TI-84 Plus CE and HP 17BII+: Euclidian Divison

 TI-84 Plus CE and HP 17BII+: Euclidian Division

What is the ├ symbol on the calculator?

(├ the symbol is a vertical line with a horizontal line coming from the center to the right)

In several scientific calculators sold in Europe such as the Texas Instruments TI-40 Galaxy:

http://www.datamath.org/Sci/Galaxy/TI40_Galaxy.htm

and the current TI-Primaire Plus, which is sold in France, 

http://www.datamath.org/Sci/Modern/TI-PrimairePlus.htm

There is a key marked [ ├ ].  Going through the manual for the TI-40 Galaxy, this is the Euclidean Division key, which returns the quotient and remainder of the division.

x ├ y returns two results:

Q = int(x/y)   (quotient)

R = x - frac(x/y) * y   (remainder)

The TI-84 Plus family has the remainder function (I think it is OS 2.5 and later).  Here are several ways to emulate the Euclidian Division function:

TI-84 Plus CE Program:  EUCDIV

Prompt X,Y

{int(X/Y),remainder(X,Y)}

HP 17BII+ Formula:

EUCDIV: 0=IF(S(Q): IP(X÷Y)-Q: MOD(X:Y)-R)

Inputs: X,Y

Outputs:  Q (quotient), R (remainder)

Examples:

25 ├ 7 ->  Q = 3, R = 4

77 ├ 6 ->  Q = 12, R = 5

Can we use the [ a b/c ] fraction key?  

Calculators such as the Casio fx-260 Solar II calculator could be used to simulate the Euclidean division with the [ a b/c ] key.  The fraction would be illustrated in the mixed fraction format.  Let us illustrate:

56 [ a b/c ] 5 [ = ]   11 _ 1 / 5    

(Q = 11,  R = 1, divisor = 5)

125 [ a b/c ] 14 [ = ]  8 _ 13 / 14

(Q = 8,  R = 14, divisor = 14)

So far so good, but...

9314 [ a b/c ] 60 [ = ]  155 _ 7 / 30

Look at the denominator:  it is 30 not in 60, the fraction has been simplified.   Of course, we know that 7/30 = 14/60, hence Q = 155, R = 14. 

Note:  if you use the [ a b/c ] key, be sure to look at the resulting denominator is not the same,  it is not in the Q _ R / D form.  

Eddie

All original content copyright, © 2011-2021.  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. 

HP 42S & Casio fx-260 Solar: Continued Fractions

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.