Don't you have a better chance of winning if you change your selection? I read this somewhere.
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Metzger MeisterIt Gets Worsebefore it gets any better.Registered Userregular
edited May 2008
I'm not good at math. Most of it is fairly easy for me, but sometimes that shit just lookes like someone throwing a bunch of symbols together to fuck with me.
Four is a pretty big number, in its way. If I had four elephants in my room right now, I'd be thinking, "What a lot of elephants".
You remind me of my calc teacher, this is the kind of stuff he would say. He's also pretty crazy, and makes racist remarks without realising it. "Me and this guy? see we don't need these (throws $100 school graphics calculator in bin). We're Korean, we're smart."
He's white.
First, I have to explain arrow notation, it's not too hard. Basically, it's like super-exponents.
4^3 = 4*4*4, right? Well, along those lines, 4^^3 = 4^4^4. This goes right to left, so 4^4^4 = 4^256 = a pretty huge number.
Anyway, it continues like that, so 6^^^^^4 = 6^^^^6^^^^6^^^^6.
These numbers get unbelievably huge really quickly. for example, there's this sequence of numbers called Ackermann numbers. it goes 1^1, 2^^2, 3^^^3, 4^^^^4, and so on.
1^1 = 1
2^^2 = 2^2 = 4
3^^^3 = 3^^3^^3 = 3^^7625597484987 = 3^3^3^3^...^3 (with 7625597484987 3's) = a number too huge to ever possibly hope to write out.
So anyway, that's Knuth Arrow notation in a nutshell.
Now, let's assume that g_0 = 3^^^^3. That's a three, four arrows, and another three. omghueg. That's g_0.
Now let's further assume that g_n = 3^^^^^...^3, with g_(n-1) ^'s. What. Yeah.
Odds are better if you switch
I know this one because I read Curious Incident of the Dog in the Nighttime
Give me a minute to type it up and I'll explain it
Since this is a Math thread, I need someone to explain the goddamn Monty Hall problem.
Monty shows you three doors. Behind one is a car. Behind the other two are goats.
You pick a door, and Monty opens another door to reveal a goat.
Monty then offers you the chance to switch your door selection to the remaining unopened door.
Now, are your odds of getting the car better if you switch or if you stay on the door you originally picked?
they are better when you switch
you can see why if you use larger numbers:
Imagine there are 100 doors, and you pick one. The guy opens 98 empty doors, leaving yours and one other.
Should you switch? Obviously; the chances of you picking the right one straight off the bat were only 1/100, meaning the other door almost certainly holds the prize, despite the fact that there are "50/50" odds.
The same holds true even when there are only 3 doors. Your chance of picking the correct one straight off the bat were 1/3; the chance of the other door being correct is 1/2.
For this example, let's say that the car is behind Door C
Say you pick Door A, and the host opens Door B to reveal a goat
Now you have to make a choice between staying with Door A or changing to Door C.
If you change to C, you get the car
Same thing happens if you pick B and change to C once A is opened. If you pick C and change, though, you get a goat.
So, out of three times, you get the car two times if you change the door
Whereas staying with the same door only gets you the car 1/3 times
I'm doing Multi-Variable calculus in my maths subject right now, it is ok I guess. At least we don't dwell on shit for weeks on end like in highschool, that was fucking lame.
"So you did the top grade maths, yeah lets just fuck around with the same subject for 5 hours or so a week for a month. Ain't no big."
I actually didn't do very well in that class (better than the majority of the people and for determine your ENTER which is to do with university entrance here it gets scaled up an assload compared with other subjects but I'm straying from the point) but I understand more of the shit I'm doing now in this more complex course because I don't just fall asleep in class or not do anything because I'm bored by the pace.
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FramlingFaceHeadGeebs has bad ideas.Registered Userregular
First, I have to explain arrow notation, it's not too hard. Basically, it's like super-exponents.
4^3 = 4*4*4, right? Well, along those lines, 4^^3 = 4^4^4. This goes right to left, so 4^4^4 = 4^256 = a pretty huge number.
Anyway, it continues like that, so 6^^^^^4 = 6^^^^6^^^^6^^^^6.
These numbers get unbelievably huge really quickly. for example, there's this sequence of numbers called Ackermann numbers. it goes 1^1, 2^^2, 3^^^3, 4^^^^4, and so on.
1^1 = 1
2^^2 = 2^2 = 4
3^^^3 = 3^^3^^3 = 3^^7625597484987 = 3^3^3^3^...^3 (with 7625597484987 3's) = a number too huge to ever possibly hope to write out.
So anyway, that's Knuth Arrow notation in a nutshell.
Now, let's assume that g_0 = 3^^^^3. That's a three, four arrows, and another three. omghueg. That's g_0.
Now let's further assume that g_n = 3^^^^^...^3, with g_(n-1) ^'s. What. Yeah.
It has infinite surface area (2 pi ln(a), as a=> infinity) but finite volume (pi * (1 - 1/a), as a=> infinity)
meaning you could fill it with paint, but you couldn't paint it
crazy, huh
My Calc teacher showed us Gabriel's Horn once and it blew my mind.
Another time he taught us about the Devil's Staircase. It's continuous between x=0 and x=1, and has a derivative of 0 almost everywhere (its slope is undefined on a countably infinite set of points, meaning 0% of the domain), and yet it goes from (0,0) to (1,1)
He was the best math teacher I ever had.
Framling on
you're = you are
your = belonging to you
their = belonging to them
there = not here
they're = they are
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Metzger MeisterIt Gets Worsebefore it gets any better.Registered Userregular
edited May 2008
That doesn't make fucking sense how can something be finite in volume but infinite in surface area this is fucking bullshit.
He was a stuffy british dude that looked like Adam West. He was so invested in all of his students, and was so genuinely sad whenever we didn't try our hardest, and just so good at teaching that everyone loved him.
One day he came into class, slammed the door behind himself and started shouting about how our test scores were down from the last unit, and we weren't paying attention in class, and we were becoming terrible students. Our class was mortified that we had upset him. He started drilling us all, yelling "TOO SLOW! NEXT PERSON! NONE OF YOU HAVE BEEN PAYING ANY ATTENTION IN CLASS!!" when people didn't have answers right away. The entire class was frozen in our chairs thinking oh fuck oh fuck why is he freaking out like this ...and then the biggest smile crossed his face and he said, "today we're going to learn about the cross product. hahaha!"
The class was so horrified that it took us about thirty seconds before we actually realised that he was fucking with us.
We also did a whole lot of note-passing during that math class. One day near the end of the year our teacher decided to actually call us on it; of course, he chose the classic route and asked me to read what I was writing out loud to everyone. So I stood up, walked to the front of the class, cleared my throat and read what I had just written:
"Why did no one like the two vectors? Because they were always co-planing!"
I walked back to my seat while the class and the teacher just stared in disbelief.
Yeah, the thing that threw me about the monty hall problem (and most people) is that they only start thinking about probability once you're down to two doors, when there's really three separate cases, since you start with 3 doors, and they're all equally probable:
case 1: you picked the right door - you shouldn't switch
case 2: you picked the wrong door - you should switch
case 3: you picked the other wrong door - you should switch
The last scene in that video belongs in the blingee thread.
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WeaverWho are you?What do you want?Registered Userregular
edited May 2008
Framling I'm sorry dude but mathematical concepts that are infinite or physically impossible excite me just about as much as showing my work for Pi, measuring the event horizon of a super-massive black hole or writing an original proof on why anonymity+internet=dick
That doesn't make fucking sense how can something be finite in volume but infinite in surface area this is fucking bullshit.
Put very simply, the function 1/x is rotated around the x-axis, giving us a horn shape. Normally 1/x has an asymptote (a singularity on the graph, usually somewhere where you've divided by 0) at 0, but Gabriel's horn has intervals that start at x>=1 it is effectively cut off from the region of the graph where it starts "stretching" to infinity (this would have been at x=0).
Well, that roughly explains the finite volume. As far as infinite surface area, I don't quite remember.
I do know that the derivative of an anti-derivative (i.e. volume) is surface area. Possible that using the fundamental theorem of calculus the derivative of integral[1/x] with interval [1,inf) is infinity.
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Kovakdid a lot of drugsmarried cher?Registered Userregular
edited May 2008
i love maths
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Kovakdid a lot of drugsmarried cher?Registered Userregular
That doesn't make fucking sense how can something be finite in volume but infinite in surface area this is fucking bullshit.
Put very simply, the function 1/x is rotated around the x-axis, giving us a horn shape. Normally 1/x has an asymptote (a singularity on the graph, usually somewhere where you've divided by 0) at 0, but Gabriel's horn has intervals that start at x>=1 it is effectively cut off from the region of the graph where it starts "stretching" to infinity (this would have been at x=0).
Well, that roughly explains the finite volume. As far as infinite surface area, I don't quite remember.
I do know that the derivative of an anti-derivative (i.e. volume) is surface area. Possible that using the fundamental theorem of calculus the derivative of int(1/x) with interval [1,inf) is infinity.
the math just kinda says it does
but to try to explain in plain words
The volume if you imagine the small end gets smaller and smaller till eventually an addition of volume is not really adding anyhting
The limit of the extra volume approaches 0. However, the surface area that provides that volume does not go to 0.
That doesn't make fucking sense how can something be finite in volume but infinite in surface area this is fucking bullshit.
Put very simply, the function 1/x is rotated around the x-axis, giving us a horn shape. Normally 1/x has an asymptote (a singularity on the graph, usually somewhere where you've divided by 0) at 0, but Gabriel's horn has intervals that start at x>=1 it is effectively cut off from the region of the graph where it starts "stretching" to infinity (this would have been at x=0).
Well, that roughly explains the finite volume. As far as infinite surface area, I don't quite remember.
I do know that the derivative of an anti-derivative (i.e. volume) is surface area. Possible that using the fundamental theorem of calculus the derivative of int(1/x) with interval [1,inf) is infinity.
the math just kinda says it does
but to try to explain in plain words
The volume if you imagine the small end gets smaller and smaller till eventually an addition of volume is not really adding anyhting
The limit of the extra volume approaches 0. However, the surface area that provides that volume does not go to 0.
I can post the math I guess.
You kind of repeated the question in that second to last sentence there. No biggie, you did explain the whole small end gets smaller business that I failed to.
That doesn't make fucking sense how can something be finite in volume but infinite in surface area this is fucking bullshit.
Put very simply, the function 1/x is rotated around the x-axis, giving us a horn shape. Normally 1/x has an asymptote (a singularity on the graph, usually somewhere where you've divided by 0) at 0, but Gabriel's horn has intervals that start at x>=1 it is effectively cut off from the region of the graph where it starts "stretching" to infinity (this would have been at x=0).
Well, that roughly explains the finite volume. As far as infinite surface area, I don't quite remember.
I do know that the derivative of an anti-derivative (i.e. volume) is surface area. Possible that using the fundamental theorem of calculus the derivative of int(1/x) with interval [1,inf) is infinity.
the math just kinda says it does
but to try to explain in plain words
The volume if you imagine the small end gets smaller and smaller till eventually an addition of volume is not really adding anyhting
The limit of the extra volume approaches 0. However, the surface area that provides that volume does not go to 0.
I can post the math I guess.
You kind of repeated the question in that second to last sentence there. No biggie, you did explain the whole small end gets smaller business that I failed to.
For those still kinda confused I thought of a much better way to explain it
if you think of the horn as a whole bunch of rings, placed right next to each other.
Like
))))))
the horn starts off with the cring radius being 1. This is the largest it ever is. The rings keep getting smaller and smaller until there is basically no "interior" they are no longer contributing to the volume of the object. However, despite being solid and having no interior. They still have an exterior which continues to add on area to the outside of the object.
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Kovakdid a lot of drugsmarried cher?Registered Userregular
edited May 2008
Wikipedia explains it far better than I do.
A non-symbolic way of saying the same thing is the following: to "paint" the surface of the horn does indeed require an infinite surface area of paint, so that the sense in which it is infinite is as a two-dimensional substance. But to "paint" the surface by filling the horn with paint is to obscure it by a three-dimensional object, so the sense in which the amount of paint is finite is as a three-dimensional substance. The paradox arises because real paint is not two-dimensional, and in fact has a discrete thickness, so that painting the surface actually requires an infinite three-dimensional quantity. However, when the horn is filled with paint it is not the outside but the inside surface that is painted. To paint the inside surface of the horn with a layer of paint having a discrete thickness is impossible; once the horn becomes too narrow the paint will not fit. In fact, it is also impossible to fill the horn with such paint, so that in both cases, only finite extent of the horn is covered and the paradox vanishes.
I amended it to the Gamma Function, do you still disagree? I'm not wrong in this case.
I don't think you understand my complaint.
I am not disagreeing with the mathematics of 0! = 1. There is no doubt in my mind, as I can prove it using sound math logic and even argue for the point of 0! = 1.
All of the math background and experience with algebra will not convince me. You must argue this point using no mathematics for the conversation to be interesting.
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I don't know what that is, but I'm all for more goats
My strength is writin'.
I look at calculus or trig now and have to shake my head sadly at all my lost skills.
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Didn't he maybe die a virgin too? So don't try and cite that shit, wiggin.
You remind me of my calc teacher, this is the kind of stuff he would say. He's also pretty crazy, and makes racist remarks without realising it. "Me and this guy? see we don't need these (throws $100 school graphics calculator in bin). We're Korean, we're smart."
He's white.
kpop appreciation station i also like to tweet some
Ronald Graham was my discrete math professor
8-)
Also, you know what's cool?
Gabriel's Horn
It has infinite surface area (2 pi ln(a), as a=> infinity) but finite volume (pi * (1 - 1/a), as a=> infinity)
meaning you could fill it with paint, but you couldn't paint it
crazy, huh
Monty shows you three doors. Behind one is a car. Behind the other two are goats.
You pick a door, and Monty opens another door to reveal a goat.
Monty then offers you the chance to switch your door selection to the remaining unopened door.
Now, are your odds of getting the car better if you switch or if you stay on the door you originally picked?
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I know this one because I read Curious Incident of the Dog in the Nighttime
Give me a minute to type it up and I'll explain it
they are better when you switch
you can see why if you use larger numbers:
Imagine there are 100 doors, and you pick one. The guy opens 98 empty doors, leaving yours and one other.
Should you switch? Obviously; the chances of you picking the right one straight off the bat were only 1/100, meaning the other door almost certainly holds the prize, despite the fact that there are "50/50" odds.
The same holds true even when there are only 3 doors. Your chance of picking the correct one straight off the bat were 1/3; the chance of the other door being correct is 1/2.
Say you pick Door A, and the host opens Door B to reveal a goat
Now you have to make a choice between staying with Door A or changing to Door C.
If you change to C, you get the car
Same thing happens if you pick B and change to C once A is opened. If you pick C and change, though, you get a goat.
So, out of three times, you get the car two times if you change the door
Whereas staying with the same door only gets you the car 1/3 times
"So you did the top grade maths, yeah lets just fuck around with the same subject for 5 hours or so a week for a month. Ain't no big."
I actually didn't do very well in that class (better than the majority of the people and for determine your ENTER which is to do with university entrance here it gets scaled up an assload compared with other subjects but I'm straying from the point) but I understand more of the shit I'm doing now in this more complex course because I don't just fall asleep in class or not do anything because I'm bored by the pace.
My Calc teacher showed us Gabriel's Horn once and it blew my mind.
Another time he taught us about the Devil's Staircase. It's continuous between x=0 and x=1, and has a derivative of 0 almost everywhere (its slope is undefined on a countably infinite set of points, meaning 0% of the domain), and yet it goes from (0,0) to (1,1)
He was the best math teacher I ever had.
your = belonging to you
their = belonging to them
there = not here
they're = they are
He was a stuffy british dude that looked like Adam West. He was so invested in all of his students, and was so genuinely sad whenever we didn't try our hardest, and just so good at teaching that everyone loved him.
One day he came into class, slammed the door behind himself and started shouting about how our test scores were down from the last unit, and we weren't paying attention in class, and we were becoming terrible students. Our class was mortified that we had upset him. He started drilling us all, yelling "TOO SLOW! NEXT PERSON! NONE OF YOU HAVE BEEN PAYING ANY ATTENTION IN CLASS!!" when people didn't have answers right away. The entire class was frozen in our chairs thinking oh fuck oh fuck why is he freaking out like this ...and then the biggest smile crossed his face and he said, "today we're going to learn about the cross product. hahaha!"
The class was so horrified that it took us about thirty seconds before we actually realised that he was fucking with us.
also: quick reply does not interrupt video watching
"Why did no one like the two vectors? Because they were always co-planing!"
I walked back to my seat while the class and the teacher just stared in disbelief.
case 1: you picked the right door - you shouldn't switch
case 2: you picked the wrong door - you should switch
case 3: you picked the other wrong door - you should switch
Put very simply, the function 1/x is rotated around the x-axis, giving us a horn shape. Normally 1/x has an asymptote (a singularity on the graph, usually somewhere where you've divided by 0) at 0, but Gabriel's horn has intervals that start at x>=1 it is effectively cut off from the region of the graph where it starts "stretching" to infinity (this would have been at x=0).
Well, that roughly explains the finite volume. As far as infinite surface area, I don't quite remember.
I do know that the derivative of an anti-derivative (i.e. volume) is surface area. Possible that using the fundamental theorem of calculus the derivative of integral[1/x] with interval [1,inf) is infinity.
the math just kinda says it does
but to try to explain in plain words
The volume if you imagine the small end gets smaller and smaller till eventually an addition of volume is not really adding anyhting
The limit of the extra volume approaches 0. However, the surface area that provides that volume does not go to 0.
I can post the math I guess.
I mean, that's bullshit.
also I know like 3 proofs for it I don't care what the maths say
It has something to do with a special case of the Zeta Function
Kind of complicated, so unless you take some upper level math courses, it is just defined as 1.
Taking the function 1/x and revolving it around the X axis, on the interval 1<=x<infinity
The volume of this solid is given by the integrand
F = pi * integral(1/x^2), [the integral is from 1 to infinity]
Which gives
lim a -> infinity pi * -1/a + 1
Thus the volume of the "horn" is pi
However, the surface area is defined as
2pi * integral[(1/x) * sqrt[1 + (1/x^4)] [evaluated from 1 to infinity]
This results in
-(1/2) Sqrt[1 + 1/x^4] + (Sqrt[1 + 1/x^4] x^2 ArcSinh[x^2])/(
2 Sqrt[1 + x^4])
which when evaluated from 1 to infinity results in infinity
You kind of repeated the question in that second to last sentence there. No biggie, you did explain the whole small end gets smaller business that I failed to.
you can do it using simple algebra, I know the proofs I've taken those classes
I DISAGREE
For those still kinda confused I thought of a much better way to explain it
if you think of the horn as a whole bunch of rings, placed right next to each other.
Like
))))))
the horn starts off with the cring radius being 1. This is the largest it ever is. The rings keep getting smaller and smaller until there is basically no "interior" they are no longer contributing to the volume of the object. However, despite being solid and having no interior. They still have an exterior which continues to add on area to the outside of the object.
I don't think you understand my complaint.
I am not disagreeing with the mathematics of 0! = 1. There is no doubt in my mind, as I can prove it using sound math logic and even argue for the point of 0! = 1.
All of the math background and experience with algebra will not convince me. You must argue this point using no mathematics for the conversation to be interesting.
N!
factorial counts how many ways you can line up N distinct objects
You know like you have 3 objects
for your first slot
you have 3 options, then one is taken so you have 2 options, then there is one left so you have 1 option
3*2*1
Now how many ways do you have to line up 0 objects
1
1 ways