TheAspiringDoc's Math Thread

[keltingmeith]

The probability unit of methods 3/4 is pretty applicable, and spesh is getting some probability from next year.

Hah, nice joke.

[GeniDoi]

Hah, nice joke.

Cmon now, all chickens secretly strategize to lay eggs whose diameter conforms to a normal distribution curve

[MathsGuru]

...
If you get done with that, and you're done with the VCE maths courses, look up some other area of maths that interests you. For instance, linear algebra, group theory, multivariate calculus, differential equations, real analysis...there's so much out there to learn.
...

If you want a nice bird's eye view of a whole swathe of maths, so you can get a feel for what areas inspire you the most, Norman Wildberger has a really cool series of videos (including university lectures) on the history of maths, a lot of which should be reasonably accessible to a sharp high school student :-)
https://www.youtube.com/playlist?list=PL55C7C83781CF4316

[Ssuper19]

If you want a nice bird's eye view of a whole swathe of maths, so you can get a feel for what areas inspire you the most, Norman Wildberger has a really cool series of videos (including university lectures) on the history of maths, a lot of which should be reasonably accessible to a sharp high school student :-)
https://www.youtube.com/playlist?list=PL55C7C83781CF4316

Great YouTube playlist. Really goes into the depth of the history of maths. Thanks

[TheAspiringDoc]

Hello
How do you find the equation of a line (a parabola) that passes through the x-axis at -2 and 4, and the y-axis at 24?
Thanks

[TheMereCat]

Start with y = a(x-b)(x-c) as the vertex equation form of a parabola

Where a is some constant, b, c and are the x-intercepts. So you have the equation, y= a(X+2)(X-4)

Now plug in (0,24) into the equation:

24= a(0+2)(0-4)

24=a(2)(-4)

24=-8a

a=-3

You can now express the equation as Y= -3(X-4)(X+2) or
-3(X^2+2x-4x-8)
y=-3x^2+2x+24

[TheAspiringDoc]

Does anyone know of some nice ways to tie chemistry and maths in together?
Things like pH involves logarithms and then I believe there's some differentiation involved in chem as well?
Thanks!

[GeniDoi]

Does anyone know of some nice ways to tie chemistry and maths in together?
Things like pH involves logarithms and then I believe there's some differentiation involved in chem as well?
Thanks!

Yup, there's a pretty tricky differential equation question in essential specialist maths 3/4:

Two chemicals, A and B, are put together in a solution where they react to form a compound, X. The rate of increase of the mass, x kg, of X is proportional to the product of the masses of unreacted A and B present at time t minutes. It takes 1 kg of A and 3 kg ofB to form 4 kg of X. Initially 2 kg of A and 3 kg ofB are put together in solution. One kg of X forms in one minute.

a) Set up the appropriate differential equation expressing dx/dt as a function of x.
b) Solve the differential equation.
c) Find the time taken to form 2 kg of X. d Find the mass of X formed after two minutes.


Our teacher did it with us in class and sent us a copy of the working, I've attached it here. According to him, definitely something that wouldn't come up in a vcaa specialist exam as it's too reliant on chemistry knowledge, but it's a good question that involves many elements to solve it like partial fractions and integrating logs. The working out ends where it does because from there it's a simple integration problem.

[lzxnl]

Does anyone know of some nice ways to tie chemistry and maths in together?
Things like pH involves logarithms and then I believe there's some differentiation involved in chem as well?
Thanks!

Here are many of the ways I can imagine tying chemistry and maths together.

1. Quantum chemistry. AKA quantum physics applied to chemistry. Maths absolutely everywhere, from Huckel theory to particle-in-the-box/ring approximations to density functional theory to solutions to Schrodinger's equation (particle in box, harmonic oscillator, hydrogen atom, time-dependent), perturbation theory (plus more)...I don't know when that's taught in Australia but on exchange it's a graduate level chemistry course. The maths involved here would be mainly linear algebra (tonnes of matrices and inner products), differential equations and multiple integrals.

2. Kinetics. Rate laws generally relate the rate of change of the concentration of some reactant with the instantaneous concentrations of the reactants, aka a differential equation

3. Thermodynamics. So things like thermodynamic potentials (Gibbs, enthalpy etc), energies of mixing, Raoult's law, equilibrium constants, electrochemistry (Nernst equation etc), ideal/non-ideal gas calculations...these would involve integrals and differentials everywhere.

Is that enough for you? This is all at a first/second year uni level at a minimum so it's ok if it doesn't currently make sense.

[keltingmeith]

For something a bit more obscure, I recently discovered that a lot of supramolecular chemistry is quite mathsy. Note that everything I'm about to talk about is third year chemistry/honours level, so if lzxnl's stuff is a bit confusing, this will DEFINITELY make you go "... lolwut". (and some of the maths is also third year/honours level, too)

1. Crystallography. Not only is this insanely physicsy (it's one of those things were it was basically invented by a physicist and chemists just use it to their advantage) in the way that it works, but a lot of the principles behind crystallography are very closely related to group theory (symmetry operations and the like.)

2. Molecular geometries. This stuff is pure geometry at its finest, all the way down to using things like pythagoras to help us find bond lengths or the dimensions of a particular molecule.

3. Supramolecular topologies. It's in the name - this stuff builds on basic ideas of topologies to classify particular molecules. In fact, to distinguish the difference between a polyrotaxane and a polycatenane, you will need a working level knowledge of basic topology (essentially just that lengths don't matter, coffee cup lid = doughnut type stuff)

[GeniDoi]

For something a bit more obscure, I recently discovered that a lot of supramolecular chemistry is quite mathsy. Note that everything I'm about to talk about is third year chemistry/honours level, so if lzxnl's stuff is a bit confusing, this will DEFINITELY make you go "... lolwut". (and some of the maths is also third year/honours level, too)

1. Crystallography. Not only is this insanely physicsy (it's one of those things were it was basically invented by a physicist and chemists just use it to their advantage) in the way that it works, but a lot of the principles behind crystallography are very closely related to group theory (symmetry operations and the like.)

2. Molecular geometries. This stuff is pure geometry at its finest, all the way down to using things like pythagoras to help us find bond lengths or the dimensions of a particular molecule.

3. Supramolecular topologies. It's in the name - this stuff builds on basic ideas of topologies to classify particular molecules. In fact, to distinguish the difference between a polyrotaxane and a polycatenane, you will need a working level knowledge of basic topology (essentially just that lengths don't matter, coffee cup lid = doughnut type stuff)

1 and 3 both sound like applications of pure mathematics, especially 3.

[keltingmeith]

1 and 3 both sound like applications of pure mathematics, especially 3.

That would be because they are.

[Floatzel98]

Probably most appropriate here. Is someone able to explain how the Mandelbrot Set works? I've watched videos on it and read up about it but it is still very confusing. I don't really know what kind of questions I have about it because though because it seems so confusing. Can anyone explain in simple terms how it works? How you can keep on going deeper into it, does it ever stop?

[TheAspiringDoc]

Probably most appropriate here. Is someone able to explain how the Mandelbrot Set works? I've watched videos on it and read up about it but it is still very confusing. I don't really know what kind of questions I have about it because though because it seems so confusing. Can anyone explain in simple terms how it works? How you can keep on going deeper into it, does it ever stop?

Hi!
I haven't really got my head around it yet as exams are approaching, but this appears to be very informative: https://m.youtube.com/watch?v=0YaYmyfy9Z4 . I like it's explaination, "Mandelbrot Set = set of numbers that display certain properties, displayed on complex plain." Also, it mainly works on this function (right?):
f(x)=x^2+c
Where c is an arbitrary original number that you chose (e.g. 4+3i [I think])
And x is the result from the previous computation.

For example, let's do the first few terms for c=1, and x starts as 0.
f(0)=(0)^2+1=1
f(1)=(1)^2+1=2
f(2)=(2)^2+1=5
f(5)=(5)^2+1=26

etc.
This above series produces exponential growth (not sre how it is relevant; but it does).
Also, the result of each function seen above is then 'plugged' into the next function  -  this is the 'iteration' concept that is continually mentioned.
That's pretty much as far as my understanding has progressed, so I'm not sure how different values on the complex plane are then assigned colours and all, or even how the shape (i.e. the madelbrot set) is produced, but yeah, at least I've bumped it and maybe someone will pick up if I've made any mistakes.
EDIT: removed LaTeX as it doesn't seem to be working *sigh*

[Floatzel98]

Hi!
I haven't really got my head around it yet as exams are approaching, but this appears to be very informative: https://m.youtube.com/watch?v=0YaYmyfy9Z4 . I like it's explaination, "Mandelbrot Set = set of numbers that display certain properties, displayed on complex plain." Also, it mainly works on this function (right?):
f(x)=x^2+c
Where c is an arbitrary original number that you chose (e.g. 4+3i [I think])
And x is the result from the previous computation.

For example, let's do the first few terms for c=1, and x starts as 0.
f(0)=(0)^2+1=1
f(1)=(1)^2+1=2
f(2)=(2)^2+1=5
f(5)=(5)^2+1=26

etc.
This above series produces exponential growth (not sre how it is relevant; but it does).
Also, the result of each function seen above is then 'plugged' into the next function  -  this is the 'iteration' concept that is continually mentioned.
That's pretty much as far as my understanding has progressed, so I'm not sure how different values on the complex plane are then assigned colours and all, or even how the shape (i.e. the madelbrot set) is produced, but yeah, at least I've bumped it and maybe someone will pick up if I've made any mistakes.
EDIT: removed LaTeX as it doesn't seem to be working *sigh*

I'll definitely watch the video, thanks! I'm pretty sure I watched the numberphile video on it. But yeah, I understand the basic iteration part, I just don't really understand how all of that translates onto the graph, how the colours work etc. 

Thanks Doc