Monthly Archives: April 2011

Towards an Expression for pi II

Continuing from this post

We start with $latex \cos( \dfrac{\xi}{2})\cos(\dfrac{\xi}{4}) … \cos(\dfrac{\xi}{2^n})$. Recall the double angle identity for sin: $latex \sin 2 \theta = 2\sin \theta \cos \theta $. We will use this a lot.

Multiply our expression by $latex \sin(\dfrac{\xi}{2^n})$. Then we have

$latex \cos( \dfrac{\xi}{2})\cos(\dfrac{\xi}{4}) … \cos(\dfrac{\xi}{2^n})\sin(\dfrac{\xi}{2^n})$

Using the double angle identity, we can reduce this:

$latex = \dfrac{1}{2} \cos( \dfrac{\xi}{2})\cos(\dfrac{\xi}{4}) … \cos(\dfrac{\xi}{2^{n-1}})sin(\dfrac{\xi}{2^{n-1}}) =$
$latex = \dfrac{1}{4} \cos( \dfrac{\xi}{2})\cos(\dfrac{\xi}{4}) … \cos(\dfrac{\xi}{2^{n-2}})\sin(\dfrac{\xi}{2^{n-2}}) =$
$latex …$
$latex = \dfrac{1}{2^{n-1}}\cos(\xi / 2)\sin(\xi / 2) = \dfrac{1}{2^n}\sin(\xi)$

So we can rewrite this as

$latex \cos( \dfrac{\xi}{2})\cos(\dfrac{\xi}{4}) … \cos(\dfrac{\xi}{2^n}) = \dfrac{\sin \xi}{2^n \sin( \dfrac{\xi}{2^n} )}$ for $latex \xi \not = k \pi$

Because we know that $latex lim_{x \to \infty} \dfrac{\sin x}{x} = 1$, we see that $lim_{n \to \infty} \dfrac{\xi / 2^n}{\sin(\xi / 2^n)} = 1$. So we see that

$latex \cos( \dfrac{\xi}{2})\cos(\dfrac{\xi}{4}) … = \dfrac{\xi}{\xi}$
$latex \xi = \dfrac{\sin(\xi)}{\cos(\dfrac{\xi}{2})\cos(\dfrac{\xi}{4})…}$

Now we set $latex \xi := \pi /2$. Also recalling that $latex \cos(\xi / 2 ) = \sqrt{ 1/2 + 1/2 \cos \xi}$. What do we get?

$latex \dfrac{\pi}{2} = \dfrac{1}{\sqrt{1/2} \sqrt{ 1/2 + 1/2 \sqrt{1/2} } \sqrt{1/2 + 1/2 \sqrt{ 1/2 + 1/2 \sqrt{1/2} …}}}$

This is pretty cool. It’s called Vieta’s Formula for $latex \dfrac{\pi}{2}$. It’s also one of the oldest infinite products.

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Towards an Expression for Pi

I have stumbled across something beautiful! I haven’t the time to write of it now, but I can allude to it without fear. Eventually, I will reproduce a very fascinating formula for $latex \pi$.

But first:

Consider the following expression:

$latex \cos \dfrac{\xi}{2} \cos \dfrac{\xi}{4} \cos \dfrac{\xi}{8} … \cos \dfrac{\xi}{2^n}$

It can be simplified into a very simple quotient of $latex sin$ in terms of $latex \xi$.

Posted in Expository, Mathematics | Tagged , , | 1 Comment

2401: Additional Examples for Test 3

In the past, I have talked about how good a supplemental source of information the Khan Academy is. Again, it is supplementary. But it seems to have lots of fully worked and fully explained examples of the concepts of chapters 17 and chapter 18 (sections 1 through 4) — the topics for your next exam. I have placed the relevant links below.

Double Integrals ( I, II, III, IV, V, VI)
Triple Integrals( I, II, III)
Line Integrals( I, II, III, IV)
Clever Line Integrals (I, II, III, IV, V, VI, VII, VIII, IX)

As always, if you have any questions let me know. I will be hosting a review session in the Math Lab at 5 – come prepared and with questions. I suspect we’ll be focusing on the iterated integrals of Chapter 17. Good luck!

Posted in Georgia Tech, Mathematics | Tagged , , | 15 Comments

Slow factoring algorithm II

I was considering the algorithm described in the parent post, and realized suddenly that the possible ‘clever method’ to speed up the algorithm is complete nonsense. In particular, this simply reduces to trial division (except slightly obscured, so still slower). But the partition thing is still pretty cool, I think.

But I’ve suddenty become interested in different factoring algorithms again, and I think that I’ll make a series on factoring methods out there.

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Blue Eyes and Brown Eyes

This is a puzzle I heard on a much smaller level while I was in my freshmen year of college. Georgia Tech has a high school mathematics competition every spring for potential incoming students. The competition comes in rounds – and those that don’t make it to the final rounds can attend fun mathematical talks. I was helping with the competition and happened to be at a talk on logic puzzles, and this came up.

I bring it up now because it has raised a lot of ruckus at Terry Tao’s blog. It doesn’t seem so peculiar to me, but the literally hundreds of comments at Terry’s blog made me want to spread it some more. There is something about this puzzle that makes people doubt the answer.

I have reposted the puzzle itself, as written by Terry. But for his included potential ‘solutions,’ I direct you back to his blog. Of course, the hundreds of comments there also merit attention.

Terry’s puzzle:

There is an island upon which a tribe resides. The tribe consists of 1000 people, with various eye colours. Yet, their religion forbids them to know their own eye color, or even to discuss the topic; thus, each resident can (and does) see the eye colors of all other residents, but has no way of discovering his or her own (there are no reflective surfaces). If a tribesperson does discover his or her own eye color, then their religion compels them to commit ritual suicide at noon the following day in the village square for all to witness. All the tribespeople are highly logical and devout, and they all know that each other is also highly logical and devout (and they all know that they all know that each other is highly logical and devout, and so forth).

Of the 1000 islanders, it turns out that 100 of them have blue eyes and 900 of them have brown eyes, although the islanders are not initially aware of these statistics (each of them can of course only see 999 of the 1000 tribespeople).

One day, a blue-eyed foreigner visits to the island and wins the complete trust of the tribe.

One evening, he addresses the entire tribe to thank them for their hospitality.

However, not knowing the customs, the foreigner makes the mistake of mentioning eye color in his address, remarking “how unusual it is to see another blue-eyed person like myself in this region of the world”.

What effect, if anything, does this faux pas have on the tribe?

Note 1:  For the purposes of this logic puzzle, “highly logical” means that any conclusion that can logically deduced from the information and observations available to an islander, will automatically be known to that islander.

Note 2: An essentially equivalent version of the logic puzzle is also given at the xkcd web site.  Many other versions of this puzzle can be found in many places.

 

Posted in Math.LO, Math.REC, Mathematics | Tagged , , , | 3 Comments

An even later pi day post

In my post dedicated to pi day, I happened to refer to a musical interpretation of pi. This video (while still viewable from the link I gave) has been forced off of YouTube due to a copyright claim. The video includes an interpretation by Michael Blake, a funny and avid YouTube artist. The copyright claim comes from Lars Erickson – he apparently says that he created a musical creation of pi first (and… I guess therefore no others are allowed…). In other words, it seems very peculiar.

I like Vi Hart’s treatment of the copyright claim. For completeness, here is Blake’s response.

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An interesting (slow) factoring algorithm

After my previous posts (I, II) on perfect partitions of numbers, I continued to play with the relationship between compositions and partitions of different numbers. I ended up stumbling across the following idea: which numbers can be represented as a sum of consecutive positive integers? This seems to be another well-known question, but I haven’t come across it before.

(more…)

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Math subject organization

This is a brief description of how my mathematical blog posts will be categorized. Although the Mathematics Subject Classification (MSC) scheme is perhaps the most official, I will be employing the same description as used by the arxiv. This means that the possible areas are the following:

  • Algebraic Geometry (math.AG)
  • Algebraic Topology (math.AT)
  • Analysis of PDEs (math.AP)
  • Classical Analysis and ODEs (math.CA)
  • Category Theory (math.CT)
  • Combinatorics (math.CO)
  • Commutative Algebra (math.AC)
  • Complex Variables (math.CV)
  • Differential Geometry (math.DG)
  • Dynamical Systems (math.DS)
  • Functional Analysis (math.FA)
  • General Mathematics (math.GM)
  • General Topology (math.GN)
  • Geometric Topology (math.GT)
  • Group Theory (math.GR)
  • History and Overview (math.HO)
  • Information Theory (math.IT)
  • K-Theory and Homology (math.KT)
  • Logic (math.LO)
  • Mathematical Physics (math.MP)
  • Metric Geometry (math.MG)
  • Numerical Analysis (math.NA)
  • Number Theory (math.NT)
  • Operator Algebras (math.OA)
  • Optimization and Control (math.OC)
  • Probability (math.PR)
  • Quantum Algebra (math.QA)
  • Representation Theory (math.RT)
  • Rings and Algebras (math.RA)
  • Spectral Theory (math.SP)
  • Statistics (math.ST)
  • Symplectic Geometry (math.SG)

Of course, I almost certainly won’t refer to all of them, but I will use many. I will also use math.REC for recreational mathematics, which I do all the time.

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