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Author Topic: Random Stuff  (Read 1162414 times)

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Kirian

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Re: Random Stuff
« Reply #1125 on: November 06, 2013, 09:30:24 am »
0

My friend just rad only came up with a formula to find prime numbers:

For whole numbers 3 and up:

sqrt(numbernumber-2). If it is irrational, it's prime. If it's rational, it's not prime.

Counterexample: 15
Nooooooo! Are there any more? 25 is rational, and so is thirty, and I think 45 works.

Huh?  If n = 2k+1 this is just n^(k-1)*sqrt(n). This is irrational whenever n is not a perfect square... 15 is the first odd composite non-perfect square.
Yeah, just figured that out. If it's even it'll be rational and if it's odd it has to be a perfect square. Sigh.

Empirical induction is fun. Our maths professor once introduced the Coaster Theorem to us, which states that when you mark a round coaster (well, a circle) with equidistant points on the perimeter, and you draw lines between each pair of points and cut the coaster along these lines, you obtain 2n-1 pieces. Works for n=0 to n=5 but is short by two on n=6.


That doesn't work for n = 5 either... I count 11.
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Witherweaver

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Re: Random Stuff
« Reply #1126 on: November 06, 2013, 09:56:39 am »
0

My friend just rad only came up with a formula to find prime numbers:

For whole numbers 3 and up:

sqrt(numbernumber-2). If it is irrational, it's prime. If it's rational, it's not prime.

Counterexample: 15
Nooooooo! Are there any more? 25 is rational, and so is thirty, and I think 45 works.

Huh?  If n = 2k+1 this is just n^(k-1)*sqrt(n). This is irrational whenever n is not a perfect square... 15 is the first odd composite non-perfect square.
Yeah, just figured that out. If it's even it'll be rational and if it's odd it has to be a perfect square. Sigh.

Empirical induction is fun. Our maths professor once introduced the Coaster Theorem to us, which states that when you mark a round coaster (well, a circle) with equidistant points on the perimeter, and you draw lines between each pair of points and cut the coaster along these lines, you obtain 2n-1 pieces. Works for n=0 to n=5 but is short by two on n=6.


That doesn't work for n = 5 either... I count 11.

I don't think it works for n=0, either~

(But n=5 does give 16, which I think is right)
« Last Edit: November 06, 2013, 10:00:15 am by Witherweaver »
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Re: Random Stuff
« Reply #1127 on: November 06, 2013, 12:06:22 pm »
0

n=0 doesn't work, right.

But for n=5 I think you forgot the pie slices outside the pentagram.
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Polk5440

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Re: Random Stuff
« Reply #1128 on: November 06, 2013, 12:28:57 pm »
+2

Number theory is full of conjectures where the smallest counterexample is quite a large number.

Here's a math overflow thread:
http://mathoverflow.net/questions/15444/the-phenomena-of-eventual-counterexamples
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Re: Random Stuff
« Reply #1129 on: November 06, 2013, 12:46:36 pm »
+2

Number theory is full of conjectures where the smallest counterexample is quite a large number.

Here's a math overflow thread:
http://mathoverflow.net/questions/15444/the-phenomena-of-eventual-counterexamples

I think that's misleading.  An integer with 15 digits only seems like "quite a large number" because our everyday experience is with numbers less than 100.

Meanwhile, my cookie count has more than 15 digits.
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shraeye

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Re: Random Stuff
« Reply #1130 on: November 06, 2013, 12:48:18 pm »
+1

Number theory is full of conjectures where the smallest counterexample is quite a large number.

Here's a math overflow thread:
http://mathoverflow.net/questions/15444/the-phenomena-of-eventual-counterexamples
This is exactly something I was looking for; I'm teaching infinite series right now, and students will say "oh look here's a pattern that's true twice.  This sequence is decreasing."  They don't understand why they should rigorously PROVE decreasing, like with a negative derivative or something.
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Polk5440

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Re: Random Stuff
« Reply #1131 on: November 06, 2013, 01:45:46 pm »
+2

Number theory is full of conjectures where the smallest counterexample is quite a large number.

Here's a math overflow thread:
http://mathoverflow.net/questions/15444/the-phenomena-of-eventual-counterexamples

I think that's misleading.  An integer with 15 digits only seems like "quite a large number" because our everyday experience is with numbers less than 100.

Meanwhile, my cookie count has more than 15 digits.

I was thinking about it more in the empirical induction sense mentioned earlier. It's large because your first guess for a lot of these is to prove by induction, so let's test some cases, starting with the base case. n=1? n=2?..... n= 15 digit number? is a lot of cases to check before coming across a counterexample....

The following conjecture doesn't have a large counterexample in the same sense because you don't start checking n=1? n=2?... -- it's easy to "see" the counterexample.

Conjecture: All natural numbers are less than 1,000,000,000,000,000.
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Re: Random Stuff
« Reply #1132 on: November 06, 2013, 03:17:44 pm »
0

Number theory is full of conjectures where the smallest counterexample is quite a large number.

Here's a math overflow thread:
http://mathoverflow.net/questions/15444/the-phenomena-of-eventual-counterexamples

I think that's misleading.  An integer with 15 digits only seems like "quite a large number" because our everyday experience is with numbers less than 100.

Meanwhile, my cookie count has more than 15 digits.

I was thinking about it more in the empirical induction sense mentioned earlier. It's large because your first guess for a lot of these is to prove by induction, so let's test some cases, starting with the base case. n=1? n=2?..... n= 15 digit number? is a lot of cases to check before coming across a counterexample....

The following conjecture doesn't have a large counterexample in the same sense because you don't start checking n=1? n=2?... -- it's easy to "see" the counterexample.

Conjecture: All natural numbers are less than 1,000,000,000,000,000.

That's a good point, and perhaps a good way of indicating why we shouldn't be so surprised about the eventual counterexample phenomena. 

Also would be fun in an undergrad class.  Tell them you're trying to prove some formula on natural numbers that involves some computation so you made a computer program that takes in a natural number and outputs True if the formula holds and False if it does not.  Ask them what they can deduce by plugging in a bunch of numbers and examining the output.
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SirPeebles

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Re: Random Stuff
« Reply #1133 on: November 06, 2013, 04:28:54 pm »
+11

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Re: Random Stuff
« Reply #1134 on: November 06, 2013, 04:33:18 pm »
+1

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Re: Random Stuff
« Reply #1135 on: November 06, 2013, 04:38:23 pm »
0

I would have learned quick sort so much faster if my comp-sci teacher had shown this video...  This is fantastic

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Re: Random Stuff
« Reply #1136 on: November 06, 2013, 05:16:22 pm »
0



21 would have been a much more awesomer movie if this was the main character
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Re: Random Stuff
« Reply #1137 on: November 06, 2013, 05:55:57 pm »
+1



The funny thing is, right before I read this, I was trying to explain the whole thing to my dad, and his first question was, "What if I want the goat?"
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   Quote from: sudgy on June 31, 2011, 11:47:46 pm

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Re: Random Stuff
« Reply #1138 on: November 07, 2013, 01:46:18 pm »
+2

Question: Should I do a cover of Chocolate Rain, as I can make my voice sound really deep?
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   Quote from: sudgy on June 31, 2011, 11:47:46 pm

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Re: Random Stuff
« Reply #1139 on: November 07, 2013, 02:43:35 pm »
0

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Re: Random Stuff
« Reply #1140 on: November 07, 2013, 03:56:16 pm »
0

Question: Should I do a cover of Chocolate Rain, as I can make my voice sound really deep?
Yeah!

I'm currently working on a vocal cover of Renai Circulation by Kana Hanazawa. I like challenges.
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Re: Random Stuff
« Reply #1141 on: November 07, 2013, 04:31:51 pm »
0



I posted this on Facebook last night, and it's already gotten 2 shares and 7 likes.
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Dsell

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Re: Random Stuff
« Reply #1142 on: November 07, 2013, 04:46:44 pm »
+1



I watched this twice. I think I understand quick-sort. :o
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Watno

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Re: Random Stuff
« Reply #1143 on: November 07, 2013, 04:48:22 pm »
0

We watched that in a lecture once.
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Re: Random Stuff
« Reply #1144 on: November 07, 2013, 04:56:41 pm »
+2

I watched it once, and a few things struck me

1) Hungarians are crazy
2) It took 6:55 minutes!! Thats not quick!
3) They all had numbers on thier chests, pretty sure they could have just looked at each other and just stood in the right place
4) If I dont own a hat does this mean I cannot do quicksort?
5) See No.4 but if I cant dance

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Re: Random Stuff
« Reply #1145 on: November 07, 2013, 05:07:38 pm »
+7

I watched it once, and a few things struck me

1) Hungarians are crazy

I find this offensive. As a mathematician, that is. All we can say for certainty is that ten people doing Hungarian dances who are crazy.

Quote
2) It took 6:55 minutes!! Thats not quick!

Wow, well what are you, some sort of sorting guru? How would you know if that's fast? I seriously doubt you could do it faster.

Quote
3) They all had numbers on thier chests, pretty sure they could have just looked at each other and just stood in the right place

...Oh.

Quote
4) If I dont own a hat does this mean I cannot do quicksort?

I believe that was quite strongly implied in the video. Honestly,these questions.

Quote
5) See No.4 but if I cant dance

See 4.
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Re: Random Stuff
« Reply #1146 on: November 07, 2013, 05:10:37 pm »
+15

2) It took 6:55 minutes!! Thats not quick!

Quicksort is only fast on average; it can be rather slow in worst-case situations, like when your data is performing a traditional hungarian dance.
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Re: Random Stuff
« Reply #1147 on: November 07, 2013, 05:13:32 pm »
+5

This was just a warm-up; the next phase involves sorting the entire population of Budapest by cooking some sort of goulash.
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Re: Random Stuff
« Reply #1148 on: November 07, 2013, 05:16:41 pm »
+6

2) It took 6:55 minutes!! Thats not quick!

Quicksort is only fast on average; it can be rather slow in worst-case situations edge cases, like when your data is performing a traditional hungarian dance.

Ahem.
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Re: Random Stuff
« Reply #1149 on: November 07, 2013, 06:59:12 pm »
+1

Contrary to popular belief, InsertionSort is the fastest sorting algorithm:

Only slightly slower is MergeSort

ShellSort is still ok:

Then there's a small gap before BubbleSort:

QuickSort is actually the second worst:

But the red lantern is taken by SelectionSort:
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