HP 42S & Casio fx-260 Solar: Continued Fractions
Introduction
Let x be a real number. Then x can be represented by the fraction:
x = n_1 + 1/(n_2 + 1/(n_3 + 1/(n_4 + 1/(n_5 + ...))))
The above form is known as a continuous fraction, which can either have a finite set of terms or infinite set of terms. In short form, continuous fractions can be written in a vector form:
x = [ n_1, n_2, n_3, n_4, n_5, ... ]
If you are given a continuous fraction (n_1, n_2, etc.), you can calculate x with the following keystrokes, starting with the last term n_k and working left to n_1:
RPN Calculators
1. Start by entering n_k, then press [ 1/x ]
2. Loop: For each n_m for 1 < m < k: enter n_m, [ + ], [ 1/x ]
3. For n_1: Enter n_1, [ + ]
Remember, we are working leftwards.
Example:
Calculate 2 + 1/( 3 + 1/(5 + 1/2)). In other words x = [2, 3, 5, 2]
2 [ 1/x ]
5 [ + ] [ 1/x ]
3 [ + ] [ 1/x ]
2 [ + ]
Result: 81/35 ≈ 2.31429
The program CF for the HP 42S (and Swiss Micros DM42 and Free42 emulator) calculates the value of a continued fraction. Instructions:
1. Run CF ( [ XEQ ] (CF) )
2. Enter n_k, press (LAST)
3. For each n_m for 1 < m < k, enter n_m, press (MID)
4. For n_1, enter n_1, press (1ST). You get the result.
HP 42S/DM42/Free 42 Program CF
00 {56-Byte Prgm}
01 LBL "CF"
02 LBL 04
03 CLMENU
04 "LAST"
05 KEY 1 GTO 01
06 "MID"
07 KEY 2 GTO 02
08 "1ST"
09 KEY 3 GTO 03
10 MENU
11 LBL 00
12 STOP
13 GTO 00
14 LBL 01
15 1/X
16 GTO 04
17 LBL 02
18 +
19 1/X
20 GTO 04
21 LBL 03
22 +
23 CLMENU
24 EXITALL
25 RTN
26 END
Classic Algebraic (AOS) Calculators
We can use the same strategy for classic algebraic calculators such as the Casio fx-260 Solar II and TI-30Xa Solar:
1. Start by entering n_k, then press [ 1/x ]
2. Loop: For each n_m for 1 < m < k: press [ + ], enter n_m, press [ = ], [ 1/x ]
3. For n_1: Enter n_1, [ + ]
Remember, we are working leftwards.
Calculate 2 + 1/( 3 + 1/(5 + 1/2)). In other words x = [2, 3, 5, 2]
2 [ 1/x ]
[ + ] 5 [ = ] [ 1/x ]
[ + ] 3 [ = ] [ 1/x ]
[ + ] 2 [ = ]
Result: 81/35 ≈ 2.31429
Eddie
All original content copyright, © 2011-2020. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.
Introduction
Let x be a real number. Then x can be represented by the fraction:
x = n_1 + 1/(n_2 + 1/(n_3 + 1/(n_4 + 1/(n_5 + ...))))
The above form is known as a continuous fraction, which can either have a finite set of terms or infinite set of terms. In short form, continuous fractions can be written in a vector form:
x = [ n_1, n_2, n_3, n_4, n_5, ... ]
If you are given a continuous fraction (n_1, n_2, etc.), you can calculate x with the following keystrokes, starting with the last term n_k and working left to n_1:
RPN Calculators
1. Start by entering n_k, then press [ 1/x ]
2. Loop: For each n_m for 1 < m < k: enter n_m, [ + ], [ 1/x ]
3. For n_1: Enter n_1, [ + ]
Remember, we are working leftwards.
Example:
Calculate 2 + 1/( 3 + 1/(5 + 1/2)). In other words x = [2, 3, 5, 2]
2 [ 1/x ]
5 [ + ] [ 1/x ]
3 [ + ] [ 1/x ]
2 [ + ]
Result: 81/35 ≈ 2.31429
The program CF for the HP 42S (and Swiss Micros DM42 and Free42 emulator) calculates the value of a continued fraction. Instructions:
1. Run CF ( [ XEQ ] (CF) )
2. Enter n_k, press (LAST)
3. For each n_m for 1 < m < k, enter n_m, press (MID)
4. For n_1, enter n_1, press (1ST). You get the result.
HP 42S/DM42/Free 42 Program CF
00 {56-Byte Prgm}
01 LBL "CF"
02 LBL 04
03 CLMENU
04 "LAST"
05 KEY 1 GTO 01
06 "MID"
07 KEY 2 GTO 02
08 "1ST"
09 KEY 3 GTO 03
10 MENU
11 LBL 00
12 STOP
13 GTO 00
14 LBL 01
15 1/X
16 GTO 04
17 LBL 02
18 +
19 1/X
20 GTO 04
21 LBL 03
22 +
23 CLMENU
24 EXITALL
25 RTN
26 END
Classic Algebraic (AOS) Calculators
We can use the same strategy for classic algebraic calculators such as the Casio fx-260 Solar II and TI-30Xa Solar:
1. Start by entering n_k, then press [ 1/x ]
2. Loop: For each n_m for 1 < m < k: press [ + ], enter n_m, press [ = ], [ 1/x ]
3. For n_1: Enter n_1, [ + ]
Remember, we are working leftwards.
Calculate 2 + 1/( 3 + 1/(5 + 1/2)). In other words x = [2, 3, 5, 2]
2 [ 1/x ]
[ + ] 5 [ = ] [ 1/x ]
[ + ] 3 [ = ] [ 1/x ]
[ + ] 2 [ = ]
Result: 81/35 ≈ 2.31429
Eddie
All original content copyright, © 2011-2020. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.
