# continued fraction convergents sqrt(2), sqrt(77).

Don McDonald parabola at paradise.net.nz
Thu May 1 13:15:36 CEST 2003

```don reble, seq fans

I use the Euclidean algorithm to generate a sequence of
continued fraction convergents.
To machine precision.

(an inferior method is  ..
integer part
fractional part
reciprocal
repeat.  )

I do not know any other software that uses my method to produce
a list?

continued fractions map the rationals 1-1 onto the natural numbers.

11 ooo 1 o 1111 ooo ,etc.
(Farey sequence, 1= right, o = left.)

/  don.mcdonald
01.05.03  22:47
>  cD.Calc.Profile.eisintegsq.Seqfan.cf-Farey

BEGIN, ENTER START NUMBER (Expression),  Q. END
?2
(1) 2  SQUARE ROOT, v
(2) 1.41421356  DISPLAY COMMA'S (,000)
1.414,213,562,373,095,15

(3) 1.41421356  ƒCTD FRACTION,
RATIO TO  DENOM 'A' (default 1)
(ENTER NO. OR EXPRESSION) ?
= 1
CALCULATE B / A =1.41421356 / 1
= 1.41421356  recipr = 0.70710678
CTD-FRAC | NUMERATOR / DENOMIN |=DECIMAL
fraction

1 1/1= 1
2 3/2= 1.5
2 7/5= 1.39999999999999991
2 17/12= 1.41666666666666674
2 41/29= 1.4137931034482758
2 99/70= 1.41428571428571437
2 239/169= 1.41420118343195256
2 577/408= 1.41421568627450989
2 1393/985= 1.41421319796954315
2 3363/2378= 1.41421362489486957
2 8119/5741= 1.41421355164605478
2 19601/13860= 1.41421356421356426
2 47321/33461= 1.41421356205732041
2 114243/80782= 1.41421356242727336
2 275807/195025= 1.41421356236379947
2 665857/470832= 1.41421356237468987
2 1607521/1136689= 1.41421356237282136
2 3880899/2744210= 1.414213562373142
2 9369319/6625109= 1.41421356237308693
2 22619537/15994428= 1.41421356237309648
2 54608393/38613965= 1.4142135623730947
1 77227930/54608393= 1.41421356237309537
loss of accuracy

1 131836323/93222358= 1.41421356237309515
1 209064253/147830751= 1.41421356237309515
2 549964829/388883860= 1.41421356237309515
LIMIT , BIG ENOUGH

(4) 1.41421356

BEGIN, ENTER START NUMBER (Expression),  Q. END
?77
(5) 77  SQUARE ROOT, v

(6) 8.77496439  ƒCTD FRACTION,
RATIO TO  DENOM 'A' (default 1)
(ENTER NO. OR EXPRESSION) ?
= 1
CALCULATE B / A =8.77496439 / 1
= 8.77496439  recipr = 0.11396058
CTD-FRAC | NUMERATOR / DENOMIN |=DECIMAL
fraction

8 8/1= 8
1 9/1= 9
3 35/4= 8.75
2 79/9= 8.77777777777777857
3 272/31= 8.77419354838709609
1 351/40= 8.77500000000000036
PREVIOUS LINE IS GOOD APPROXIMATION

16 5888/671= 8.77496274217585714
1 6239/711= 8.77496483825597728
3 24605/2804= 8.77496433666191145
2 55449/6319= 8.77496439310017351
3 190952/21761= 8.77496438582785743
1 246401/28080= 8.77496438746438834
PREVIOUS LINE IS GOOD APPROXIMATION

i think the last group repeats.

16 4133368/471041= 8.77496438738878304
1 4379769/499121= 8.77496438739303741
3 17272675/1968404= 8.77496438739201956
2 38925119/4435929= 8.77496438739213325
4 172973151/19712120= 8.77496438739212259
PREVIOUS LINE IS GOOD APPROXIMATION

6 1.07676402E9/122708649= 8.77496438739212259
1 1.24973718E9/142420769= 8.77496438739212259
LIMIT , BIG ENOUGH
(7) 8.77496439

Q. PROGRAM  SERIcalc4S  E N D.

```