not over time, just distance from the ground, it would be a parabola with an exact equation, but no matter what, I would never actually be at 0 distance from the ground yet I could still get imaginary numbers for 0s. And then another problem with it is you can generate real numbers from imaginary numbers by raising them to an exponent, or perhaps there actually is something multiplied by itself to give a quantity that is less than 0 that we don't know of.

Edited by questionposter

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I guess its really just a by product of how our math works, but If I swing on the swing and I graphed the position off the ground, not over time, just distance from the ground, it would be a parabola with an exact equation, but no matter what, I would never actually be at 0 distance from the ground yet I could still get imaginary numbers for 0s. And then another problem with it is you can generate real numbers from imaginary numbers by raising them to an exponent, or perhaps there actually is something multiplied by itself to give a quantity that is less than 0 that we don't know of.

 

Plenty of things.

They're intimately linked to trigonometric equations.

Quantum wavefunctions are, in general, imaginary. Although it's the real magnitude of the complex number that represents a probability.

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Plenty of things.

They're intimately linked to trigonometric equations.

Quantum wavefunctions are, in general, imaginary. Although it's the real magnitude of the complex number that represents a probability.

 

But I mean, there still just imaginary, there just a by product. And with wave functions even those are real things, it's just that they oscillate. If I make a ripple in the water, is there really some imaginary point outside of the pool of water that's rippling? Not really, there's some water who's position is now below and above where water would be if it was stagnant, and even then, the level of water where it's stagnant exists in the middle of the ripples.

 

I guess it seems like they don't have a purpose in reality, they are just a by product of math is more of what I'm getting at. Like they aren't really a part of nature, and yet all these mathematical equations we developed use them.

Edited by questionposter

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But I mean, there still just imaginary, there just a by product. And with wave functions even those are real things, it's just that they oscillate. If I make a ripple in the water, is there really some imaginary point outside of the pool of water that's rippling? Not really, there's some water who's position is now below and above where water would be if it was stagnant, and even then, the level of water where it's stagnant exists in the middle of the ripples.

 

I guess it seems like they don't have a purpose in reality, they are just a by product of math is more of what I'm getting at. Like they aren't really a part of nature, and yet all these mathematical equations we developed use them.

 

Well, you can't really point to the number 2.7183... either. But it pops up all the time, too.

Generally we formulate all of our equations around non-imaginary things being the ones we can measure.

Whenever we have an equation that outputs something imaginary, we juggle it around a bit until it's real.

 

The one exception I can think of is quaternions/clifford algebras.

Directions in three dimensions can be represented as different roots of -1.

In much the same way i can be used to represent a direction in 2d.

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The typical use of imaginary numbers in physics and engineering is to represent phases. This is deeply linked to trigonometric functions as Schrödinger's hat has noted.

 

Are complex numbers part of nature? Well, it is true that one cannot measure a complex number, so in that sense they are not part of nature. However, complex numbers are fundamental in building our theories, and this is not just quantum mechanics. So in that respect complex numbers are part of nature, or at least part of the physicists tool kit.

 

From a mathematics point of view the power in the complex numbers is that they are algebraically closed. The statement of being closed is often refereed to as the fundamental theorem of algebra.

 

When one looks as geometry, you see that complex manifolds are far more beautiful than than real ones. There are a lot of nice theorems and results due to the use of complex, rather than real coordinates.

Edited by ajb

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I wonder how much of this reluctance or reticence towards imaginary numbers is due to the use of the term 'imaginary'? In that, if they were just called 'complex' or even a new word, would people have as much of a problem with them? I mean, I think that most people don't believe in most everything else that is called imaginary, so it is only natural to question imaginary numbers, too.

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I wonder how much of this reluctance or reticence towards imaginary numbers is due to the use of the term 'imaginary'? In that, if they were just called 'complex' or even a new word, would people have as much of a problem with them? I mean, I think that most people don't believe in most everything else that is called imaginary, so it is only natural to question imaginary numbers, too.

 

People would still have a problem even with a new name because you can't see them on normal graphs. They don't represent real quantities no matter what you call them. What I would really find strange though, is if you "needed" an actual non existent value in order to have this universe itself, although I suppose scientists could easily just call that "an unknown particle".

 

Le'ts call them "transformative-reality numbers"

 

Then someone asks "Hey, why don't transformative reality numbers appear on graphs?"

 

And then someone answers

 

"Because they don't actually exist, they are essentially imaginary."

Edited by questionposter

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You can graph complex values in the complex plane if you'd like. I think the fundamental issue is more that nobody has ever held 3i apples.

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You can graph complex values in the complex plane if you'd like. I think the fundamental issue is more that nobody has ever held 3i apples.

 

That's what I mean, not on normal graphs, but even in a complex plane they still don't represent real values which is why you never see them when you graph a function on a Cartesian graph.

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I don't see real values when I graph a function on the imaginary line either. Why should it be a problem that graphs not designed to show a certain kind of value don't show that kind of value?

 

I'm not sure what you're looking for. What sort of "purpose" for complex numbers would be satisfactory?

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I don't see real values when I graph a function on the imaginary line either. Why should it be a problem that graphs not designed to show a certain kind of value don't show that kind of value?

 

I'm not sure what you're looking for. What sort of "purpose" for complex numbers would be satisfactory?

 

Well I suppose if the universe could not operate without imaginary numbers, then that would satisfy their purpose, but otherwise they just seem to be a byproduct of the fact that math is just our interpretation of patterns found in nature, not nature itself.

.

Edited by questionposter

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You can graph complex values in the complex plane if you'd like. I think the fundamental issue is more that nobody has ever held 3i apples.

 

But I can imagine 3i apples.

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But I can imagine 3i apples.

 

You can imagine the concept of them but they don't look like anything. You probably can't even imagine them because there's no real thing to compare them too, there's no visual quantity that represents them.

Edited by questionposter

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You can imagine the concept of them but they don't look like anything. You probably can't even imagine them because there's no real thing to compare them too, there's no visual quantity that represents them.

 

You cannot square an apple either.

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Imaginary numbers are used because they are useful. As ajb said they can represent phases. You can use them to represent attenuation by having a complex index of refraction.

 

 

 

 

(3i apples ≠ 3 iApples though both are imaginary)

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I wonder how much of this reluctance or reticence towards imaginary numbers is due to the use of the term 'imaginary'? In that, if they were just called 'complex' or even a new word, would people have as much of a problem with them?

 

I don't think the terms "imaginary" or "complex" really does them justice. Imaginary makes them sound to "made up" or "artificial" and complex makes then sound too "hard" or "difficult" to work with.

 

But I can imagine 3i apples.

 

I think the point is that we know how to deal abstractly with 3i apples. I am not sure I can really "see" this in my head. :D

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I think the point is that we know how to deal abstractly with 3i apples. I am not sure I can really "see" this in my head. :D

 

Easy. It is the square root of minus nine apples.

 

The problem is that you cannot even imagine the square root of a single positive apple. And as I posted before, you cannot square an apple either.

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You cannot square an apple either.

 

You can't square an apple because an apple isn't a number or even a variable. We are talking about quantities and you can certainly square the quantity of apples.

 

 

Numbers aren't a real thing in of themselves, they are more of a simple catalog of what you see.

Edited by questionposter

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The most practical application that for me makes it seemingly understandable, is electronics. When gauging power supplies, the mathematics involved identify anomalous resistance values as imaginary. This relates to a variety of effects that produce what is known as negative resistance(imaginary.) One cause of this is internal impedance in the power supply. Not to mention it's uses in Z Impedance.

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You can't square an apple because an apple isn't a number or even a variable. We are talking about quantities and you can certainly square the quantity of apples.

 

 

Numbers aren't a real thing in of themselves, they are more of a simple catalog of what you see.

 

Right. But we use square speed all the time without anybody having trouble with it. Almost anybody.

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Right. But we use square speed all the time without anybody having trouble with it. Almost anybody.

 

That's because if it's squared it's usually dealing with more than 1 dimension such as in a vector, which uses 2 dimensions. I'm telling you; there really is something imaginary about imaginary numbers.

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Any maths is just a 'brain diagram' of events that occur in the real world and that we which to measure. Maths does not really exist outside of our brain.

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