Mind = blown
Post me your hardest maths question you know
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Mind = blown
From my physics undergrad... Can't believe I used to be able to do this
Brings back memories
Was QM your least favourite/toughest module? I found it very challenging in third year. I disliked statistical physics (tougher thermodynamics mostly), and some of third year electromagnetism was ridiculous (Lienard-Wiechert potentials- steer clear!).
Soldato
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Grr, your 'faster' comment on the first page of this thread actually had me confused for a while.
my bad lol
Brings back memories
Was QM your least favourite/toughest module? I found it very challenging in third year. I disliked statistical physics (tougher thermodynamics mostly), and some of third year electromagnetism was ridiculous (Lienard-Wiechert potentials- steer clear!).
I think QM was probably second or third hardest after GR and our third year maths modules. It's one of those things that when you live and breathe the physics - i.e. you're "in the zone" answering/working through problems, and understanding the maths day to day it's a lot clearer. I think I'd struggle to know where to begin on almost any of them now and it was only 4 years ago.
I got bored and attempted a proof of the 0.9r = 1 thing using geometric sequences:
Consider that 0.9r can be found by finding the sum of the following geometric sequence to infinity.
Un = 0.9*0.1^(n-1)
Which is in the form ar^(n-1)
So what this says is the sum of this geometric series can be found effectively by doing this:
Sn = (0.9/1) + (0.9/10) + (0.9/100) + (0.9/1000) + (0.9/10^4) ... (0.9/10^n)
So we could write a table for this that looks a bit like this:
S1 = 0.9
S2 = 0.99
S3 = 0.999
Sn = 0.[n number of 9s]
etc.
And we know that the sequence will converge at some point as the value r, which is 0.1, in this geometric sequence is between less than 1 and greater than -1.
Hence we can find the value at which the sequence will converge by using:
S = (a/1-r)
Which is a formula we can use to derive the sum to infinity of any series which converges.
where a is 0.9 and r = 10.
So we get (0.9/1-0.1) = (.9/.9) = 1.
So 0.9r = 1.
(in case this turns out to be wildly invalid or already shown in the thread: problem?)
Consider that 0.9r can be found by finding the sum of the following geometric sequence to infinity.
Un = 0.9*0.1^(n-1)
Which is in the form ar^(n-1)
So what this says is the sum of this geometric series can be found effectively by doing this:
Sn = (0.9/1) + (0.9/10) + (0.9/100) + (0.9/1000) + (0.9/10^4) ... (0.9/10^n)
So we could write a table for this that looks a bit like this:
S1 = 0.9
S2 = 0.99
S3 = 0.999
Sn = 0.[n number of 9s]
etc.
And we know that the sequence will converge at some point as the value r, which is 0.1, in this geometric sequence is between less than 1 and greater than -1.
Hence we can find the value at which the sequence will converge by using:
S = (a/1-r)
Which is a formula we can use to derive the sum to infinity of any series which converges.
where a is 0.9 and r = 10.
So we get (0.9/1-0.1) = (.9/.9) = 1.
So 0.9r = 1.
(in case this turns out to be wildly invalid or already shown in the thread: problem?
)
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Man of Honour
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If you wrote 0.9 followed by an infinite number of 9s you would still have a sqillion gillion 9's (and then another squillion gillion 9's and so on).
It will never reach 1 and no amount of maths will prove otherwise.
HOWEVER, if some great Mathemeticians declared that the symbol 0.9r = 1 to make maths a bit easier then that is a different matter.
Soldato
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But infinity is a human construct. Infinity does not exist in nature, and therefore any application of 0.9r = 1 is wrong.
That is if r represents recurring, rather than some variable like radius of a circle, which is what I thought when I read the post on page 1.
you make it sound like it's not true because you can't write the number in finite time. Just take the number 0.9r as 'there' already. Now differentiate it from 1.0. You can't. You can't 'draw a circle around' 0.9r that doesn't also include 1.0.
Soldato
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But this is why some people say 0.9r=1 and some say it doesn't. Those for 0.9r=1 are dealing in theoretical maths, and those that don't are dealing in real maths.
You can not just take 0.9r as being there, because it can not exist.
Does that also mean pi can't exist?
After all, you can't write it down in decimal notation.
P.S. no-one with a suitably advanced education in maths asserts 0.9r!=1.
The entire numeral system is ultimately a human construct, that's why it's imperfect in its representation of many values that we use daily and why we have progressively been 'adding bits' on to it over the centuries (e.g. real numbers, complex numbers, quaternions, countless irrational constants, etc.). This is entirely the reason we have the occasional weird occurrence such as 0.9r = 1.
Come up with a quick non numerical way of solving the Navier-Stokes Equations for any flow and get back to me.
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Soldato
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As far as I am aware pi is not proven to be of an infinite number of decimal places, therefore it is not the same as 0.9r. Just because humans have not fully calculated pi, does not mean it does or does not exist.
