There have been many people that disagree with Lang Lang's piano playing, and I too share some of those criticisms. He's pretty much the devil in any conservative musician's mind for his exaggerated movements and incorrect musical interpretations. However, I am very impressed with his performance in this video (just don't watch his face):
The piece is Liszt's Réminiscences de Don Juan, a operatic transcription of Mozart's opera Don Giovanni. It is full of extreme technical difficulties and highly dramatic. What a great piece.
Saturday, October 24, 2009
Thursday, October 22, 2009
Using arrow-notation - Graham's number
You might be wondering what the previous post on arrow notation can be used for. It is mainly a useless convention that is inapplicable in most contexts, because the numbers involved are so big as demonstrated by the 3↑↑↑↑3 case.
However, it is an integral part in explaining the largest named number in existence. Obviously, there is no such thing as a biggest number, but this one has actual meaning. And, it has a name: Graham's number.
Named after Ronald Graham, the number arose from solving a problem in a branch of mathematics called Ramsey theory that involved coloring graphs on a hypercube (4-dimensional).
Now, how do we calculate Graham's number?
We first start with 3↑↑↑↑3. Calculate this number (which as we found last time, was really really HUGE), and call it "G1".
Now, find 3↑↑...↑↑3, where there are "G1" arrows. Let this new giant number be G2.
Calculate 3↑↑...↑↑3, where there are "G2" arrows. Let this be G3.
And so on, until you get to G64=G, Graham's number.
Yes it is very large, larger than anyone can ever imagine, and nothing can come close to express the magnitude of the number in our universe (even 3↑↑↑↑3 was impossible, so G is just beyond impossible!).
However, it is an integral part in explaining the largest named number in existence. Obviously, there is no such thing as a biggest number, but this one has actual meaning. And, it has a name: Graham's number.
Named after Ronald Graham, the number arose from solving a problem in a branch of mathematics called Ramsey theory that involved coloring graphs on a hypercube (4-dimensional).
Now, how do we calculate Graham's number?
We first start with 3↑↑↑↑3. Calculate this number (which as we found last time, was really really HUGE), and call it "G1".
Now, find 3↑↑...↑↑3, where there are "G1" arrows. Let this new giant number be G2.
Calculate 3↑↑...↑↑3, where there are "G2" arrows. Let this be G3.
And so on, until you get to G64=G, Graham's number.
Yes it is very large, larger than anyone can ever imagine, and nothing can come close to express the magnitude of the number in our universe (even 3↑↑↑↑3 was impossible, so G is just beyond impossible!).
Wednesday, October 21, 2009
"Up-arrow" notation
As you know, multiplication is repeated adding:
5 x 4 = 5+5+5+5 = 20.
So a x b = a+a+a+... +a, where there are b a's.
Raising to an exponent is repeated multiplication:
5^4 = 5x5x5x5 = 625.
So a^b = a x a x a x ... x a, where there are b a's (see a similarity?).
How about repeated exponentiation? This is possible through something called the "up-arrow" notation:
5↑↑4 = 5^(5^(5^5)).
This means to do 5^5=25. Then do 5^25 = 298023223876953125. Finally, do 5^298023223876953125 = [really big number]. We start from the inside and move out.
In general, "a↑↑b" means a^(a^(a^(a^...)))), where there are b a's (again!)*.
And we can go further. a↑↑↑b means a↑↑a↑↑a...a↑↑a, where there are b a's.
a↑↑↑↑b = a↑↑↑a↑↑↑a...a↑↑↑a (b a's).
etc.
Let's just imagine the value of 3↑↑↑↑3, which seems rather simple. Well, we'll see after the following:
3 ↑↑↑↑ 3 = 3↑↑↑3↑↑↑3 = 3↑↑↑(3↑↑↑3) (start on the inside and move out).
= 3↑↑↑(3↑↑3↑↑3), the term in the parenthesis is equal to 3↑↑↑3.
So first we have to calculate 3↑↑3↑↑3 = 3↑↑(3↑↑3).
3↑↑3=3^(3^3) = 3^27 = 7625597484987
So 3↑↑(3↑↑3) = 3↑↑(7625597484987). So we have 3^(3^(3^...^(3^3))...)), where there are 7625597484987 three's. That is HUGE. Let's start calculating:
3^3 = 27.
3^(3^3) = 3^27 = 7625597484987
3^(3^(3^3)) = 3^7625597484987 = [really big number]
Now we have to do this 7625597484983 more times! There is no real way to imagine this number in our current universe.
So let's have that giant number 3↑↑(7625597484987) = [HUGE].
We had to calculate 3↑↑↑(3↑↑3↑↑3) = 3↑↑↑ [HUGE]. This means
3↑↑3↑↑3↑↑...↑↑3 [HUGE] number of times. I don't even know where to start. If there was no way to express 3↑↑(7625597484987) in our universe, 3↑↑↑ [HUGE] is just too big to fathom.
Moral of the story: 3↑↑↑↑3 really isn't that simple.
*You might be wondering what 5↑4 means? This is actually just another way of writing 5^4 = 625 ! Double arrows are where things get interesting.
5 x 4 = 5+5+5+5 = 20.
So a x b = a+a+a+... +a, where there are b a's.
Raising to an exponent is repeated multiplication:
5^4 = 5x5x5x5 = 625.
So a^b = a x a x a x ... x a, where there are b a's (see a similarity?).
How about repeated exponentiation? This is possible through something called the "up-arrow" notation:
5↑↑4 = 5^(5^(5^5)).
This means to do 5^5=25. Then do 5^25 = 298023223876953125. Finally, do 5^298023223876953125 = [really big number]. We start from the inside and move out.
In general, "a↑↑b" means a^(a^(a^(a^...)))), where there are b a's (again!)*.
And we can go further. a↑↑↑b means a↑↑a↑↑a...a↑↑a, where there are b a's.
a↑↑↑↑b = a↑↑↑a↑↑↑a...a↑↑↑a (b a's).
etc.
Let's just imagine the value of 3↑↑↑↑3, which seems rather simple. Well, we'll see after the following:
3 ↑↑↑↑ 3 = 3↑↑↑3↑↑↑3 = 3↑↑↑(3↑↑↑3) (start on the inside and move out).
= 3↑↑↑(3↑↑3↑↑3), the term in the parenthesis is equal to 3↑↑↑3.
So first we have to calculate 3↑↑3↑↑3 = 3↑↑(3↑↑3).
3↑↑3=3^(3^3) = 3^27 = 7625597484987
So 3↑↑(3↑↑3) = 3↑↑(7625597484987). So we have 3^(3^(3^...^(3^3))...)), where there are 7625597484987 three's. That is HUGE. Let's start calculating:
3^3 = 27.
3^(3^3) = 3^27 = 7625597484987
3^(3^(3^3)) = 3^7625597484987 = [really big number]
Now we have to do this 7625597484983 more times! There is no real way to imagine this number in our current universe.
So let's have that giant number 3↑↑(7625597484987) = [HUGE].
We had to calculate 3↑↑↑(3↑↑3↑↑3) = 3↑↑↑ [HUGE]. This means
3↑↑3↑↑3↑↑...↑↑3 [HUGE] number of times. I don't even know where to start. If there was no way to express 3↑↑(7625597484987) in our universe, 3↑↑↑ [HUGE] is just too big to fathom.
Moral of the story: 3↑↑↑↑3 really isn't that simple.
*You might be wondering what 5↑4 means? This is actually just another way of writing 5^4 = 625 ! Double arrows are where things get interesting.
Monday, October 19, 2009
sporcle
I recently found this site, that probably most of the world has already heard of one time or another. However, I have seen that there are several useful ways to use the quiz features.
It is very easy to make quizzes to test yourself, whether it is studying for tests or just reinforcing knowledge. Enter the questions in one column, and the answers in another. When you play, the system will show the answers you get right as you type them in.
While a lot of the games are either frivolous or completely useless trivia (which is actually the point of the site to begin with), there are a couple of science games, such as the periodic table, triangular numbers, and electromagnetic spectrum that have educational value. The good thing is that the site becomes rather addicting, so you will be gaining knowledge without pain!
Elementary school teachers can also make quick arithmetic drills for their students on this site because they can utilize the time limit wisely; I remember doing a lot of "mad minutes" when I was young, which involve solving a certain number of addition or subtraction problems within a minute. In fact, there is already a version of it on the site.
It is very easy to make quizzes to test yourself, whether it is studying for tests or just reinforcing knowledge. Enter the questions in one column, and the answers in another. When you play, the system will show the answers you get right as you type them in.
While a lot of the games are either frivolous or completely useless trivia (which is actually the point of the site to begin with), there are a couple of science games, such as the periodic table, triangular numbers, and electromagnetic spectrum that have educational value. The good thing is that the site becomes rather addicting, so you will be gaining knowledge without pain!
Elementary school teachers can also make quick arithmetic drills for their students on this site because they can utilize the time limit wisely; I remember doing a lot of "mad minutes" when I was young, which involve solving a certain number of addition or subtraction problems within a minute. In fact, there is already a version of it on the site.
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