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#51 Re: Puzzles and Games » Spot the error » 2007-02-28 21:38:28
Wow - I'd totally forgotten about this. I'm glad somebody got it! 
I'd be interested in carrying on this thread, if you've got a false proof you can offer Jane?
#52 Re: Jai Ganesh's Puzzles » 10 second questions » 2007-02-28 21:18:31
Dross wrote:
Thank you for pointing that out Jane - I'll try not to be such a numpty in future! 
#53 Re: Help Me ! » Question by antgaudi » 2007-02-28 08:26:35
#54 Re: Help Me ! » LaTeX - A Crash Course » 2007-02-28 04:25:48
\! removes some space between charachters:
y = m x + c \\\\
y \! = \! m \! x \! + \! c(remember latex ignores white-space)
You can put more spacing between charachters using (in increacing order of the amount of space it'll give you) "\,", "\:", "\;", "\quad" and "\qquad".
a b \, c \: d \; e \quad f \qquad gEdit: Oh yeah, strange spacing is before the first line, but only if there's more than one line 
It's just a forum thing - put a double-backslash as the first thing in your math-tags and it's not a problem.
#55 Re: This is Cool » Robot Walks Like Us » 2007-02-28 03:50:27
I am SO getting one of those next christmas. 
#57 Re: This is Cool » 0.9999....(recurring) = 1? » 2007-02-27 23:53:25
I think the discussion cannot be resumed until your 0.111... becomes (become?) constant.
Note that 0.1111... is constant - it has a constant value all the time.
Don't be fooled into thinking that the value of 0.1111... changes value as you go along. What does happen, however, is that as a human you would write down 0.1, then adding another digit you'd have written 0.11, then 0.111, then 0.1111 and so on and so forth.
While it is indeed true that the numbers 0.1, 0.11, 0.111, ... are an increacing sequence and all of them are different, it's important to note that none of them are the number we are concerned with, namely 1.1111... .
The value of 0.1111... is indeed constant - it's only our approximations to it that may change as they get more accurate.
#58 Re: Help Me ! » 3 step stairs investigation » 2007-02-27 22:38:02
Could you set the problem out more clearly? What's the problem you've been given?
Although nobody on here will hand you a perfect GCSE paper on a plate, I'm sure we'll point you in the right direction if you can be clearer as to what problem you've been set.
#59 Re: Help Me ! » 2nd deriv of circle versus parabola » 2007-02-27 08:52:30
And if anyone knows of some equations with inflexion points in them, post those equations, so I can try to graph them. Post your first and second derivatives, if they are too hard for me to calculate, bear in mind, my calculus is too rusty, but I could use pointers and links to lessons on calculus.
Well, a point of inflection is any point where the second derivative changes sign. So start with the second derivative and intigrate. As a simple example of what I mean, let:
Then we know that
must change sign at x = 0 and hence this is a point of inflection. Intigrating, you can add any constants of intigration you want:You could also start with trigonometric functions and you'd end up with something like
with inflection points at for any .#60 Re: Help Me ! » Rolling Dice » 2007-02-27 08:28:44
And you make it look so easy - I can see exactly why that solution works, but it would never have occured to me to put it down like that. Thanks, Jane Fairfax! 
#61 Re: Help Me ! » 2nd deriv of circle versus parabola » 2007-02-27 05:24:41
On the other hand, I would regard a point of inflexion (where the curvature is also 0) as having zero radius of curvature.
I wouldn't say so - I instinctively envisage the radius of curvature as a circle with an edge on the curve and corresponding radius. As the circle approaches the point of inflection from one side, it will grow and grow, expanding to a straight line (in my mind, anyway 
) as it hits the point of inflection, and then appearing as a circle of decreacing radius when it's on the other side of the point of inflection.
I would think of something that has a zero radius of curvature as a corner (and hence non-differentiable) rather than a straight line/point of inflection.
I think this explains what I mean - I'll do some diagrams when I get home to illustrate how I see it. I agree it does depend on how you visualise it, though.
#62 Re: This is Cool » 0.9999....(recurring) = 1? » 2007-02-27 04:33:47
Unfortunately, you might be wrong.
Start with an equivalent of 2, which might be 1.999...
Now go to the end, which you may have difficulty getting there.
However, when you find an inhuman way to reach the last 9,
then work your way leftward toward the 9.1 going backwards.
You'll never get there, unless you use an inhuman method we
don't know of. So you might conclude on the way back, that
the 9.1 doesn't exist, until you unlock the inhuman way
of getting to it. Warp drive super power active.
But think about that - go to the end? Even by inhuman means, this is impossible; not because it's very far away, but because there is no end. There are an infinite amount of 9's after the decimal place, which means that for every "9" you point to there will be one after it. Since the last digit would be that digit that has no digits after it, there is therefore no last digit to get to.
All the inhuman devices in the world won't help you tie a knot in a string that doesn't exist.
#63 Re: Help Me ! » 2nd deriv of circle versus parabola » 2007-02-27 03:48:19
If you were being all "proper" and such, you'd probably just state that there was no radius of curvature. Of course, if you said the radius of curvature was infinite, everyone would know what you really meant...
#64 Re: This is Cool » 0.9999....(recurring) = 1? » 2007-02-27 00:46:25
1 - Infinite 0.9 = 0 is a Contradiction! because the Calculation is saying the .9's End!!
A.R.B
So what is 1 - 0.9999... equal to, then?
#65 Re: Help Me ! » LaTeX - A Crash Course » 2007-02-26 21:30:50
George,Y wrote:Latex is getting more and more complex.
Indeed.
#66 Re: Help Me ! » Rolling Dice » 2007-02-26 21:04:52
So it really is much simpler than I had it made out to be - thanks very much, all!
#67 Re: Dark Discussions at Cafe Infinity » Future job; what do you want to do? » 2007-02-26 05:24:52
When I was at school, I wanted to be a maths or physics teacher. Don't know why it never panned out, but I like my current job because it involves a hefty amount of maths that's actually applied to something - I got a little frustrated with the abstractness of some of the maths at uni.
At the moment I help to develop mathematical models of groundwater, heat and contaminant flow and transport through the ground using numerical analysis (specifically the finite element method). I enjoy it, though I never thought I'd be doing it when I left uni!
#68 Re: Help Me ! » Rolling Dice » 2007-02-26 02:22:33
But then you're saying that if you roll six dice, you're garunteed that one of them will be a six, and if you roll 9 dice the probability is 1.5 that one of them will be six
#69 Re: Help Me ! » 2nd deriv of circle versus parabola » 2007-02-26 02:20:14
The second derivative of a circle is constant, if only you use polar coordinates:
#70 Help Me ! » Rolling Dice » 2007-02-26 02:09:51
- Dross
- Replies: 12
Here's a problem someone asked me if I could do. I'm no good at probability (was never really interested, to be honest) but I gave it a shot and got an answer I'm fairly confident of - just wanted to see other approaches (and if I was right or not ).
Here's the problem:
You roll m dice. What is the probability that any one of them is a six?
#71 Re: Introductions » Introducing myself » 2007-02-26 00:55:07
Hello! 
Hope you find lots you like here - sounds like you might!
#72 Re: This is Cool » 0.9999....(recurring) = 1? » 2007-02-26 00:49:08
However it is naive induction, philosophically flawed....
What? It's mathematical induction - not flawed at all, and made rigorous through the properties of the real numbers.
And the last section of your post merely demonstrates that:
... well... yeah. I know.
#73 Re: This is Cool » 0.9999....(recurring) = 1? » 2007-02-25 23:08:19
How many points does a linesegment of length 1 have? How many points does a linesegment of length 1/1000 have?
The same? That's indeed the most absurd part of Cantor's theory.
This may be highly counter-intuitive, but then again so is the whole of quantum mechanics, and relativity. Yet these are well established scientific theories with many real-life experiments to back-up those theories. Just because it sounds absurd, doesn't mean it's not true.
Likewise, two segments of the real line having different lengths but the same number of points sounds ridiculous - but that doesn't mean you should dismiss it, and it is in fact demonstratably true.
Question for you - how many natural numbers are there? Infinitely many, of course!
Now, how many even numbers are there? Well, there are infinitely many. Does that mean there are the same amount of even numbers as there are natural numbers?
Well, it seems so - and we can demonstrate this by pairing them up. Write out the set of ordered pairs, the first member of which is the next natural number, the second member is the next even number. It'll look like:
(1, 2), (2, 4), (3, 6), (4, 8), (5, 10), ...
(I've started the natural numbers from 1 purely for convenience - it's irrelevant whether they start from 1 or 0 in this case)
Now you can hopefully see that there are the same amount of even numbers as there are natural numbers - every even number gets paired with a natural number, and vice-versa. No magic in it, either.
Similarly, there are indeed the same number of points on a unit-interval of the real line as there are in an interval of length 1/1000.
#74 Re: This is Cool » 0.9999....(recurring) = 1? » 2007-02-21 06:32:09
Me and one of my friends somehow started talking about this a few days ago.
I said that 0.999... = 1, and told him about the proof that involves taking x away from 10x to give 9, meaning x is 9/9 which is 1. He didn't believe me, and said that that couldn't be true on the grounds that 1 is a natural number. If 0.999... = 1, then 0.999... should be a natural number as well. It clearly isn't though, because its decimal form has things other than a bunch of zeroes after the point.
That isn't enough to convince me that 0.999... <> 1, but at the same time, I'm not entirely sure why that argument isn't valid.
Thoughts?
In the context of what your friend is talking about, 0.9999... is not a number. And neither is 1 - they are just funny little symbols that represent numbers. Just because the funny little symbol that looks like 0.9999... or 0.9999999999999.......... or looks like a circle and then a dot and a bunch of the same numbers and then some more dots, does not change what it represents, and that is the number that is the smallest positive integer - however you write it.
#75 Re: Help Me ! » help me in factors!?!!??? » 2007-02-13 22:32:57
I am stuck in this sum of factors please someone help me:(
x^2+x^5
What exactly is the task you've been set?