AetherFox

2008-11-24, 03:51 PM

So, today in my class we worked with imaginary numbers, the basis of which is 'I', or the square root of -1. Does anyone know how this can be applied in the real world? :smallconfused:

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AetherFox

2008-11-24, 03:51 PM

So, today in my class we worked with imaginary numbers, the basis of which is 'I', or the square root of -1. Does anyone know how this can be applied in the real world? :smallconfused:

Dallas-Dakota

2008-11-24, 03:54 PM

Nope.

But I'l bet you that Player Zero does:smallwink:

Good luck for trying to find a use of it in the real world........

Shun the heretic...

But I'l bet you that Player Zero does:smallwink:

Good luck for trying to find a use of it in the real world........

Shun the heretic...

Flame of Anor

2008-11-24, 03:55 PM

For one thing, it's not "I", it's "i". Always lowercase. And I think it does have some applications but I don't know what.

Player_Zero

2008-11-24, 03:56 PM

There's a wikipedia page (http://en.wikipedia.org/wiki/Imaginary_number#Applications_of_imaginary_numbers ) on all the different applications, but quantum mechanics is by far the most interesting.

Also, it's 'i'. Not 'I'. Never capitalised.

In addition, there's quaternion (http://en.wikipedia.org/wiki/Quaternion), the next step up.

Also, it's 'i'. Not 'I'. Never capitalised.

In addition, there's quaternion (http://en.wikipedia.org/wiki/Quaternion), the next step up.

Coidzor

2008-11-24, 03:57 PM

Being able to conceptualize imaginary numbers helps with plotting certain things and with pushing the brain to the limit?

Project_Mayhem

2008-11-24, 03:58 PM

I asked the same question at a-Level, and my teacher said that they used complex numbers in making some massive pipeline. No Idea how.

ah the days of xi+yj. Just wait till you have to integrate it.

ah the days of xi+yj. Just wait till you have to integrate it.

Player_Zero

2008-11-24, 04:00 PM

I asked the same question at a-Level, and my teacher said that they used complex numbers in making some massive pipeline. No Idea how.

ah the days of xi+yj. Just wait till you have to integrate it.

Integrating imaginary numbers? :smallconfused: It's just the same as normal numbers... What are you on about?

ah the days of xi+yj. Just wait till you have to integrate it.

Integrating imaginary numbers? :smallconfused: It's just the same as normal numbers... What are you on about?

Project_Mayhem

2008-11-24, 04:11 PM

Integrating imaginary numbers? It's just the same as normal numbers... What are you on about?

... I'm sure it was harder

...erm

... shut up. I do English lit OK. :smallredface:

damn kids from Warrick, think theyre so smart, I oughta bkah blah blah

... I'm sure it was harder

...erm

... shut up. I do English lit OK. :smallredface:

damn kids from Warrick, think theyre so smart, I oughta bkah blah blah

oreganoe

2008-11-24, 04:15 PM

The most common use I know of is oscillation, both in AC circuits and springs and the like.

Player_Zero

2008-11-24, 04:17 PM

I believe they may also be used in specific instances of general relativity, but don't quote me on that, I ain't studied that particular topic further than a ten minute glance over a couple of websites.

Tirian

2008-11-24, 04:21 PM

One nice thing about them is that you sort of get to squash two real numbers into a single "number", and one of the things you'll so soon enough if you haven't already is identifying the number 3-2i with the Euclidean point (3, -2). This can be handy in things like fluid dynamics, where you can specify the flow of a two-dimensional liquid with a single function. The Mandlebrot set and other similar fractals is another example of a function that only really makes sense in terms of complex numbers.

Mostly, though, it's important in the philosophy of mathematics, because the complex numbers have a property called closure under the arithmetic operators. From the simple counting numbers, there kept on being new classes of numbers added in to solve equations like x + 1 = 0 (the negative numbers), 2x = 1 (the rational numbers), x² = 2 (the algabraic irrational numbers), and 2ⁿ = 3 (the transcendental numbers), and finally x² = -1 (the complex numbers). But this is the end of that road. Thanks to what is called the Fundamental Theorem of Algebra, it turns out that you can't write a polynomial function that doesn't have a solution in complex numbers.

Mostly, though, it's important in the philosophy of mathematics, because the complex numbers have a property called closure under the arithmetic operators. From the simple counting numbers, there kept on being new classes of numbers added in to solve equations like x + 1 = 0 (the negative numbers), 2x = 1 (the rational numbers), x² = 2 (the algabraic irrational numbers), and 2ⁿ = 3 (the transcendental numbers), and finally x² = -1 (the complex numbers). But this is the end of that road. Thanks to what is called the Fundamental Theorem of Algebra, it turns out that you can't write a polynomial function that doesn't have a solution in complex numbers.

Ponce

2008-11-24, 05:42 PM

It appears sometimes in practical linear algebra. For example, there was a study conducted that analyzed the population demographics of sea turtles (if I recall correctly). It turned out that some of the eigenvalues of the matrix were imaginary numbers. The model wasn't ruined as a result, but it is nevertheless interesting that this occurred. I think this particular example is somewhat iconic.

Just about anything involving differential equations has the potential to require knowledge of imaginary numbers.

Project_Mayhem, I seem to recall that imaginary numbers need a little more TLC if you start involving repeating functions. Is that what you were thinking of? I might be mistaken.

Just about anything involving differential equations has the potential to require knowledge of imaginary numbers.

Project_Mayhem, I seem to recall that imaginary numbers need a little more TLC if you start involving repeating functions. Is that what you were thinking of? I might be mistaken.

Player_Zero

2008-11-24, 05:50 PM

Generally speaking here, purely conjecture and not based upon specific cases mind, imaginary numbers are perfectly valid solutions to any particular equation you care to name. It may or may not have real interpretations which are valid but in some cases it does.

You may rnd up with a 5i growth in animal population over time, but if, say, the speed at which the growth is increasing over time is a function of the growth over time, say, growth squared, then you end up with a -25 change in gowth over time. The interpretation for this would naturally be a decay in animal growth over time.

Mmm... Purposefully confusing.

I think it's funny that we call them imaginary... Because they bloody well aren't imaginary unless you say that all numbers are, in that they conceptualise a particular dealie.

You may rnd up with a 5i growth in animal population over time, but if, say, the speed at which the growth is increasing over time is a function of the growth over time, say, growth squared, then you end up with a -25 change in gowth over time. The interpretation for this would naturally be a decay in animal growth over time.

Mmm... Purposefully confusing.

I think it's funny that we call them imaginary... Because they bloody well aren't imaginary unless you say that all numbers are, in that they conceptualise a particular dealie.

Moff Chumley

2008-11-24, 05:55 PM

I think they're just for gits and shiggles, and mathematicians invented some kind of justification for them later.

Totally Guy

2008-11-24, 06:08 PM

I'm pretty sure elecricians use it when modelling a circuit in a 3d space like a building. But I'm still not sure why a regular 3D space is insufficient.

unstattedCommoner

2008-11-24, 06:11 PM

Because writing A*exp(i*w*t) is faster than writing C*cos(w*t)+D*sin(w*t).

Also, being linear, Maxwell's equations lend themselves to solution by Fourier transform.

Also, being linear, Maxwell's equations lend themselves to solution by Fourier transform.

Project_Mayhem

2008-11-24, 06:15 PM

Project_Mayhem, I seem to recall that imaginary numbers need a little more TLC if you start involving repeating functions. Is that what you were thinking of? I might be mistaken.

Science knows. I have no idea

Science knows. I have no idea

Player_Zero

2008-11-24, 06:18 PM

I think they're just for gits and shiggles, and mathematicians invented some kind of justification for them later.

This is how everything was invented.

This is how everything was invented.

averagejoe

2008-11-24, 06:20 PM

Integrating imaginary numbers? :smallconfused: It's just the same as normal numbers... What are you on about?

Technically yes, but there are a lot of conveniences present when you integrate on the complex plane; for example, the fact that you can integrate over closed curves and the value of those integrals are completely specified by the values of the functions inside of them (an oversimplification, but the essence of it.) This actually makes certain real integrals possible, because you can extend the real line to the complex plane, integrate over a different path (which, hopefully, is easier to integrate), then integrate over the whole loop (which tends to be easy enough to be nearly trivial, except in really contrived examples.) This allows you to do certain integrals which have no indefinite integral or very messy ones.

On complex numbers in physics: it's actually not necessary as often as you see it; complex numbers are just preferred in certain cases because it makes the math nicer.

Technically yes, but there are a lot of conveniences present when you integrate on the complex plane; for example, the fact that you can integrate over closed curves and the value of those integrals are completely specified by the values of the functions inside of them (an oversimplification, but the essence of it.) This actually makes certain real integrals possible, because you can extend the real line to the complex plane, integrate over a different path (which, hopefully, is easier to integrate), then integrate over the whole loop (which tends to be easy enough to be nearly trivial, except in really contrived examples.) This allows you to do certain integrals which have no indefinite integral or very messy ones.

On complex numbers in physics: it's actually not necessary as often as you see it; complex numbers are just preferred in certain cases because it makes the math nicer.

EvilDMMk3

2008-11-24, 06:46 PM

Am I correct in my understanding that whilst i has no "real world" use (ie you cannot have i metres of string) it lets you solve things that do that otherwise could not be solved?

averagejoe

2008-11-24, 06:56 PM

Am I correct in my understanding that whilst i has no "real world" use (ie you cannot have i metres of string) it lets you solve things that do that otherwise could not be solved?

More or less. It also lets you solve things that can be solved, only much easier.

Also, this (http://xkcd.com/410/) seems somewhat obligatory, cuz I'm trendy. :smallcool:

(And this (http://xkcd.com/179/).)

More or less. It also lets you solve things that can be solved, only much easier.

Also, this (http://xkcd.com/410/) seems somewhat obligatory, cuz I'm trendy. :smallcool:

(And this (http://xkcd.com/179/).)

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