[jndiii]

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, that Kisai mentioned above.

Perhaps its most salient point comes from explaining what "i" means functionally: it's a rotation. If 1 points straight ahead, and -1 points straight back, then i is a 90-degree rotation. As a matter of convention +i is 90-degrees to the left and -i is 90-degrees to the right. And that bit of intuitive understanding gets you to where e^(i*theta) makes sense, and it becomes obvious that e^i*pi = -1. (pi is a 180-degree rotation).

[Kenny]

	Originally Posted by Monte314 To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	I'm using the term "split" here to mean factored, that is, written as a product of at least two whole numbers all bigger than 1. For example: 210 = 2*3*5*7 36 = 2*2*3*3 12 = 2*2*3 6 = 2*3 4 = 2*2 OK, I've done some hard ones, DeaconSyre. You do an easy one: show us how to "split" 2 by writing it as a product of at least two whole numbers all bigger than 1: 2 = ?

It's very silly because in the real world there's only one thing that there's only two of anyway, everything else is only "two" because you're ignoring probably most of the universe.
If you want the measurements to make sense just chunk the units of what you're measuring up, instead of 2cm use 20mm.
Imaginary numbers indeed! Cartesian coordinates I call them.

[Latro]

	Originally Posted by Kenny To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	Imaginary numbers indeed! Cartesian coordinates I call them.

I have a bit of trouble accepting "C is the coordinate plane" when its multiplication and division give it such drastically different properties from R^2, including the ability to differentiate with respect to a complex variable.

[nacht]

	Originally Posted by Latro To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	I have a bit of trouble accepting "C is the coordinate plane" when its multiplication and division give it such drastically different properties from R^2, including the ability to differentiate with respect to a complex variable.

Basically it's a vector space where multiplication has been defined. You can
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, but I'm not sure it serves much of a point to say "well, we can't take the square root of a negative number in R, but we can in this 2D vector space with R-algebra!" versus "but we can use complex numbers!"

[thod]

	This process gives us a sequence of nested sets: N is contained in Z, which is contained in Q

One might assume that the set of R{1,2} is contained within the set R{0,3}. Yet Mr Cantor put a stop to all that nonsense since both sets are exactly the same size. Thus what is meant by the term 'contained'?

[Kenny]

	Originally Posted by nacht To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	Basically it's a vector space where multiplication has been defined. You can To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts. , but I'm not sure it serves much of a point to say "well, we can't take the square root of a negative number in R, but we can in this 2D vector space with R-algebra!" versus "but we can using complex numbers!"

I wouldn't limit it to two dimensions, or three. This is how I understand it, although I'm not using for any other purpose than understanding Riemann.

[nacht]

	Originally Posted by Kenny To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	I wouldn't limit it to two dimensions, or three. This is how I understand it, although I'm not using for any other purpose than understanding Riemann.

Only when generalized or extended to include more than one real/imaginary axis (e.g., Clifford algebras). C by itself can be entirely represented with a 2-dimensional real vector space.

[Kenny]

	Originally Posted by nacht To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	Only when generalized or extended to include more than one real/imaginary axis (e.g., Clifford algebras). C by itself can be entirely represented with a 2-dimensional real vector space.

I've come to the right place lmao!

I need the concepts to make geometric sense. To me there's still no square root of two, it's just a trick achieved by manipulating the meaning of 2?

[scorpiomover]

	Originally Posted by Monte314 To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	A high school student asked me about the "imaginary numbers" today, and I wrote this short article to answer his question. Perhaps you will enjoy it.

It's OK. But overly long, to say that imaginary numbers are just concepts that allow us to solve problems that often later turn out to be problems in real life, just like numbers, words, theories, and everything else that we think of.

Originally Posted by jndiii To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.

Perhaps its most salient point comes from explaining what "i" means functionally: it's a rotation. If 1 points straight ahead, and -1 points straight back, then i is a 90-degree rotation. As a matter of convention +i is 90-degrees to the left and -i is 90-degrees to the right. And that bit of intuitive understanding gets you to where e^(i*theta) makes sense, and it becomes obvious that e^i*pi = -1. (pi is a 180-degree rotation).

That is just the observation that our description of i happens to also fit a 1/4-circle rotation, and so one of the applications of i, would be to geometry, and those areas of science where such rotations are part of the equations.

[thod]

	Originally Posted by Kenny To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	I need the concepts to make geometric sense. To me there's still no square root of two, it's just a trick achieved by manipulating the meaning of 2?

The area of a rectangle is length*width. Since a square has all sides the same its area is just length*length = length^2. That's why it is called a square root. Given the area of the square you are finding the length of a side.

[jndiii]

	Originally Posted by scorpiomover To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	That is just the observation that our description of i happens to also fit a 1/4-circle rotation, and so one of the applications of i, would be to geometry, and those areas of science where such rotations are part of the equations.

Yeah, but you're missing the point, here.

Imagine (pun intended), that no one else in the world understands complex numbers, but you do. Now, try to explain i = sqrt(-1), and how this has anything to do with anything they might care about. Explaining it as a "rotation" is step one. "Those areas of science where rotations are part of the equations" includes all wave equations and all but the simplest differential equations ... and allows those simplest differential equation solutions to be smoothly mapped (e^kx -> e^-ikx) to their sinusoidal aspects.

Once it's applied to rotations, then other uses become acceptable, rather than just regarded as weird number theory. Yes, it's more than "just rotations", just as positive integers do so much more than count apples, but never underestimate the value of the concrete explanations.

[roninpro]

	Originally Posted by Kenny To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	I've come to the right place lmao! I need the concepts to make geometric sense. To me there's still no square root of two, it's just a trick achieved by manipulating the meaning of 2?

The square root of two is geometrically constructible. All you have to do is draw an isosceles right triangle whose legs are 1. What is the length of the hypotenuse?

[K27]

Are there any holes found in the complex number system?

[nacht]

	Originally Posted by K27 To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	Are there any holes found in the complex number system?

Define a "hole."

[K27]

	Originally Posted by nacht To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	Define a "hole."

Which? What?

If this is a pun, I don't get it.

[nacht]

	Originally Posted by K27 To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	Which? What? If this is a pun, I don't get it.

I considered making a pun based on algebraic rings, but opted against it. You asked "Are there any holes found in the complex number system?" What do you mean by "hole"?

[thod]

Deep man. A hole is defined by what it is not rather than what it is. Thus a hole in a copper sheet is defined by a lack of copper and a hole in an iron sheet by a lack of iron. Does this mean the holes are somehow different? If the copper sheet has an iron insert, is this a hole?

[K27]

	Originally Posted by nacht To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	I considered making a pun based on algebraic rings, but opted against it. You asked "Are there any holes found in the complex number system?" What do you mean by "hole"?

Like the holes get filled in by R in Q.

---------- Post added 07-21-2012 at 01:22 AM ----------

	Originally Posted by thod To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	Deep man. A hole is defined by what it is not rather than what it is. Thus a hole in a copper sheet is defined by a lack of copper and a hole in an iron sheet by a lack of iron. Does this mean the holes are somehow different? If the copper sheet has an iron insert, is this a hole?

Great. You don't see holes in Q now because R is filling it. But before we define R properly, there was a lack of representation of certain values, eg. sqrt(2). In that sense, there was holes in Q.

Similarly, since, to me, I don't know anything filling C at the moment, any value that lacks a representation is a hole.

[Latro]

	Originally Posted by K27 To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	Like the holes get filled in by R in Q.

C is complete; the underlying vector space is isomorphic to R^2.

Also, the sense in which the holes in the rationals exist is a bit more delicate than "there is no representation for sqrt(2)", because sqrt(2) is an absurdity in the language of the rationals. We think of them as holes because you can make sequences of rationals that build up on each other but do not converge to any rational number.

[nacht]

	Originally Posted by K27 To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	Like the holes get filled in by R in Q.

That sort of system breaks down once you get away from a single dimension.

There are three possible 2 dimensional unital algebras over the set of real numbers. Complex numbers, split complex numbers, and dual numbers. But there are
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as we increase dimensionality.

[Latro]

	Originally Posted by nacht To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	That sort of system breaks down once you get away from a single dimension. There are three possible 2 dimensional unital algebras over the set of real numbers. Complex numbers, split complex numbers, and dual numbers. But there are To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts. as we increase dimensionality.

I don't think she was addressing the issue of actually getting an algebra, but rather just the underlying vector space (or more precisely inner product space).

[K27]

Originally Posted by Latro To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.

C is complete; the underlying vector space is isomorphic to R^2. Also, the sense in which the holes in the rationals exist is a bit more delicate than "there is no representation for sqrt(2)", because sqrt(2) is an absurdity in the language of the rationals. We think of them as holes because you can make sequences of rationals that build up on each other but do not converge to any rational number.

Would it be more comprehensible if I write,

	There is no representation for sqrt(2) in Q.

?

---------- Post added 07-21-2012 at 01:39 AM ----------

	Originally Posted by Latro To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	I don't think she was addressing the issue of actually getting an algebra, but rather just the underlying vector space (or more precisely inner product space).

Yay. Generating algebra is easy. You just cook some rules up.
But I am still understanding what vector space is ... I have just learned what isomorphism is in graphs.
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[Latro]

	Originally Posted by K27 To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	Would it be more comprehensible if I write, ?

I know what you meant, but in the rationals there is no such object. This was the point. It's a little bit philosophical, but "the problem with sqrt(2)" isn't "special" when the only things you have written down are the rationals. It becomes "special" when you throw in, say, the Euclidean geometry axioms. Rather, with the rationals alone, the "problem" that the irrationals introduce is Cauchy sequences without limits, or bounded sets without least upper bounds and greatest lower bounds. In this setting sqrt(2) (and pi, and e, and ...) are just one of an astonishing infinity of little holes.

As for your vector space question, I can rephrase it by saying that if I take away C's multiplication (but leave behind multiplication by reals) and its division, I get R^2.

[Vermillion]

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[Monte314]

I think I might have it in a coffee can back at the dumpster...