From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: fractional representations of decimals
Date: 14 Jan 1999 17:39:36 GMT
Keywords: Measures of quality of approximations by rationals
Michael Hamm wrote:
>The number 0.43 can be represented by several fractions: 43/100 is perfect
>in accuracy, but is unwieldy in that the denominator has three digits; 1/2
>is very wieldy (that's not a word, but you get my point), but is not very
>accurate; and 3/7 is a good compromise.
>
>If there is some way of measuring unwieldiness (say, proportional to the
>value of the denominator, or proportional to the number of its digits, or
>something of the sort), and some way of measuring accuracy (say, percent
>difference from actual value or something), then the two values could be
>combined to form some sort of criterium for which is the best fractional
>representation of a decimal.
You want fractions p/q so that |x - p/q| is small relative to p and q.
You can ask for having |x - p/q| smaller than |x - p'/q'| for every
fraction with q' < q ; those are the Farey approximants of x.
Note that you can certainly have |x - p/q| be less than 1/q simply by
choosing the right p, so perhaps what's more remarkable is when the
ratio E(p/q) = ( |x - p/q| ) / ( 1/q ) is small. The fractions
having E(p/q) less than E(p'/q') for every fraction with q' < q
are the continued fraction approximants of x.
Farey approximants include continued-fraction approximants.
One can show that |x - p/q| < 1/q^2 for all continued-fraction approximants
p/q of a number x. Indeed, the difference is approximately (1/N)*(1/q^2),
where N is the next term in the continued-fractions expansion of x, so
you can set your level of attention to, say, occurences of term greater than
10 in the continued-fractions expansion of x.
Examples:
Pi = 3.14159... has the Farey sequence
3/1 4/1 7/2 10/3 13/4 16/5 19/6 22/7 25/8 47/15 69/22 ...
If you recognize "freshman addition", you might be able to discern the
algorithm (Hint: which numbers are too big? Which too small?)
The continued-fraction approximants abbreviate this sequence by jumping
to the next change between too-big and too-small:
3/1 22/7 333/106 355/113
The terms following these in the continued-fraction exansion of Pi are
7, 15, 1, and 292 respectively, which is why people remember the "3",
"22/7", and "355/113" but not the "333/106".
log(3)/log(2) = 1.5849625... has Farey sequence
1/1 2/1 3/2 5/3 8/5 11/7 19/12 27/17 46/29 65/41 84/53
and continued-fraction sequence
1/1 2/1 3/2 8/5 19/12 65/41 84/53 485/306 1054/665
followed by 1 2 2 3 1 5 2 23
respectively, that is, none except the last is really remarkable, I guess.
The denominators 5, 12, 41, ... suggest useful partitions of a
musical octave into 5, 12, 41, ... equal tones, if you want transposable
music which has a good approximation of a pure "fifth".
Your number 0.43 has a Farey sequence
0/1 1/1 1/2 1/3 2/5 3/7 4/9 7/16 10/23 13/30 ... 43/100
among which we may select the continued-fraction approximants
0/1 1/2 3/7 43/100
Since the continued-fraction expansion is just [0,2,3,14], the penultimate
term is rather good (and of course the last is perfect).
Exercise: which real number in the interval [0,1] is the worst in terms
of finding "remarkable" rational approximations?
dave
Continued fractions:
http://www.math.niu.edu/~rusin/known-math/index/11AXX.html
Quality of rational approximations in general:
http://www.math.niu.edu/~rusin/known-math/index/11JXX.html
==============================================================================
[Answer to exercise: 1/phi=[1,1,1,...]=0.618... --djr]