Maths, Science and Technology Thread

[CVT Temp]

Okay, I can see that being simpler. I was just basing my answer on a brief perusal of the Wikipedia article.

I haven't had a rigorous introduction to the properties of vector spaces, but that should be coming up soon. We're just ploughing through topology at the moment, moving into multivariable calculus.

I assume you're a graduate student?

I'm a grad student in physics specializing in string theory. However, not only does string theory require a lot of math, but I'm interested enough in mathematics that I've studied a lot of it on my own, just for it's own sake. I especially like abstract algebra and algebraic topology.

What class is it that you're doing all this different stuff in? Is it a math methods class?

[The Joseon Dynasty]

Nope. i, j, k are not in the set of real numbers. I'd write it in matrix form, and just calculate it from there.

I meant right so far.

Oh, I'm sure the algebra is correct. I just wasn't sure if you understood the context of the question.

[The Joseon Dynasty]

Okay, I can see that being simpler. I was just basing my answer on a brief perusal of the Wikipedia article.

I haven't had a rigorous introduction to the properties of vector spaces, but that should be coming up soon. We're just ploughing through topology at the moment, moving into multivariable calculus.

I assume you're a graduate student?

I'm a grad student in physics specializing in string theory. However, not only does string theory require a lot of math, but I'm interested enough in mathematics that I've studied a lot of it on my own, just for it's own sake. I especially like abstract algebra and algebraic topology.

What class is it that you're doing all this different stuff in? Is it a math methods class?

Multivariable Calculus (Analysis), but we're being provided a fairly rigorous introduction to Topology beforehand. Then I'll be taking my second course in Linear Algebra next term, after finishing up Partial Differential Equations.

I'm still being introduced to mathematical topics, so the subjects are a bit all over the place at the moment.

[Typhlochactas]

Pulsars and magnetars?

Pulsars and magnetars.

[CVT Temp]

Pulsars and magnetars?

Pulsars and magnetars.

What about them?

[Typhlochactas]

Pulsars and magnetars?

Pulsars and magnetars.

What about them?

They're fucking cool.

[CVT Temp]

They're fucking cool.

Nah, they're actually rather hot.

[Chinese Regions]

It provides you medicine and computers... and yet it serves you no purpose in life?

I barely use medicine, if I'm sick, tough shit.

And computers, most don't last long anyways.

Probably because you had vaccines as a kid, vaccines discovered by scientists.
You still use them though along with cameras, tvs, phones, games etc.

[Strykla]

Okay, I can see that being simpler. I was just basing my answer on a brief perusal of the Wikipedia article.

I haven't had a rigorous introduction to the properties of vector spaces, but that should be coming up soon. We're just ploughing through topology at the moment, moving into multivariable calculus.

I assume you're a graduate student?

I'm a grad student in physics specializing in string theory. However, not only does string theory require a lot of math, but I'm interested enough in mathematics that I've studied a lot of it on my own, just for it's own sake. I especially like abstract algebra and algebraic topology.

What class is it that you're doing all this different stuff in? Is it a math methods class?

Huh. Yeah, here is the part that I prefer to look at pre-existing work and ignore the math.... As much as I think physics is awesome, I am not a mathematician.

[The New World Oceania]

Okay, I can see that being simpler. I was just basing my answer on a brief perusal of the Wikipedia article.

I haven't had a rigorous introduction to the properties of vector spaces, but that should be coming up soon. We're just ploughing through topology at the moment, moving into multivariable calculus.

I assume you're a graduate student?

I'm a grad student in physics specializing in string theory. However, not only does string theory require a lot of math, but I'm interested enough in mathematics that I've studied a lot of it on my own, just for it's own sake. I especially like abstract algebra and algebraic topology.

What class is it that you're doing all this different stuff in? Is it a math methods class?

Woo hoo, physics! Same path here, though first I just have to trudge through the rest of high school...

From the little I know about string theory, I'm under the impression that it's rather wrong to some degree. I can't make any conclusions until I actually get the chance to study it in depth, though. With the number of people studying it, it's hard to simply say that it's incorrect.

[CVT Temp]

Woo hoo, physics! Same path here, though first I just have to trudge through the rest of high school...

From the little I know about string theory, I'm under the impression that it's rather wrong to some degree. I can't make any conclusions until I actually get the chance to study it in depth, though. With the number of people studying it, it's hard to simply say that it's incorrect.

Out of curiosity, what aspects of it lead you to think that's it's wrong? I might be able to either clear up misunderstandings, or I might be able to agree that those are actual unsolved problems in the discipline.

[CVT Temp]

Bumpulation.

[Azrael]

Well, I'm currently a Physics Honors student and an AP Calculus BC student (I've been two years ahead of my grade in math since middle school).

[The Joseon Dynasty]

Well, I'm currently a Physics Honors student and an AP Calculus BC student (I've been two years ahead of my grade in math since middle school).

I don't know what any of that means. But the curriculum in mathematics in many western countries is needlessly slow. Skipping ahead should be simple for most reasonably adept students.

[AiliailiA]

(1+2i+3j+4k)²
(1+2i+3j+4k)(1+2i+3j+4k) show within brackets
1+2i+3j+4k+2i+4i²+6ij+8ik+3j+6ij+9j²+12jk+4k+8ik+12jk+16k² multiply brackets
1+4i+6j+8k+4i²+12ij+9j²+24jk+16k² group like terms
Did I do that right?

Yeah, except i*j = -j*i = k, j*k = -k*j = i, k*i = -i*k = j, and i² = j² = l² = i*j*k = -1.

Once you use those simplifications, the product is complete.

"Simplifications" ??

It took me a while to accept the usefulness of the imaginary number i. I'm comfortable with it now, but ...

Perhaps you can explain the usefulness of j and k for me ?

[Forsher]

Well, I'm currently a Physics Honors student and an AP Calculus BC student (I've been two years ahead of my grade in math since middle school).

I don't know what any of that means. But the curriculum in mathematics in many western countries is needlessly slow. Skipping ahead should be simple for most reasonably adept students.

I think AP is accelerant.

That said, don't ask me anything other than essentially primary school maths stuff at the moment. As I said, I currently feel incapable of doing anything I am meant to be able to do at the moment.

[The Joseon Dynasty]

Yeah, except i*j = -j*i = k, j*k = -k*j = i, k*i = -i*k = j, and i² = j² = l² = i*j*k = -1.

Once you use those simplifications, the product is complete.

"Simplifications" ??

It took me a while to accept the usefulness of the imaginary number i. I'm comfortable with it now, but ...

Perhaps you can explain the usefulness of j and k for me ?

Simply, i, j and k are the unit vectors in three space, forming the Cartesian co-ordinate system. (1, 0, 0), (0, 1, 0), (0, 0, 1) respectively. Any vector can be written as the sum of these units vectors. For example ai + bj + ck = a(1, 0, 0) + b(0, 1, 0) + c(0, 0, 1) = (a, b, c), for any real a, b, c.

Quaternions are extended into the complex numbers, though (and exist in four space, as a vector space over the real numbers).

[New Rogernomics]

2+2 = 5

[The Joseon Dynasty]

I don't know what any of that means. But the curriculum in mathematics in many western countries is needlessly slow. Skipping ahead should be simple for most reasonably adept students.

I think AP is accelerant.

That said, don't ask me anything other than essentially primary school maths stuff at the moment. As I said, I currently feel incapable of doing anything I am meant to be able to do at the moment.

Is it an American thing?

At my university, there are very, very few students from Western countries pursuing mathematically intensive degrees (engineering partially excepted). Chinese nationals are over-whelmingly the majority.

[Genivaria]

I want my Fusion-powered bombers already.

[Forsher]

I think AP is accelerant.

That said, don't ask me anything other than essentially primary school maths stuff at the moment. As I said, I currently feel incapable of doing anything I am meant to be able to do at the moment.

Is it an American thing?

At my university, there are very, very few students from Western countries pursuing mathematically intensive degrees (engineering partially excepted). Chinese nationals are over-whelmingly the majority.

It's American.

We call it MAA at my school. It means Advanced Maths and the only difference is we do one more external (a standard/paper assessed in the end of year exams) at a higher level. As far as I can tell it's essentially the same thing.

My school is 14% Asian. My maths class, even assuming there are 32 students (there are at least 24 and 32 is pretty much the maximum), has far more than that with six (restricted to Chinese-type) or nine (with Indian-type) included.

The comparison isn't quite the same as by Asian it's ethnicity as opposed to "Chinese nationals" but they are, for the most part, raised in the same sort of climate.

[CVT Temp]

"Simplifications" ??

It took me a while to accept the usefulness of the imaginary number i. I'm comfortable with it now, but ...

Perhaps you can explain the usefulness of j and k for me ?

The algebra of quaternions with unit length is the same algebra as that of three dimensional rotations. The Lie algebras are the same, but the Lie groups are different, as every three dimensional rotation corresponds to two different unit quaternions. This is why the group SU(2), the unit quaternions, is said to be the "double cover" of SO(3), the group of three dimensional rotations. They are both representations of the same Lie algebra.

Basically, Hamilton discovered the quaternions before the ideas of vectors or linear transformations had been discovered. Thus, he formulated three dimensional physics using quaternions in order to account for rotations. All you have to do is remove the double-counting nature of the quaternions and you can recover normal 3D rotations.

Also, some physical phenomena in nature actually transform under quaternion rotations instead of ordinary rotations. These objects which transform under SU(2) rotations are the spin 1/2 particles such as electrons and quarks.

[CVT Temp]

In a less abstract sense, the product of two quaternions contains both dot and cross products in it, so all the 3D vector calculus stuff can be hidden in there.

[Azrael]

Well, I'm currently a Physics Honors student and an AP Calculus BC student (I've been two years ahead of my grade in math since middle school).

I don't know what any of that means. But the curriculum in mathematics in many western countries is needlessly slow. Skipping ahead should be simple for most reasonably adept students.

http://en.wikipedia.org/wiki/Advanced_Placement

Basically, it's college-level courses offered to high school students in the United States.

[Birkinghamia]

"Simplifications" ??

It took me a while to accept the usefulness of the imaginary number i. I'm comfortable with it now, but ...

Perhaps you can explain the usefulness of j and k for me ?

Simply, i, j and k are the unit vectors in three space, forming the Cartesian co-ordinate system. (1, 0, 0), (0, 1, 0), (0, 0, 1) respectively. Any vector can be written as the sum of these units vectors. For example ai + bj + ck = a(1, 0, 0) + b(0, 1, 0) + c(0, 0, 1) = (a, b, c), for any real a, b, c.

Quaternions are extended into the complex numbers, though (and exist in four space, as a vector space over the real numbers).

In which class do you learn about this? Calculus?