1. ## Re: Interesting mathematical statements

Originally Posted by braintic
There is a lot of highly theoretical mathematics here.
Does anyone have more real-world mathematical statements?
(The birthday problem was a good one)
The Kakeya Needle Problem: What is the smallest area of a parking lot in which you can have a needle of length 1 turn around 180 degrees and return to its starting position, pointing in the other direction?

Answer: The area can be made arbitrarily small through a series of divisions and transformations of the shape required for the needle to turn around. Hence, no smallest area exists.

2. ## Re: Interesting mathematical statements

The Kakeya Needle Problem: What is the smallest area of a parking lot in which you can have a needle of length 1 turn around 180 degrees and return to its starting position, pointing in the other direction?

Answer: The area can be made arbitrarily small through a series of divisions and transformations of the shape required for the needle to turn around. Hence, no smallest area exists.
This isn't very "practical" though, since the car / needle needs to be made arbitrarily thin if you want the area to be arbitrarily small. For anyone interested, there is a Numberphile video on the Kakeya Needle Problem: www.youtube.com/watch?v=j-dce6QmVAQ

3. ## Re: Interesting mathematical statements

Originally Posted by InteGrand
This isn't very "practical" though, since the car / needle needs to be made arbitrarily thin if you want the area to be arbitrarily small. For anyone interested, there is a Numberphile video on the Kakeya Needle Problem: www.youtube.com/watch?v=j-dce6QmVAQ
Well the problem never stated any width. It is a mathematical solution, after all.

$\noindent Simpson's Paradox: When two sets of data are analysed separately, there is a common trend between both sets of data. But when the data is combined together and analysed as a whole, the trend reverses.$

$\noindent One of the famous real life examples was the Berkeley Gender Bias case. The entire university's admissions were significantly biased towards men. However, when the data from the individual faculties were looked at separately, each one had a small, but statistically significant bias towards women.$

$Not sure if your mind has exploded yet, but I'll go get the sponge.$

4. ## Re: Interesting mathematical statements

Another good one is that there are always two opposite points on Earth with exactly the same temperature. This is proved quickly and accessibly in this video by Dr James Grime:

5. ## Re: Interesting mathematical statements

Well the problem never stated any width. It is a mathematical solution, after all.

$\noindent Simpson's Paradox: When two sets of data are analysed separately, there is a common trend between both sets of data. But when the data is combined together and analysed as a whole, the trend reverses.$

$\noindent One of the famous real life examples was the Berkeley Gender Bias case. The entire university's admissions were significantly biased towards men. However, when the data from the individual faculties were looked at separately, each one had a small, but statistically significant bias towards women.$

$Not sure if your mind has exploded yet, but I'll go get the sponge.$
Haha yeah, it is an interesting result, I just meant that it's not really real-world as braintic wanted.

6. ## Re: Interesting mathematical statements

Well the problem never stated any width. It is a mathematical solution, after all.

$\noindent Simpson's Paradox: When two sets of data are analysed separately, there is a common trend between both sets of data. But when the data is combined together and analysed as a whole, the trend reverses.$

$\noindent One of the famous real life examples was the Berkeley Gender Bias case. The entire university's admissions were significantly biased towards men. However, when the data from the individual faculties were looked at separately, each one had a small, but statistically significant bias towards women.$

$Not sure if your mind has exploded yet, but I'll go get the sponge.$
It can't be that EACH of the individual faculties has a bias towards women, just that most of them do. At least one faculty must have male bias if the overall bias is male.

7. ## Re: Interesting mathematical statements

This is the nature of weighted averages.

8. ## Re: Interesting mathematical statements

Originally Posted by InteGrand
Another good one is that there are always two opposite points on Earh with exactly the same temperature. This is proved quickly and accessibly in this video by Dr James Grime:

I love intermediate value theorem consequences like this. You can get quite a lot out of such a seemingly common sense result.

Another one is the one that colloquially states that you can always rotate a square four legged table on wobbly ground into a position in which it balances.

9. ## Re: Interesting mathematical statements

Originally Posted by glittergal96
I love intermediate value theorem consequences like this. You can get quite a lot out of such a seemingly common sense result.

Another one is the one that colloquially states that you can always rotate a square four legged table on wobbly ground into a position in which it balances.
I'd just rather have the triangular legged table in the first place. Stable on almost any readily available terrain.

$\noindent Most X2 Students know the harmonic series is divergent, but consider the infinite sum: \sum_{k=1}^{\infty} \frac{1}{k^r} \\ For r \leq 1, the series is divergent, but for r>1 the series converges. This means that \sum_{k=1}^{\infty} \frac{1}{k^{1+\epsilon}} where \epsilon >0 is convergent, for arbitrarily small values of \epsilon. To put it this way: \sum_{k=1}^{\infty} \frac{1}{k} is a divergent sum, but \sum_{k=1}^{\infty} \frac{1}{k^{1.0000000000000000000000001}} is a convergent sum.$

10. ## Re: Interesting mathematical statements

Originally Posted by glittergal96
I love intermediate value theorem consequences like this. You can get quite a lot out of such a seemingly common sense result.

Another one is the one that colloquially states that you can always rotate a square four legged table on wobbly ground into a position in which it balances.
Haha yeah that table one is really nice. Here's a math / physics paper about it: http://arxiv.org/pdf/math-ph/0510065.pdf

Here's the Numbephile video about it:

11. ## Re: Interesting mathematical statements

I don't have the mathematical vocabulary or ability to precisely describe it but I find Benford's Law fascinating and very strange

https://en.m.wikipedia.org/wiki/Benford%27s_law

12. ## Re: Interesting mathematical statements

Originally Posted by glittergal96
It can't be that EACH of the individual faculties has a bias towards women, just that most of them do. At least one faculty must have male bias if the overall bias is male.
Imagine that
(a) In 2013/14, Steve Smith has a batting average of 30 in 5 innings, while David Warner has a batting average of 31 in 20 innings
(b) In 2014/15, Steve Smith has a batting average of 40 in 50 innings, while David Warner has a batting average of 41 in 15 innings

In both years, Warner's average was higher than Smith's.
Yet when you work out the combined average for the two seasons, Smith's average beats Warner's 39.1 to 35.3

So it doesn't have to be simply a majority. It can be all.

13. ## Re: Interesting mathematical statements

I love the "wobbly circle" and "dragoncurve" videos on Numberphile.

14. ## Re: Interesting mathematical statements

Originally Posted by InteGrand
Haha yeah, it is an interesting result, I just meant that it's not really real-world as braintic wanted.
No, that is good enough for me. Statistical paradoxes are definitely real-world.

I thought someone might give a chaos theory example. That seems to be a topic that is never given the attention it deserves.

15. ## Re: Interesting mathematical statements

Originally Posted by braintic
No, that is good enough for me. Statistical paradoxes are definitely real-world.

I thought someone might give a chaos theory example. That seems to be a topic that is never given the attention it deserves.
I was referring to the Kakeya Needle one in those posts.

16. ## Re: Interesting mathematical statements

Originally Posted by InteGrand
I was referring to the Kakeya Needle one in those posts.
Oops - sorry. Nevertheless, I wonder if someone can come up with an example of the "regression to the mean" fallacy.

17. ## Re: Interesting mathematical statements

Originally Posted by braintic
Imagine that
(a) In 2013/14, Steve Smith has a batting average of 30 in 5 innings, while David Warner has a batting average of 31 in 20 innings
(b) In 2014/15, Steve Smith has a batting average of 40 in 50 innings, while David Warner has a batting average of 41 in 15 innings

In both years, Warner's average was higher than Smith's.
Yet when you work out the combined average for the two seasons, Smith's average beats Warner's 39.1 to 35.3

So it doesn't have to be simply a majority. It can be all.
I think glittergal96 might have thought that the bias thing with men and women was referring to the percentage out of all those accepted who are male. The study actually considered the acceptance rates of women and men applicants separately (like your cricket example basically).

18. ## Re: Interesting mathematical statements

By the way, my example with averages partially shows why, when your maths teacher is calculating your yearly mark based on say 4 tests, the individual test marks should NOT be artificially scaled to the same mean before adding.

19. ## Re: Interesting mathematical statements

Originally Posted by braintic
By the way, my example with averages partially shows why, when your maths teacher is calculating your yearly mark based on say 4 tests, the individual test marks should NOT be artificially scaled to the same mean before adding.

20. ## Re: Interesting mathematical statements

Originally Posted by braintic
Imagine that
(a) In 2013/14, Steve Smith has a batting average of 30 in 5 innings, while David Warner has a batting average of 31 in 20 innings
(b) In 2014/15, Steve Smith has a batting average of 40 in 50 innings, while David Warner has a batting average of 41 in 15 innings

In both years, Warner's average was higher than Smith's.
Yet when you work out the combined average for the two seasons, Smith's average beats Warner's 39.1 to 35.3

So it doesn't have to be simply a majority. It can be all.
Actually, let me explain why people have trouble seeing how this works.

People try to compare 30 to 31 and 40 to 41.
Instead, they should be comparing diagonally: 30 to 41 in Warner's favour, and 40 to 31 in Smith's favour.
The differences are 11 and 9 - roughly the same.
But the weightings of the two pairs are 20 to 70.
So the slightly lower difference which is in Smith's favour has a much higher weighting.

So the key is the large discrepancy between the number of innings each year, and the turnaround in those numbers between the two years.

21. ## Re: Interesting mathematical statements

Originally Posted by InteGrand
They call in the maths staff

22. ## Re: Interesting mathematical statements

Originally Posted by braintic
Oops - sorry. Nevertheless, I wonder if someone can come up with an example of the "regression to the mean" fallacy.
braintic is having trouble with his mobile data connection. He tries walking around the room until the signal strength improves enough for him to be able to view the latest BOS forum posts on his phone. Incidentally, braintic subconsciously picks up on background details, and his subconscious observation is that he is near a pineapple which is there for no apparent reason. A short while later, Integrand enters the room and removes the pineapple for some nefarious purpose. Later braintic is in a pyrotechnics lab, which happens to be the room where Integrand placed the pineapple. It turns out Integrand is making a pineapple bomb. braintic tries to view the latest posts, but is unable to load the page, and walks around the room to get a better signal. The page loads as soon as braintic walks by the pineapple. braintic realises that the pineapple was near his phone when he was able to load the previous time. braintic concludes the pineapple caused his signal to improve, and kills Integrand to prevent him from destroying the pineapple.

23. ## Re: Interesting mathematical statements

Originally Posted by braintic
No, that is good enough for me. Statistical paradoxes are definitely real-world.

I thought someone might give a chaos theory example. That seems to be a topic that is never given the attention it deserves.
Two celestial bodies orbiting each other, assuming there exists no other sources of gravity in their entire universe, are comprised by a perfectly deterministic system where the exact location of each body can be predicted with 100% accuracy infinitely far into the future.

Introduce a third body into the simulation, and it is analytically impossible to determine with any amount of accuracy where any of the bodies will be located infinitely far into the future. This has to do with the resulting calculations giving rise to an integral that cannot be expressed in terms of elementary functions, and can only be resolved numerically.

24. ## Re: Interesting mathematical statements

Originally Posted by braintic
Imagine that
(a) In 2013/14, Steve Smith has a batting average of 30 in 5 innings, while David Warner has a batting average of 31 in 20 innings
(b) In 2014/15, Steve Smith has a batting average of 40 in 50 innings, while David Warner has a batting average of 41 in 15 innings

In both years, Warner's average was higher than Smith's.
Yet when you work out the combined average for the two seasons, Smith's average beats Warner's 39.1 to 35.3

So it doesn't have to be simply a majority. It can be all.
Yeah, sorry about that. Misinterpreted the original post and what kind of bias was entailed. (overall students vs acceptance rates)

25. ## Re: Interesting mathematical statements

$\\ 2016 = {2}^{5}+{2}^{6}+ \dots +{2}^{10}$

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