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14 May 2009 02:59
in discussion Ideas and Questions / Student Ideas » Cool Puzzle

Ha! I have no idea where to start that problem.

Re: Cool Puzzle by , 14 May 2009 02:59
13 May 2009 22:42
in discussion Ideas and Questions / Student Ideas » Cool Puzzle

So a friend told me this today. I want to know if any of you guys can get it. It took me a good while to figure it out.

If 5 = 4, 7 = 17, 9 = 25 and 35 = 2, what does 14 equal?

Cool Puzzle by , 13 May 2009 22:42

Did you also wikipedia paradoxes?? haha

this isn't math related, but did anyone read xkcd last week? it pretty brilliantly made fun of the show "firefly".

Re: XKCD Friendly Numbers?? by , 12 May 2009 18:55

Does love even have an identity? And if so, what do the inverses look like?

To a certain degree it depends on what you mean by logic. In particular the incompleteness results depend on a particular kind of logic that uses axioms assumptions basically, and the provability of any particular statement is depend on the axioms you choose. So if you have a statement you want to prove, and you can't under your current logical system, you can always change the system in such a way that the statement becomes provable (or disprovable).

I guess the way I've always thought about this is by picturing myself standing in a dark room with a flashlight. Whatever logical system you're using determines where you stand in the room, and from that spot you can use the flashlight to study things in the room. However, the flashlight will also cast shadows, so to see what lies in the shadows, you have to move (ie change your logical system).

11 May 2009 22:38
in discussion Ideas and Questions / Student Ideas » Cardinality

Yes, that's right. \$\aleph_0\$ is the smallest infinite cardinal, and so the set of primes and the rationals and all those countably infinite sets all have the same cardinality.

Re: Cardinality by , 11 May 2009 22:38

This definitely isn't a legit proof, but I thought it was a funny "paradox."
Claim: All natural numbers are interesting
Proof (by contradiction): Assume there are natural numbers that aren't interesting. Then by the Well-Ordering Principle, there must be a smallest "uninteresting" number. However, the smallest uninteresting number is actually interesting in of itself. Contradiction!

Let's not forget about "fortunate" and "lucky" numbers!

According to wikipedia, fortunate numbers can be described as the following:
Given a positive integer n, we say that m>1 is a fortunate number if it is the smallest integer m such that p(n)+m is prime, where p(n) is the product of the first n primes.
For example, take n=1. Then p(1) = 2 since 2 is the smallest prime. We need the smallest m so that 2+m is prime. m=3 is our answer, so 3 is our first fortunate number!
As of today, all known fortunate numbers are also prime.

A "lucky" number can be found through a sort of sieve process. Take the list of all positive integers:
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25…
First, we cross every second number off of our list:
1,3,5,7,9,11,13,15,17,19,21,23,25…
The second number left on our list is 3. This means that every 3rd number must be crossed off of our list:
1,3,7,9,13,15,19,21,25…
The third number left on our list is 7. This means that every 7th number can be crossed off of our list:
1,3,7,9,13,15,21,25,…

As you keep repeating this process, here is a list of the "lucky" numbers that you are left with:
1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79, 87, 93, 99, …

A lucky prime is a lucky number that is also a prime. It is unknown if there are infinitely many lucky primes.

For more information, check out these links!
http://en.wikipedia.org/wiki/Fortunate_number
http://en.wikipedia.org/wiki/Lucky_number

Re: XKCD Friendly Numbers?? by , 11 May 2009 20:43

I actually looked up a little information on women in mathematics. A weird name that came up was Danica McKellar who was an actress in The Wonder Years but has also made appearances on the shows The West Wing and NYPD Blue. However, it turns out that she graduated from UCLA majoring in mathematics. I have found out that she helped come up with and prove a theorem. However, the only information that I could really find out about it was that it is called the Chayes-McKellar-Winn theorem. So I guess she didn't really come up with all of it herself. If anyone has any interesting information on this, that would be great.

Also, to all the future teacher out there. Danica McKellar did write a book called Math Doesn't Suck which is aimed towards pre-teen girls in hopes of motivating them to learn mathematics. This could be a great resource in the future.

http://terrytao.wordpress.com/2007/08/20/math-doesnt-suck-and-the-chayes-mckellar-winn-theorem/

I remember Andy talking about this in class. The story that he told us was that Nobel's wife had an affair with mathematician Mittag-Leffler so Nobel wanted revenge and would never give this award for mathematics. However, I found some information saying that this is almost completely false since Mittag-Leffler was 'a life-long bachelor.' The article states that Mittag-Leffler was a leading Swedish mathematician who antagonized Nobel (For what? The article doesn't say). Because of this Nobel decided not to give a prize in mathematics simply because he didn't want Mittag-Leffler to be a winner. Which story is true? Either one of them? I have no idea but I understand why math people like to say that Nobel's wife had an affair.

http://www.agnesscott.edu/Lriddle/women/prizes.htm#nobel
scroll down to the section labled 'Nobel Prize'

Nobel Prize for Mathematics by , 11 May 2009 19:47

haha…I like that comment Josh. But what happens when you love yourself? Is that the identity?

11 May 2009 19:31
in discussion Ideas and Questions / Student Ideas » Sexy Primes

Actually, the sexy primes come from the sex- which means six. Another example would be sextuplets - six kids. So that name really does make sense.

But, Alex does make a good point. Cousin primes differ by four and I don't know why they are called that.

Re: Sexy Primes by , 11 May 2009 19:31

I actually looked into some paradoxes after posting this and found something by the name of the Berry paradox.

Berry paradox: The phrase "the first number not nameable in under ten words" appears to name it in nine words.

Contradiction!!

11 May 2009 19:08
in discussion Ideas and Questions / Student Ideas » Sexy Primes

Haha!! Katie, your comment cracked me up.

Who decides to name these???

Can I just declare primes that contain three prime numbers between them and differ by more than 20 are called "unlucky primes." Like.. it seems like some of the names and characteristics of prime numbers are so arbitrary.

Re: Sexy Primes by , 11 May 2009 19:08

Edward Witten can prove anything… just wait.

Another example: if you look at your passport (on your information page at the bottom), the last number in the second row is the check digit. The two rows are called a machine readable passport. So, if you input your information incorrectly on an embassy website and they email you, you will be asked for it. Woot! (and whoops!)

Re: Illinois Drivers License by , 11 May 2009 15:38
11 May 2009 15:27
in discussion Ideas and Questions / Student Ideas » Cardinality

I am pretty sure the naturals has the smallest infinite cardinality… It is designated \$\aleph_0\$.

Re: Cardinality by , 11 May 2009 15:27
11 May 2009 08:29
in discussion Ideas and Questions / Student Ideas » Sexy Primes

haha. I checked them out though. They're not very sexy.

Re: Sexy Primes by , 11 May 2009 08:29

Haha, I am with you Alex. The idea that there is somewhere in between true and false in mathematics sort of blows my mind.

It also got me wondering about theory outside of mathematics. Like the String Theory, for example. Could it be unprovable? And if it were, how could someone go about proving that the it is unprovable? Woa.

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