[Project Euler] – Problem 57

May 19, 2011
By

(This article was first published on YGC, and kindly contributed to R-bloggers)

It is possible to show that the square root of two can be expressed as an infinite continued fraction.

√ 2 = 1 + 1/(2 + 1/(2 + 1/(2 + … ))) = 1.414213…

By expanding this for the first four iterations, we get:

1 + 1/2 = 3/2 = 1.5
1 + 1/(2 + 1/2) = 7/5 = 1.4
1 + 1/(2 + 1/(2 + 1/2)) = 17/12 = 1.41666…
1 + 1/(2 + 1/(2 + 1/(2 + 1/2))) = 41/29 = 1.41379…

The next three expansions are 99/70, 239/169, and 577/408, but the eighth expansion, 1393/985, is the first example where the number of digits in the numerator exceeds the number of digits in the denominator.

In the first one-thousand expansions, how many fractions contain a numerator with more digits than denominator?

?View Code RSPLUS
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
26
27
28
29
30
31
32
f <- function(i) {
	#x <- 1/2
	x <- c(1,2) ## numerator, denominator
	x <- gmp::as.bigz(x) # to support large sized integer values
	if (i > 1) {
		for (j in 2:i) {
			#x= 1/(2+x)
			x <- rev(c(x[2] * 2 + x[1], x[2]))
		}
	}
	#x <- 1 + x
	x <- c(x[2] + x[1], x[2])
	return(x)
}
 
dec.len <- function(d) {
	l <- length(unlist(strsplit(as.character(d), split="")))
	return(l)
}
 
is.longer <- function(x) {
	# x is a vector of length 2 c(numerator, denomiator)
	num.len <- dec.len(x[1])
	denom.len <- dec.len(x[2])
	if (num.len > denom.len) {
		return(TRUE)
	} else {
		return(FALSE)
	}
}
 
idx <- sapply(1:1000, function(x) is.longer(f(x)))
> sum(idx)
[1] 153

Related Posts

To leave a comment for the author, please follow the link and comment on his blog: YGC.

R-bloggers.com offers daily e-mail updates about R news and tutorials on topics such as: visualization (ggplot2, Boxplots, maps, animation), programming (RStudio, Sweave, LaTeX, SQL, Eclipse, git, hadoop, Web Scraping) statistics (regression, PCA, time series, trading) and more...



If you got this far, why not subscribe for updates from the site? Choose your flavor: e-mail, twitter, RSS, or facebook...

Tags: , ,

Comments are closed.