FunWithNumbers (version 1.2)

bestFrac: Generate a fraction close to the input value.

Description

Inspired by the well-known approximations to pi, i.e. 22/7 and 355/113, this function allows the user to find the best-match fraction for any number, within the specified maximum magnitude of the numerator and denominator

Usage

bestFrac(x, intrange)

Value

bestmatch

The numerator and denominator of the best-matching fraction

goodmatch

An N-by-2 array of the progressively better matches found (numerators and denominators in the columns)

matcherr

A vector of the differences between the 'matcherr' fractions and the input value. This is limited in precision to the machine limit for doubles (floats).

Arguments

x

A character string representing a number to be "converted" to a fraction of nearly equal value.

intrange

If a single value, the function tests all combinations of numerator and denominator between one and intrange . If two values, the 'testing range' is intrange[1]:intrange[2]. Otherwise, whatever vector of values is supplied will be used.

Author

Carl Witthoft,

Details

For irrationals and the like, the simplest way to generate the input parameter string x is to use sprintf with as many digits to the right of the decimal point as desired. The returned values are in reduced form, i.e. the numerator and denominator are relatively prime.

Examples

gpi <- sprintf("%1.30f", pi) 
bestFrac(gpi, 100)
# $bestmatch
# [1] 22  7
# $goodmatch
#           [,1] [,2]
# goodmatch    0    0
#              1    1
#              2    1
#              3    1
#             13    4
#             16    5
#             19    6
#             22    7
# $matcherr
# [1] 1.000000e+02 6.816901e-01 3.633802e-01 4.507034e-02 3.450713e-02
#     1.859164e-02 7.981306e-03 4.024994e-04
bestFrac(gpi, 100:400)
# $bestmatch
# [1] 355 113
# $goodmatch
#           [,1] [,2]
# goodmatch    0    0
#            100   31
#            100   32
#            101   32
#            104   33
#            107   34
#            110   35  # notice this is 22/7 
#            179   57
#            201   64
#            223   71
#            245   78
#            267   85
#            289   92
#            311   99
#            333  106
#            355  113
# $matcherr
#  [1] 1.000000e+02 2.680608e-02 5.281606e-03 4.665578e-03 3.158429e-03
#      1.739936e-03 4.024994e-04
#  [8] 3.952697e-04 3.080137e-04 2.379631e-04 1.804857e-04 1.324752e-04 
#      9.177057e-05 5.682219e-05
# [15] 2.648963e-05 8.491368e-08)