2028: "Complex Numbers"
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2028: "Complex Numbers"
Title text: I'm trying to prove that mathematics forms a metaabelian group, which would finally confirm my suspicions that algebreic geometry and geometric algebra are the same thing.
pfft… that's with vectors? Are scientists and engineers too cool for regular quaternions?
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 GlassHouses
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Re: 2028: "Complex Numbers"
edo wrote:Are scientists and engineers too cool for regular quaternions?
Absolutely. The really cool kids are into octonions.
Re: 2028: "Complex Numbers"
GlassHouses wrote:Absolutely. The really cool kids are into octonions.
I've been saying for a while, that if a fifth dimension is discovered, that there will be at least 8 total, if for no other reason that the math describes it (there's a squareroot in several physics equations, your going to need a division math to solve it)
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Re: 2028: "Complex Numbers"
tehre was a recent new story about how the staandard model vould maybe be reforumated using octonions instead of matrices
Forrest Cameranesi, Geek of All Trades
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Re: 2028: "Complex Numbers"
What are all the differences between ℝ2 and ℂ?
Also complex numbers are cooler
 First the complex plane is a connected set whereas two dimensional real numbers are not.
 Second, the complex plane is capable of definite integration for systems which don't have definite solutions using only real numbers.
 Third, complex numbers have quasi ordered systems like the mandelbrot set which don't exist in in ℝ2.
Also complex numbers are cooler
Re: 2028: "Complex Numbers"
here's that news story i mentioned
https://www.quantamagazine.org/theocto ... 20180720/
Moonfish, cannot any ordered pair of reals be mapped onetoone to a complex number just by turning (x,y) into (x + yi)? With such a onetoone mapping, how are the sets not identical, i.e. from whence come the differences you describe?
https://www.quantamagazine.org/theocto ... 20180720/
Moonfish, cannot any ordered pair of reals be mapped onetoone to a complex number just by turning (x,y) into (x + yi)? With such a onetoone mapping, how are the sets not identical, i.e. from whence come the differences you describe?
Forrest Cameranesi, Geek of All Trades
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Re: 2028: "Complex Numbers"
Pfhorrest wrote:Moonfish, cannot any ordered pair of reals be mapped onetoone to a complex number just by turning (x,y) into (x + yi)? With such a onetoone mapping, how are the sets not identical, i.e. from whence come the differences you describe?
Complex numbers have arithmetic operations defined on them that differ from the arithmetic operations defined on RxR.
(1,0)*(0,1) doesn't have a standard interpretation, while 1*i = i. You can take the set RxR and add operations to it to get something isomorphic with C, but they don't emerge naturally from treating it as a set of vectors.

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Re: 2028: "Complex Numbers"
So paired reals are more general than complex numbers?
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Re: 2028: "Complex Numbers"
If I'm not mistaken, the multiplication operation means that complex numbers are not vectors. The product of two complex numbers has an angle which is the sum of the angles of the the two original numbers relative to a fixed frame of reference. Rotating the reference frame gives a different result. I believe it's part of what vectors are that vector operations have to be invariant under a change of axes.
Is there any multiplication operation defined on a 2vector that yields a 2vector as a result? (Also can anyone point me at a good explanation of tensors that might make sense to an engineer?)
Is there any multiplication operation defined on a 2vector that yields a 2vector as a result? (Also can anyone point me at a good explanation of tensors that might make sense to an engineer?)
xtifr wrote:... and orthogon merely sounds undecided.
 MartianInvader
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Re: 2028: "Complex Numbers"
Moonfish wrote:What are all the differences between ℝ2 and ℂ?
 First the complex plane is a connected set whereas two dimensional real numbers are not.
 Second, the complex plane is capable of definite integration for systems which don't have definite solutions using only real numbers.
 Third, complex numbers have quasi ordered systems like the mandelbrot set which don't exist in in ℝ2.
I'm not even sure what you mean by the second two, but the twodimensional reals are definitely connected.
The real difference between the two is that the complex numbers have a multiplication defined on them which makes them into an algebraicallyclosed field.
Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!
Re: 2028: "Complex Numbers"
Yes, but what's the imaginary difference?MartianInvader wrote:The real difference between the two is that the complex numbers have a multiplication defined on them which makes them into an algebraicallyclosed field.
Jose
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Re: 2028: "Complex Numbers"
I just came here to point out that metaabelian groups are already a thing, and it's not an abelian group that's so, like, meta.
Also:
Which topology are you placing on R^2? The product topology? The standard topology given by open balls? (These are the same.) This is connected, as if x and y lie in X and Y (R^2 is the disjoint union of X and Y), then draw a path form x to y: this starts in X and ends in Y, so must cross from X to Y at some point. Since X and Y are both closed, they contain their limit points, whence this crossing from X to Y must lie in both X and Y, as it is a limit point of both.
Also:
Moonfish wrote:First the complex plane is a connected set whereas two dimensional real numbers are not.
Which topology are you placing on R^2? The product topology? The standard topology given by open balls? (These are the same.) This is connected, as if x and y lie in X and Y (R^2 is the disjoint union of X and Y), then draw a path form x to y: this starts in X and ends in Y, so must cross from X to Y at some point. Since X and Y are both closed, they contain their limit points, whence this crossing from X to Y must lie in both X and Y, as it is a limit point of both.
 Soupspoon
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Re: 2028: "Complex Numbers"
Have you tried logarithms?
Re: 2028: "Complex Numbers"
Do these problems continue in higher dimensions, eg is an ordered pair of complexes not enough to make a quaternion?
Forrest Cameranesi, Geek of All Trades
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Re: 2028: "Complex Numbers"
Seems like some people in this thread aren't quite familiar with the idea of a "structured set". Statements like "the multiplication operation means that complex numbers are not vectors" are somewhat missing the point. The complex numbers are R² with a multiplication operator defined on it. It's the operations (the structure) that sit alongside the underlying set that matter, not the set itself (the set R²).
Again, this isn't a meaningful question because the set doesn't matter. So you have this set CxC (the set of ordered pairs of complex numbers). You can slap an operation on that which is basically quaternion multiplication, and bang, you've got, basically, the quaternions. I mean listen, as long as the cardinality is right, I can make any set into the quaternions. I could take the set of real numbers and make it into the quaternions by putting the right operation onto it. That is to say, I could define a pair of operations on R such that the resulting structure would be isomorphic to the quaternions. The set is just a pile of meaningless labels for the elements, what matters are the operations.
Anyway, I've definitely had this exact thought before. Unless you're using complex multiplication somewhere, calling a vector a "complex number" is just muddying the waters. People do this with matrices, too... often, a "matrix" is just a grid of numbers. If you're not planning on doing matrix multiplication or taking a determinant or something, just call it a grid.
Pfhorrest wrote:Do these problems continue in higher dimensions, eg is an ordered pair of complexes not enough to make a quaternion?
Again, this isn't a meaningful question because the set doesn't matter. So you have this set CxC (the set of ordered pairs of complex numbers). You can slap an operation on that which is basically quaternion multiplication, and bang, you've got, basically, the quaternions. I mean listen, as long as the cardinality is right, I can make any set into the quaternions. I could take the set of real numbers and make it into the quaternions by putting the right operation onto it. That is to say, I could define a pair of operations on R such that the resulting structure would be isomorphic to the quaternions. The set is just a pile of meaningless labels for the elements, what matters are the operations.
Anyway, I've definitely had this exact thought before. Unless you're using complex multiplication somewhere, calling a vector a "complex number" is just muddying the waters. People do this with matrices, too... often, a "matrix" is just a grid of numbers. If you're not planning on doing matrix multiplication or taking a determinant or something, just call it a grid.
Re: 2028: "Complex Numbers"
Soupspoon wrote:Have you tried logarithms?
This may be the only thing about this cartoon (and all the comments) that I understand, just can't recall the original reference. So I'll give you an internets for it.
Re: 2028: "Complex Numbers"
Old Bruce wrote:Soupspoon wrote:Have you tried logarithms?
This may be the only thing about this cartoon (and all the comments) that I understand, just can't recall the original reference. So I'll give you an internets for it.
Forrest Cameranesi, Geek of All Trades
"I am Sam. Sam I am. I do not like trolls, flames, or spam."
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Re: 2028: "Complex Numbers"
orthogon wrote:If I'm not mistaken, the multiplication operation means that complex numbers are not vectors.
To be a vector space over a field like R, there needs to be a form of scalar multiplication. This definitely applies to the complex numbers; you can multiply by a real number, and it works exactly like R^{2}. So from a vector space perspective they are the same thing as R^{2}.
But as said they are also something much more, a field, because there is also a multiplication operation for things that aren't scalars. In most contexts asking if they're just vectors wouldn't make any sense. Still I have seen them used in cases where you basically only do care about R^{2}, like for 2D graphics, which is kind of annoying because they have no 3D equivalent so they're obviously not what you mean.
Gaiajack wrote:If you're not planning on doing matrix multiplication or taking a determinant or something, just call it a grid.
How does calling it a "rank 2 tensor" sound?
Re: 2028: "Complex Numbers"
Imaginary numbers are pretty theoretical until you get to calculating laglead times on power transmission. Then they're a huge pain in the ass.
Re: 2028: "Complex Numbers"
chenille wrote:Still I have seen them used in cases where you basically only do care about R^{2}, like for 2D graphics, which is kind of annoying because they have no 3D equivalent so they're obviously not what you mean.
Well, if you need to rotate an object in 2D then it's a bit of a pain, unless you write R^{2}=C, and then it's just a multiplication by e^{iθ}.
Re: 2028: "Complex Numbers"
Gaiajack wrote:Seems like some people in this thread aren't quite familiar with the idea of a "structured set". Statements like "the multiplication operation means that complex numbers are not vectors" are somewhat missing the point. The complex numbers are R² with a multiplication operator defined on it. It's the operations (the structure) that sit alongside the underlying set that matter, not the set itself (the set R²).
It was me who said that, and I'm not getting how it's missing the point. You say "it's the operations that sit alongside the set that matter", and yet the statement you quote refers specifically to the operation. I can only make sense of your statement if you're saying that R² is the same thing as the 2vectors. I'm saying that vectors are also more than just a pair of reals: when represented as such, the components have to co or contravary in just the right way with a change of axes, and the multiplication operation on the complexes disqualifies them from being 2vectors.
xtifr wrote:... and orthogon merely sounds undecided.
Re: 2028: "Complex Numbers"
DavCrav wrote:chenille wrote:Still I have seen them used in cases where you basically only do care about R^{2}, like for 2D graphics, which is kind of annoying because they have no 3D equivalent so they're obviously not what you mean.
Well, if you need to rotate an object in 2D then it's a bit of a pain, unless you write R^{2}=C, and then it's just a multiplication by e^{iθ}.
It's not clear whether it's easier to multiply a complex number by cosθ + i sinθ or a vector by {{cosθ sinθ} {sinθ cosθ}}.
Re: 2028: "Complex Numbers"
Dylan wrote:Imaginary numbers are pretty theoretical until you get to calculating laglead times on power transmission. Then they're a huge pain in the ass.
They're still easier than being stubborn and not using complex numbers to solve ODEs at all. In that case, I hope you like trig identities!

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Re: 2028: "Complex Numbers"
Complex numbers do have a 3dimensional analog for rotations. Quaternions, where the imaginary part is the 3d part.
More generally, you can use clifford algebras to get the generic Ndimensional generalization of complex numbers and quaternions, in a way which is a bit less adhoc. You can view complex numbers as the even part of the clifford algebra generated by R^2 and the usual metric, and quaternions as the clifford algebra generated by R^3 with its usual metric.
Octonions are a completely different beast since they aren't associative, and are related to the exceptional lie groups instead of the orthogonal group.
More generally, you can use clifford algebras to get the generic Ndimensional generalization of complex numbers and quaternions, in a way which is a bit less adhoc. You can view complex numbers as the even part of the clifford algebra generated by R^2 and the usual metric, and quaternions as the clifford algebra generated by R^3 with its usual metric.
Octonions are a completely different beast since they aren't associative, and are related to the exceptional lie groups instead of the orthogonal group.
Re: 2028: "Complex Numbers"
So as MartianInvader and DavCrav point out: Two dimensional real space is connected.
I didn't know that.
I'm a programmer, not a mathematician. I find imaginary numbers fascinating, probably too fascinating.
A lot of the time I'll see a problem, usually in statistics, and think maybe there is a better solution if I allow complex numbers.
Complex numbers are too mysterious and alluring sometimes.
I didn't know that.
MartianInvader wrote:I'm not even sure what you mean by the second two, but the twodimensional reals are definitely connected.
The real difference between the two is that the complex numbers have a multiplication defined on them which makes them into an algebraicallyclosed field.
DavCrav wrote:Which topology are you placing on R^2? The product topology? The standard topology given by open balls? (These are the same.) This is connected, as if x and y lie in X and Y (R^2 is the disjoint union of X and Y), then draw a path form x to y: this starts in X and ends in Y, so must cross from X to Y at some point. Since X and Y are both closed, they contain their limit points, whence this crossing from X to Y must lie in both X and Y, as it is a limit point of both.
I'm a programmer, not a mathematician. I find imaginary numbers fascinating, probably too fascinating.
A lot of the time I'll see a problem, usually in statistics, and think maybe there is a better solution if I allow complex numbers.
Complex numbers are too mysterious and alluring sometimes.
Re: 2028: "Complex Numbers"
Dylan wrote:Imaginary numbers are pretty theoretical until you get to calculating laglead times on power transmission. Then they're a huge pain in the ass.
Well, one can go way down the tree that way.
Irrationals are theoretical, since we can't write them out in any finite decimal notation.
Integers are theoretical: you can't show me "two." You can show me two objects, and you can make a symbol that represents "two" (Hint: try "2" ), but "two" is just a concept. [feel free to use that argument to troll any metamath discussion you like]
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Re: 2028: "Complex Numbers"
cellocgw wrote:Dylan wrote:Imaginary numbers are pretty theoretical until you get to calculating laglead times on power transmission. Then they're a huge pain in the ass.
Well, one can go way down the tree that way.
Irrationals are theoretical, since we can't write them out in any finite decimal notation.
Integers are theoretical: you can't show me "two." You can show me two objects, and you can make a symbol that represents "two" (Hint: try "2" ), but "two" is just a concept. [feel free to use that argument to troll any metamath discussion you like]
Yeah, one of the biggest leaps in the entire history of mathematics  one of the weirdest ideas ever introduced  is the whole concept of "counting".
Of course, another is the idea of negative numbers. Once you have negative numbers, complex numbers are a fairly small step...
Re: 2028: "Complex Numbers"
rmsgrey wrote:Of course, another is the idea of negative numbers. Once you have negative numbers, complex numbers are a fairly small step...
Yeah ... negative numbers have to be introduced in order for all linear equations to have solutions. Complex numbers are needed for quadratics to have solutions, but weirdly they do the job for any order of polynomial: you don't need a new type of number for cubics, etc. What's that all about?
Pseudoedit: I suppose the difference isn't between linear and quadratic, but between addition and multiplication, in some way.
xtifr wrote:... and orthogon merely sounds undecided.
Re: 2028: "Complex Numbers"
You know, I've been watching xkcd for like 10 years. I gave up on Simpsons around season 11, never looked back. Simpsons should have died then. When it was great. When there are these delays in introducing a new comic I think maybe xkcd is finally done. Then... a new one appears.
When I think maybe the comic is over I feel a sense of relief. The need to see what appears every Monday, Wednesday, Friday. It's work. I know in my heart I'd be glad to be relieved of this burden.
Sometimes the new comic is a delight. Much more often I have to face the reality. It's just tired. Randall is out of ideas. No one can keep going forever. Maybe it's time to retire before the quality is completely gone.
Thank you Randall for XKCD. It has been a huge part of my life for a long time, in ways difficult to even quantify. If you need to move on, do so with my blessing.
When I think maybe the comic is over I feel a sense of relief. The need to see what appears every Monday, Wednesday, Friday. It's work. I know in my heart I'd be glad to be relieved of this burden.
Sometimes the new comic is a delight. Much more often I have to face the reality. It's just tired. Randall is out of ideas. No one can keep going forever. Maybe it's time to retire before the quality is completely gone.
Thank you Randall for XKCD. It has been a huge part of my life for a long time, in ways difficult to even quantify. If you need to move on, do so with my blessing.
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 GlassHouses
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Re: 2028: "Complex Numbers"
orthogon wrote:rmsgrey wrote:Of course, another is the idea of negative numbers. Once you have negative numbers, complex numbers are a fairly small step...
Yeah ... negative numbers have to be introduced in order for all linear equations to have solutions. Complex numbers are needed for quadratics to have solutions, but weirdly they do the job for any order of polynomial: you don't need a new type of number for cubics, etc. What's that all about?
Pseudoedit: I suppose the difference isn't between linear and quadratic, but between addition and multiplication, in some way.
Starting out from the "counting numbers," i.e. the natural numbers, you need to add negative numbers to N in order to obtain a set that is closed under subtraction. Next, you need to add rational numbers in order to obtain a set that is closed under division (except for division by zero).
In order to extend the set to be closed under the operation of finding real roots of polynomials with integer coefficients, you need to add the algebraic numbers, and I guess if you want the set to be closed under the operation of finding all roots of polynomials with integer coefficients, you'll end up with {algebraic numbers}^2.
The discovery of transcendental numbers meant that even the set of algebraic numbers didn't include "all" numbers, and R is designed to fix that once and for all, although formally defining it is not trivial.
Re: 2028: "Complex Numbers"
GlassHouses wrote:orthogon wrote:rmsgrey wrote:Of course, another is the idea of negative numbers. Once you have negative numbers, complex numbers are a fairly small step...
Yeah ... negative numbers have to be introduced in order for all linear equations to have solutions. Complex numbers are needed for quadratics to have solutions, but weirdly they do the job for any order of polynomial: you don't need a new type of number for cubics, etc. What's that all about?
Pseudoedit: I suppose the difference isn't between linear and quadratic, but between addition and multiplication, in some way.
Starting out from the "counting numbers," i.e. the natural numbers, you need to add negative numbers to N in order to obtain a set that is closed under subtraction. Next, you need to add rational numbers in order to obtain a set that is closed under division (except for division by zero).
In order to extend the set to be closed under the operation of finding real roots of polynomials with integer coefficients, you need to add the algebraic numbers, and I guess if you want the set to be closed under the operation of finding all roots of polynomials with integer coefficients, you'll end up with {algebraic numbers}^2.
The discovery of transcendental numbers meant that even the set of algebraic numbers didn't include "all" numbers, and R is designed to fix that once and for all, although formally defining it is not trivial.
Rational numbers are at a similar level of abstraction to natural numbers  it's easy to find/create half apples or quarter circles, and generally dividing things among groups is a common experience. Negative numbers and imaginary numbers are both much harder to find realworld examples of. Real numbers are also out there as soon as you start trying to deal with anything continuous...
Re: 2028: "Complex Numbers"
rmsgrey wrote:Of course, another is the idea of negative numbers. Once you have negative numbers, complex numbers are a fairly small step...
Oddly enough, the historical order was the opposite of that.
Mathematicians used negative numbers for a long while, but they weren't considered to be proper numbers. They were just a bookkeeping trick that simplified calculation and algebra. (Mind you, at that stage, the convention of using single letter variables hadn't been established, people used words instead, so algebra was rather verbose, but that's another story).
One of the barriers to their acceptance was that it seemed nonsensical to extract the square root of a negative quantity. That was kind of ok: people had been solving quadratics for ages, and it was geometrically obvious that quadratics with negative discriminant didn't have a solution. But it meant that algebra was treated with suspicion, and you could only trust proofs that were done geometrically. (That attitude was still prevalent in Newton's day, hence all the proofs in the Principia are done geometrically).
Negative numbers still had that status when people were figuring out how to solve cubic equations, and a few rather clever mathematicians discovered that you could do useful tricks by working with the square root of negative one. With cubics, you can't just dismiss the complex terms as invalid like you can with quadratics. The general algebraic techniques for solving any cubic involve intermediate complex terms that are unavoidable when the cubic has 3 real solutions.
This caused some consternation in the small circle of mathematians privy to these techniques. (The method of solving cubics was kept semisecret for quite a few years: you could make good money back then by solving algebra challenges). But eventually the word got around and gradually mathematicians came to accept complex numbers as valid entities, although they certainly weren't proper numbers.
After a few more centuries, familiarity made them more acceptable, and they were obviously useful. There was increasing pressure to consider them as legitimate numbers. But to do that, negative numbers would also have to be legitimized. And so that's what happened: the usefulness of complex numbers forced mathematicians to accept that negative numbers were more than just a bookkeeping device.
I've simplified things a little in the above account. Wikipedia has a good introduction to the history of the cubic and the colourful characters associated with it.
Re: 2028: "Complex Numbers"
But that's what numbers are. The lot of them!PM 2Ring wrote:Mathematicians used negative numbers for a long while, but they weren't considered to be proper numbers. They were just a bookkeeping trick that simplified calculation and algebra.
But you can't create half a piece of paper.rmsgrey wrote:it's easy to find/create half apples or quarter circles
Jose
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Re: 2028: "Complex Numbers"
ucim wrote:But that's what numbers are. The lot of them!PM 2Ring wrote:Mathematicians used negative numbers for a long while, but they weren't considered to be proper numbers. They were just a bookkeeping trick that simplified calculation and algebra.
Jose
Well, sure. But that's a fairly modern notion, and it took a lot of mathematical development before we could get to that position.
Re: 2028: "Complex Numbers"
I was counting on somebody to get to the root of that complex concept, but only in my imagination.PM 2Ring wrote:Well, sure. But that's a fairly modern notion, and it took a lot of mathematical development before we could get to that position.
Jose
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Re: 2028: "Complex Numbers"
If you want to get into it, for quite a while, one wasn't a number either. Or, alternatively, people talked about "mutliplicities" where we would talk about numbers  it's a question of translation and whether you translate the term used as "number" because it's used where we would use "number" in English, or as "multiplicity" (or some other term) because it only applies to plural values, not to 1 (nor fractions, nonpositive values, or anything else).
And "zero" as a definite quantity rather than as an absence (I would refer to it as a "positive quantity" rather than a "negative" but that invites ambiguity...) has its own storied history.
And "zero" as a definite quantity rather than as an absence (I would refer to it as a "positive quantity" rather than a "negative" but that invites ambiguity...) has its own storied history.
Re: 2028: "Complex Numbers"
So you've got nothing, which is not, initially, seen as a number.
Then something, which is, likewise, not seen as a number yet.
Then multiplicities, the first real "numbers", which are the products of adding several somethings together.
Then comes the thought of what do you call it when you don't have any multiplicity of something; you just have the, er, the one, thing, itself. The multiplicative identity, the product of not multiplying at all.
Then comes the thought of what do you call it when you don't add anything at all; you just have nothing, zero, the additive identity.
And now we finally have the counting numbers.
Then something, which is, likewise, not seen as a number yet.
Then multiplicities, the first real "numbers", which are the products of adding several somethings together.
Then comes the thought of what do you call it when you don't have any multiplicity of something; you just have the, er, the one, thing, itself. The multiplicative identity, the product of not multiplying at all.
Then comes the thought of what do you call it when you don't add anything at all; you just have nothing, zero, the additive identity.
And now we finally have the counting numbers.
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 GlassHouses
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Re: 2028: "Complex Numbers"
rmsgrey wrote:Real numbers are also out there as soon as you start trying to deal with anything continuous...
The real numbers were always out there, of course, but it took a long time before anyone started to realize that not all nonintegers are rational, i.e. that irrational numbers exist, and it took even longer to figure out that not all irrational numbers are algebraic, i.e. that transcendental numbers exist.
Re: 2028: "Complex Numbers"
orthogon wrote:If I'm not mistaken, the multiplication operation means that complex numbers are not vectors.
R^{2} tuples are not vectors either. They are coordinates.
orthogon wrote:he product of two complex numbers has an angle which is the sum of the angles of the the two original numbers relative to a fixed frame of reference. Rotating the reference frame gives a different result. I believe it's part of what vectors are that vector operations have to be invariant under a change of axes.
A linear operation on 2 vectors that returns yet another vector is a rank 3 tensor (or pseudotensor). Tensor components are generally not true scalars in the sense of coordinate transformation; they change according to tensor transformation laws. A 2dimensional rank 3 tensor (or pseudotensor) that correspond to complex multiplication in one coordinate system will not correspond to complex multiplication with another choice of basis vectors e_{1} and e_{i}.
orthogon wrote:(Also can anyone point me at a good explanation of tensors that might make sense to an engineer?)
This is surprisingly not bad: https://en.wikipedia.org/wiki/Hooke%27s_law
Re: 2028: "Complex Numbers"
Pfhorrest wrote:Do these problems continue in higher dimensions, eg is an ordered pair of complexes not enough to make a quaternion?
kind of, and kind of exactly that
if you say for example that q= (1,i)*(1,j)= (1+i)*(1+j) = 1+i+j+ij
= 1+i+j+k
and there you go
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