# Rational representation of arbitrary fractions.

Discussion in 'General Math' started by qWake, May 24, 2004.

1. ### qWakeGuest

Given an arbitrary fraction, is there a quick way to determine the most
accurate denominator in a given range (say from 2 to 9999999) to represent
the fraction in "numerator/denominator" format?

By "quick" I mean faster than testing all possible denominators in that
range for the smallest error in representation...

qWake, May 24, 2004

2. ### Horst KraemerGuest

Google for "FAREY sequence". A FAREY sequence is a sequence of optimal
approximations to a given rational number with increasing
denominators. Chose the element with highest denominator <=9999999. It
will be the best approximation for all denominators up to 9999999.

Horst Kraemer, May 25, 2004

3. ### Don McDonaldGuest

it is called the euclidean algorithm
continued fraction convergents.
And you don't need anything else.

as follows...
elementary/ primary school math.

don.lotto.
1.6.04..

From: (Don McDonald)
Newsgroups: alt.math
Subject: Re: Changing these fractions into decimals
Date: 3 Feb 2004 18:24:40 -0800
Lines: 259
Message-ID: <>
References: <>
<>
<401e45d1$0$18303\$>
....

From: Priscilla Periut ( )
ate: 2004-02-01 13:18:29 PST

:fractions are 71/177 69/190 69/170 51/158 and 28/92. thanks a lot

03.02.04 22:06
Thanks Priscilla.

Well, I guess that is not the proper question.
Not the question that may be in the book.

A mathematician (well me) wouldn't put it that way.
In my humble opnion.

2x fractions are approx 2/5ths =.40
which is greater, 71/177 or 69/170 ?? -- a very good problem.

2 x fractions are approx .36. the problem may have been
to arrange the fractions in increasing order.

the problem could be the
numerator is per cent mark in exam A and the
denominator is sum of percent in exam papers A AND B.

begin, enter start number (expression), q. end

CALCULATE B / A =158 / 58 *****************

= 2.72413793 recipr = 0.36708861
CTD-FRAC | NUMERATOR / DENOMIN |=DECIMAL
fraction

2 1/2= 0.5
1 1/3= 0.33333333333333331
2 3/8= 0.375
1 4/11= 0.36363636363636365
1 7/19= 0.36842105263157893
1 11/30= 0.36666666666666664
2 ** 29/79=*** 0.36708860759493672 ** REDUCES EXACT.
(2) 58
BEGIN, ENTER START NUMBER (Expression), Q. END

CALCULATE B / A =92 / 28 ** *************
= 3.28571429 recipr = 0.30434783
CTD-FRAC | NUMERATOR / DENOMIN |=DECIMAL
fraction

3 1/3= 0.33333333333333331
3 3/10= 0.29999999999999999
2 ** 7/23= 0.30434782608695654 ** reduces exact
(4) 28
BEGIN, ENTER START NUMBER (Expression), Q. END

?71
(5) 71 ƒCTD FRACTION,
CALCULATE B / A =177 / 71 ******************
= 2.49295775 recipr = 0.40112994
CTD-FRAC | NUMERATOR / DENOMIN |=DECIMAL
fraction
2 1/2= 0.5
2 2/5=********** 0.40000000000000002
PREV. LINE IS(very) Economical APPROXN.
ƒ

35 71/177= 0.40112994350282488
(6) 71 -?2

(7) 69 ƒCTD FRACTION,
RATIO TO DENOM 'A' (default 1)
(ENTER NO. OR EXPRESSION) ?190
190 = 190

( NEEDED TO SWAP SO THAT 0< A <= B )
CALCULATE B / A =190 / 69 ****************
= 2.75362319 recipr = 0.36315789
CTD-FRAC | NUMERATOR / DENOMIN |=DECIMAL
fraction

2 1/2= 0.5
1 1/3= 0.33333333333333331
3 *** 4/11***= 0.36363636363636365
PREV. LINE IS(very) Economical APPROXN.
ƒ

17 69/190= 0.36315789473684212
(8) 69 ƒCTD FRACTION,
RATIO TO DENOM 'A' (default 1)
(ENTER NO. OR EXPRESSION) ?170
170 = 170

( NEEDED TO SWAP SO THAT 0< A <= B )
CALCULATE B / A =170 / 69 ***
= 2.46376812 recipr = 0.40588235
CTD-FRAC | NUMERATOR / DENOMIN |=DECIMAL
fraction

2 1/2= 0.5
2 2/5=*** 0.40000000000000002
PREV. LINE IS(very) Economical APPROXN.
ƒ

6 13/32= 0.40625
2 28/69= 0.40579710144927539
2 69/170= 0.40588235294117647

(11) 69
BEGIN, ENTER START NUMBER (Expression), Q. END

?Q. PROGRAM SERIcalc4S E N D.

From: Henry ( )
Subject: Re: Changing these fractions into decimals
....
:
riscilla - are you any relation of Lillian?
:
:Try with your calculator: for example press "77" then the "divide
key"
:then "177" then "=" to get something like 0.4350282.

What do I learn from that? / don.l

From: Peter Webb ( )
Subject: Re: Changing these fractions into decimals

: Date: 2004-02-02 04:42:58 PST
: If you want an approximate answer, use a calculator like Henry said.
:
: If you want an exact answer with
repeating decimals then you should use long
: division.
:
: Divide (say) 71.00000000... by 177 using long division. Keep
carrying the
: remainder until it starts to repeat. You might be doing this for a
while; if
: you divide by 177 there are 177 possible remainders each time (0 up
to 176)
: so you may have to do up to 177 carries until you get a repeat ....
:
: Maybe you should try on 2/7 to see how it works ... you should get a
repeat
: after 6 carries, giving 0.285714, so you know the full answer is
0.285714
: 285714 285714 285714 285714 ....

Quite good.
I think I just posted.. 03.02.04 today.

Pi = 3.1 ... ... 713 999 9999 317 .... +(11 even digits.)
Many palindromes. (Pi-search. angio net?)
call it 5/7ths.

but pi does not repeat with a fixed length.

don.mcdonald.
03.02.04 23:05

Don McDonald, Jun 1, 2004
4. ### Don McDonaldGuest

very helpful. great. I learnt something liitle.

Search Result 6
From: Dave Rusin ()
Subject: Re: fractional representations of decimals
Original Format
Newsgroups: sci.math
Date: 1999/01/14

-...
the article includes (Dr?) Rusin's websites?

/ don.lotto
1.6.04

Don McDonald, Jun 1, 2004