Showing posts with label large numbers. Show all posts
Showing posts with label large numbers. Show all posts
20 March 2009
Billions and millions
Yes, I'm still alive. I got out of the blogging groove somehow.
Today's xkcd makes an interesting point about the difference between "billion" and "million".
And although this isn't about math, Carl Sagan's Cosmos can be watched online at hulu.com. (Thanks to Blake Stacey for the pointer.)
Today's xkcd makes an interesting point about the difference between "billion" and "million".
And although this isn't about math, Carl Sagan's Cosmos can be watched online at hulu.com. (Thanks to Blake Stacey for the pointer.)
10 August 2008
Big numbers confuse the New York Times
From Chinese basketball builds towards podium (August 9):
"With 300,000 million people playing basketball across the country — roughly the same number as the population of the United States..."
Really? I mean, it's almost a cliche by this point that There Are Lots Of People In China -- but I didn't know there were quite that many.
"With 300,000 million people playing basketball across the country — roughly the same number as the population of the United States..."
Really? I mean, it's almost a cliche by this point that There Are Lots Of People In China -- but I didn't know there were quite that many.
18 December 2007
Particularly numerically egregious typos
Language Log's Mark Liberman wrote this morning about The biggest typo in history, from this New York Times article, which referred to 10500 instead of 10500. (They've fixed the mistake now; I wonder if they found it independently or if someone pointed out the Language Log post to them?) The number here refers to the size of the string theory landscape, which I don't pretend to understand.
The Times made a similar mistake in The Death of Checkers, December 9, writing:
(James Kushner, a reader of this blog, pointed this out to me; I meant to mention it at the time but didn't.)
This particular mistake is actually so common that I barely notice it any more, especially because I usually have a decent idea of what the number should be. But to someone who doesn't know, it could make a big difference in how they perceive the article. For one thing, if you interpret the checkers quote literally, you get the idea that checkers is more complicated than chess: after all, 5 × 1020 =(削除) 5010 (削除ここまで) 5100.
Another paper that's been known to make these mistakes is The Tech, MIT's student newspaper. The difference here is that The Tech ought to know better; their entire staff is MIT students, who ought to be familiar with exponents. The Times at least has an excuse. (Disclaimer: I went to MIT as an undergraduate.)
Now if only I could get the Times to write about tilings of the plane, where they could say that an 80 by 80 rectangle has "10800" domino tilings. (That's "ten thousand eight hundred", as opposed to the correct number, which is around "ten to the eight hundredth".) Or just in general to write about statistical mechanics, which is where numbers like this come up.
(The Times article which provoked all this -- on the question of whether the laws of nature are inherently mathematical or whether we've just gotten lucky -- is interesting as well. The way I see it, even if a nonmathematical physics is possible, the mathematical physics is so far ahead that we might as well just act as if physics is ultimately governed by mathematics. But I don't think a nonmathematical physics that has the same explanatory power as the physics we have is possible. That might just be a matter of faith.)
The Times made a similar mistake in The Death of Checkers, December 9, writing:
The brute-force method is slow, which is its big limit. Schaeffer says he suspects you couldn’t use it to solve chess, because that game — with between 1040 and 1050 possible arrangements of pieces — is far more complicated than checkers, which has 5 × 1020 positions. “Chess won’t be solved in my lifetime,” he predicts. “We need some new breakthrough in technology to do that.
(James Kushner, a reader of this blog, pointed this out to me; I meant to mention it at the time but didn't.)
This particular mistake is actually so common that I barely notice it any more, especially because I usually have a decent idea of what the number should be. But to someone who doesn't know, it could make a big difference in how they perceive the article. For one thing, if you interpret the checkers quote literally, you get the idea that checkers is more complicated than chess: after all, 5 × 1020 =
Another paper that's been known to make these mistakes is The Tech, MIT's student newspaper. The difference here is that The Tech ought to know better; their entire staff is MIT students, who ought to be familiar with exponents. The Times at least has an excuse. (Disclaimer: I went to MIT as an undergraduate.)
Now if only I could get the Times to write about tilings of the plane, where they could say that an 80 by 80 rectangle has "10800" domino tilings. (That's "ten thousand eight hundred", as opposed to the correct number, which is around "ten to the eight hundredth".) Or just in general to write about statistical mechanics, which is where numbers like this come up.
(The Times article which provoked all this -- on the question of whether the laws of nature are inherently mathematical or whether we've just gotten lucky -- is interesting as well. The way I see it, even if a nonmathematical physics is possible, the mathematical physics is so far ahead that we might as well just act as if physics is ultimately governed by mathematics. But I don't think a nonmathematical physics that has the same explanatory power as the physics we have is possible. That might just be a matter of faith.)
Labels:
large numbers,
New York Times,
Overbye,
physics,
string theory
10 September 2007
Asimov on large numbers
Asimov on large numbers, including Skewes' number. (By the way, Skewes is pronounced in two syllables, which I didn't know, and Littlewood's middle name was apparently Edensor.)
Asimov also writes xy as x\y, which is a nice trick when the exponents get complicated, as they do here. Why is there not a non-subscript notation for powers, other than exp(x) for ex when the base happens to be e?
You can take any real number greater than 1 and write it as 1010円\...10円\k for some number of 10's and some constant 1 < k < 10; Skewes' number is 1010円10円34,円 or 1010円10円10円1円.53, so Asimov refers to this as a "quadruple-ten number". If I remember correctly Douglas Hofstadter uses this same idea of quantifying numbers by the number of exponentiations needed in one of his Metamagical Themas columns, although more correctly Asimov had it first.
Essentially all "real" numbers are either "single-ten" numbers (between 10 and 1010) or "double-ten" numbers (between 1010 = 1010円1円 and 101010 = 1010円10円)); note that the number of particles in the observable universe is much less than 1010円2円. (It's something like 1010円1円.90, which is actually much smaller than 1010円2円 even though it doesn't sound like it.)
There's a sense in which, say, 10\x and 10\y are close to each other if x is close to y. This is reflected in some asymptotic notation I've seen, where one writes f(x) ~ g(x) if (log f(x)/log g(x)) approaches 1 as x approaches infinity, so, for example, exp(x2) and exp(x2 + x) are "close together" in this sense. I don't know of cases where one can only show that log log f(x) and log log g(x) are asymptotically equal; this seems like too coarse a definition of "equality" to be of much use.
Asimov also writes xy as x\y, which is a nice trick when the exponents get complicated, as they do here. Why is there not a non-subscript notation for powers, other than exp(x) for ex when the base happens to be e?
You can take any real number greater than 1 and write it as 1010円\...10円\k for some number of 10's and some constant 1 < k < 10; Skewes' number is 1010円10円34,円 or 1010円10円10円1円.53, so Asimov refers to this as a "quadruple-ten number". If I remember correctly Douglas Hofstadter uses this same idea of quantifying numbers by the number of exponentiations needed in one of his Metamagical Themas columns, although more correctly Asimov had it first.
Essentially all "real" numbers are either "single-ten" numbers (between 10 and 1010) or "double-ten" numbers (between 1010 = 1010円1円 and 101010 = 1010円10円)); note that the number of particles in the observable universe is much less than 1010円2円. (It's something like 1010円1円.90, which is actually much smaller than 1010円2円 even though it doesn't sound like it.)
There's a sense in which, say, 10\x and 10\y are close to each other if x is close to y. This is reflected in some asymptotic notation I've seen, where one writes f(x) ~ g(x) if (log f(x)/log g(x)) approaches 1 as x approaches infinity, so, for example, exp(x2) and exp(x2 + x) are "close together" in this sense. I don't know of cases where one can only show that log log f(x) and log log g(x) are asymptotically equal; this seems like too coarse a definition of "equality" to be of much use.
09 September 2007
the xkcd number
Check out this xkcd comic, which includes among the "things that xkcd means":
"[xkcd] means calling the Ackermann function with Graham's number as the arguments just to horrify mathematicians."
The Ackermann function is defined as follows: A(m,n) = n+1 if m=0, A(m-1, 1) if m>0 and n=0, A(m-1, A(m,n-1)) if m>0 and n>0.
A(0,n) = n+1, which is obvious.
A(1,n) = A(0, A(1, n-1)) = A(1, n-1) + 1. Thus we have A(1,n) = n+k for some constant k. Now, A(1,0) = A(0,1) = 2, we have A(1,n) = n+2.
The same sort of reasoning leads to A(2,n) = 2n+3, A(3,n) = 2n+3 - 3.
A(4,n) can be written down it's 2^(2^(2^(2^...2)))...) - 3, where there are n+3 twos. (Incidentally, power towers should be evaluated from the right to the left; p^q^r is p^(q^r), not (p^q)^r. The reason I say this is because it's clear we should be consistent and either evaluate from left to right or from right to left, and there's already a perfectly good way of writing (p^q)^r, namely p^(qr).)
But A(5,n) and A(6,n) can't be written nicely.
And Graham's number, denoted g64, is already obscenely large; I'm not going to bother writing out how to define it, you can read Wikipedia. It's an upper bound for the answer to the following problem:
and the correct answer to this problem was suspected, for a long time, to be 6.
The author of xkcd claims that A(g64, g64) is probably the largest number ever concisely defined. Of course, one can always define a larger number. I nominate gk, where k = A(g64, g64); this number has the unfortunate property that it requires double-subscripts to write. In my opinion, good mathematical notation avoids double subscripts when at all possible.
Of course, this is all just a more mathematically sophisticated version of the "infinity plus one" that makes an appearance every so often on playgrounds. There's always a bigger number. (Although, if you want to get technical, all the numbers I mentioned are finite!)
And the xkcd number A(g64, g64) is only "concisely defined" if you've defined Graham's number and the Ackermann function, which themselves take some work. So what's the largest number you can write with, say, four characters, chosen from some "reasonable" set of characters in our standard mathematical notation? The two obvious candidates are A = 9^(9^(9^9))) or B = 9!!!. (Note that the first of these can be written in four characters if properly formatted.) In fact, A>B. We can see this by taking the logarithm of each of them twice
log A = 9^(9^9) log 9
log log A = log 9^(9^9) + log log 9
log log A = 9^9 log 9 + log log 9 = 851249821
Now, to take the log of a factorial, we recall Stirling's approximation, which I've used before. We have the approximate equality
log n! ~ 1/2 log (2π) - n + (n+1/2) log n
for our purposes we can use the much cruder approximation log n! < 2n log n. Then we have
log B < 2(9!!) log 9!!
log log B < log 2 + log(9!!) + log log (9!!)
log log B < log 2 + 2(9! log 9!) + log ( 2(9! log 9!) ) = 9291071.
Finally, all this business reminds me of a talk I went to about large numbers. Every time the speaker was about to prove that one ridiculously large number was larger than another, the fire alarm went off. It seemed as if the building didn't want us to talk about such large numbers that day, as if we were going to give away the secret of the universe.
"[xkcd] means calling the Ackermann function with Graham's number as the arguments just to horrify mathematicians."
The Ackermann function is defined as follows: A(m,n) = n+1 if m=0, A(m-1, 1) if m>0 and n=0, A(m-1, A(m,n-1)) if m>0 and n>0.
A(0,n) = n+1, which is obvious.
A(1,n) = A(0, A(1, n-1)) = A(1, n-1) + 1. Thus we have A(1,n) = n+k for some constant k. Now, A(1,0) = A(0,1) = 2, we have A(1,n) = n+2.
The same sort of reasoning leads to A(2,n) = 2n+3, A(3,n) = 2n+3 - 3.
A(4,n) can be written down it's 2^(2^(2^(2^...2)))...) - 3, where there are n+3 twos. (Incidentally, power towers should be evaluated from the right to the left; p^q^r is p^(q^r), not (p^q)^r. The reason I say this is because it's clear we should be consistent and either evaluate from left to right or from right to left, and there's already a perfectly good way of writing (p^q)^r, namely p^(qr).)
But A(5,n) and A(6,n) can't be written nicely.
And Graham's number, denoted g64, is already obscenely large; I'm not going to bother writing out how to define it, you can read Wikipedia. It's an upper bound for the answer to the following problem:
Consider an n-dimensional hypercube, and connect each pair of vertices to obtain a complete graph on 2n vertices. Then colour each of the edges of this graph using only the colours red and black. What is the smallest value of n for which every possible such colouring must necessarily contain a single-coloured complete sub-graph with 4 vertices that lies in a plane?
and the correct answer to this problem was suspected, for a long time, to be 6.
The author of xkcd claims that A(g64, g64) is probably the largest number ever concisely defined. Of course, one can always define a larger number. I nominate gk, where k = A(g64, g64); this number has the unfortunate property that it requires double-subscripts to write. In my opinion, good mathematical notation avoids double subscripts when at all possible.
Of course, this is all just a more mathematically sophisticated version of the "infinity plus one" that makes an appearance every so often on playgrounds. There's always a bigger number. (Although, if you want to get technical, all the numbers I mentioned are finite!)
And the xkcd number A(g64, g64) is only "concisely defined" if you've defined Graham's number and the Ackermann function, which themselves take some work. So what's the largest number you can write with, say, four characters, chosen from some "reasonable" set of characters in our standard mathematical notation? The two obvious candidates are A = 9^(9^(9^9))) or B = 9!!!. (Note that the first of these can be written in four characters if properly formatted.) In fact, A>B. We can see this by taking the logarithm of each of them twice
log A = 9^(9^9) log 9
log log A = log 9^(9^9) + log log 9
log log A = 9^9 log 9 + log log 9 = 851249821
Now, to take the log of a factorial, we recall Stirling's approximation, which I've used before. We have the approximate equality
log n! ~ 1/2 log (2π) - n + (n+1/2) log n
for our purposes we can use the much cruder approximation log n! < 2n log n. Then we have
log B < 2(9!!) log 9!!
log log B < log 2 + log(9!!) + log log (9!!)
log log B < log 2 + 2(9! log 9!) + log ( 2(9! log 9!) ) = 9291071.
Finally, all this business reminds me of a talk I went to about large numbers. Every time the speaker was about to prove that one ridiculously large number was larger than another, the fire alarm went off. It seemed as if the building didn't want us to talk about such large numbers that day, as if we were going to give away the secret of the universe.
Subscribe to:
Comments (Atom)