/sci/ - Science & Math

Creating models

>>6617355
WOW!!! My eyes have opened to a whole new world I've never thought about before! Wow, thank you!!

>>6617349
math isn't about numbers, it's about structure, how structures relate to each other, and how structures change with respect to each other
also if you're into logic and stuff it's about formal systems

>>6617349
>Calculus studies the rate of change of perhaps arithmetic
What? Calculus is about two operators on functions (differentiation and integration) and their relation. I'm really curious: where do you see arithmetic when you take a derivative? Where do you see arithmetic when you take an integral? (Aside, of course, from the coincidental "oh and 3*2 is 6 so write 6x instead of 3*2*x")

>>6617349

>I do not notice any other problems that exist beyond that

Uhhh, how about proving Calculus and making it rigorous?

Differential equations arise from Calculus

Abstract algebra is used for the grounding of Calculus

How about the problem of probability? Need a whole branch for that.

You seriously don't see the need for more mathematics beyond Calculus? You do realize that you don't even know what Calculus is, you're just plugging and chugging, right?

>>6617349
I am a mathematician. My two central areas of investigation are Symmetries and the relationship between Discrete and Infinite. Symmetries are best described using numbers, as they are very basic "thought units", distinguishable from one another in lots of useful ways but fundamentally similar, from which you can create more complex patterns. As for the discrete/infinite question, it is often a triumph in mathematics if you can take something infinite and "count" it; revealing a discrete operation within a continuous, infinite model can have great explanatory power.

The real question that fascinates me is low-dimension geometry. That is, there are lots of incredibly odd and interesting things that can happen in low-dimensional space that do not have higher-dimensional analogues. For example, there is no such thing as "left-handed" and "right-handed" in dimensions other than 2, 3 or 7. Given how much the handedness of everything adds to the richness of interactions, from molecules to the laws of physics, perhaps this goes some way of explaining why the universe we experience is 3D.

Anyway, mathematics is a lot like science, in that a little knowledge of high practical use is usually enough for most people, but nerds will obsess over the details.

>>6617889
chirality, correct? what is the determining factor if n-dimensions exhibit chirality?

I make sweet, sensual love to numbers.

>>6617938
Oh god, I didn't realize how much I need an answer to this.

>>6617938
The answer is a bit complicated, but I'll give it a go.

Represent 1D as the number line. We can talk about the "size" of a number <span class="math">x\in R[/spoiler] using the absolute value. Crucially, this "size" commutes with multiplication:
<span class="math">|a|\cdot |b| = | a\cdot b |[/spoiler]
If we move to 2D and the complex numbers, we have a similar way to talk about "size":
<span class="math">|a+ib|=\sqrt{a^2+b^2}[/spoiler]
This size function ALSO commutes with multiplication, and produces a REAL number answer.
If we try to do the same thing with 3D, we fail. We cannot describe a "size" function that commutes with multiplication producing a real number, no matter what function we choose.
[Note: if you try to prove the complex case, you will note that the property <span class="math">i^2=-1[/spoiler] is very important. Even if you make a new i and j that square to different values in 3D, the problem remains.]
But in 4D, there is one: this corresponds to the size function on the quaternions, and is essentially the complex numbers "doubled". We can "double" again to make 8D work, but after that any attempts at "doubling" fail to work - the octonions are NOT associative, and doubling relies on the associativity of the original number field.
In 3D, if you consider two non-parallel arrows of any length and join them at their tails, there is a single direction parallel to both of them. If you want to produce a third arrow from these two along that direction, it seems like you have two choices - if the arrows are lying in the plane of your monitor, there is the arrow pointing at you and the arrow pointing away from you. You can make the *consistent* choice to always pick the "right-handed" version - if you do, you can now produce a specific third arrow perpendicular to the first two in any given situation.
In 4D, picking two directions means you have 2D worth of "perpendicular directions" to choose from (as opposed to just the 1D line we had before).
>cont

>>6618046
...but there is no way to pick just one direction out of these 2D possibilities in a consistent fashion.

What do these two things have in common? Well, if you *can* make such a consistent choice, you can use it to define a "size" function like the one I described on the (n+1)D space.

In practice, we model 3D using i, j and k that satisfy
ij = k, jk=i, ki=j
uv=-vu
This "multiplication" of 3D basis elements allows us to define a "size" on 4D arithmetic by saying it is spanned by 1, i, j and k with the property that i^2=j^2=k^2=-1.

In short,
Chirality in n => size function on n+1
size functions exist only for n=0,1,3,7
therefore chirality exists exist only for n=0,1,3,7.

A crude sketch; but this is only the beginning. Another example of low-dimensional weirdness are the platonic solids.
For every n, there is an nD analogue of the tetrahedron, cube and octohedron.
For 2D, there is a "platonic solid" for each number of sides of a regular figure (triangle, square, pentagon, hexagon, ...)
But for n=3,4 there are additional weird ones. In 3D there are the icosahedron and dodecahedron (d12 and d20). Each of these have 4D analogues, but there are none in 5D and above.
Also, 4D has a weird one called the 24-cell that has no analogue in any other dimension.
Just look at cellular life and you will see d12s and d20s everywhere. The geometrical richness of 3D and 4D (in a weak anthropology way) part of the reason the universe is the way it is and why we experience it the way we do.

>>6618046
but you can only make that choice for n many pairs of lines, because aoc is for shitters.

>>6617766
I have such a hard time believing complex numbers have any relevance at all.

>>6618079
>complex analysis is fundamental in circuit design and analysis. explaining electrical phenomena requires complex analysis and it's insights.

>>6617779
Well, the rate of change of acceleration seems pretty clear and concrete to me. As time approaches zero, the rate of change at any given moment is more precise.

I;ve even derived a derivative. I get it. I just don't see any problems beyond that primitive knowledge of calculus

>>6618083
What empirical aspect of circuit design and analysis renders complex numbers useful?

>>6618086
that they exist

>>6618088
Well, from a newbie standpoint, a voltage is what moves the electricity from point a to point b along a conductor. It's that "want" from the polar opposite charges in a battery or other source that create this electricity. How can you manipulate this simple phenomenon abstractly to render complex numbers useful?

>>6618068
Hmmm, not sure what you mean.
>aoc
>wut

>>6618091
Voltage is all you need if you are modelling DC.
When modelling AC, you have the extra variables "frequency" and "phase shift"
You can model these easily using complex numbers
http://en.wikipedia.org/wiki/Electrical_impedance

>>6618091
Ok then, consider a circuit composed of a capacitor, a resistor, and a power source delivering an AC current at 60 Hz.

Solve this differential equation assuming there is no such thing as complex numbers and get back to me.

>>6618107
(the important thing here is that it is AC)
(if you want to answer this guy then you need to know the frequency too)
(use either 50Hz or 60Hz, it will be one of those depending on where you live)

>>6618133
hnnnnnnnnng
for some reason I read the original as
>AC current at 60V
please disregard my shitpost

>>6618046
thanks mate. great explanation. gives me a lot to think about.

>>6618079
they are fundamental in all of algebra. they are the algebraic closure of the reals, which allows for the "fundamental theorem of algebra", and most of the major components of algebraic geometry. the multivariate version of FTA falls from the nullstellensatz, one of the core findings which drives classical AG. people who think "i" is useless maybe had a chance 200 years ago, but that is a completely pointless position now.

>>6617349

btw pic is subway in Moscow, Russia
fun times
;_;

>>6618079
See
>>6618177

But also google the topic of what happens when you raise an exponential to an imaginary or complex power.