HP 71B, TI 84 Plus CE: Sum of Egyptian Fractions
Introduction
An Egyptian fraction consists of a sum of unit fractions. A unit fraction is a fraction with 1 in its numerator and a positive integer in its denominator.
Today's blog entry will calculate the Egyptian fraction given unit fractions, full simplified. On tomorrow's entry, we will attempt the reverse process, breaking down an Egyptian fraction into a sum of unit fractions.
N/D = 1/A1 + 1/A2 + 1/A3 + 1/A4 + ....
The Algorithm
Start out with two unit fractions 1/A1 and 1/A2.
Let N2/D2 = 1/A1 + 1/A2
Then:
N2/D2 = 1/A1 + 1/A2
N2/D2 = (A1 + A2)/(A1*A2)
Let N3/D3 = 1/A3 + N2/D2
Then: N3/D3 = (A3*N2 + D2)/(N2*D2)
For any general point:
N_x+1/D_x+1 = (N_x * A_x + D_x)/(A_x * D_x)
This allows the user to input as many unit fractions as allowed.
TI-84 Plus CE Program EGYPTSUM
Disp "EGYPT FRACTION SUM"
"2022-09-03 EWS"
Input "NO. TERMS? ",S
Input "1/A1? ",A
Input "1/A2? ",B
A+B→N
A*B→D
For(I,3,S)
"1/A"+toString(I)+"? "→Str1
Input Str1,A
A*N+D→N
A*D→D
End
gcd(N,D)→G
N/G→N
D/G→D
Disp "NUM= "+toString(N)
Disp "DEC= "+toString(D)
Disp "FRAC="+toString(N/D)
* This assumes your TI-84 Plus CE is at least has the 5.5 operating system.
HP 71B Program EGYPTSUM
100 DISP "EGYPT FRACTION SUM" @ WAIT .5
105 DESTROY S,A,B,I,N,D,G
110 INPUT "# TERMS? ";S
115 INPUT "1/A1? ";A
120 INPUT "1/A2? ";B
125 N=A+B @ D=A*B
130 IF S=2 THEN 300
200 FOR I=3 TO S:
205 DISP "1/A"&STR$(I)&"?" @ WAIT .5
210 INPUT "1/A? ";A
215 N=A*N+D
220 D=A*D
225 NEXT I
300 A=N @ B=D
305 G=MOD(A,B)
310 A=B @ B=G
315 IF G THEN 305
320 N=N/A @ D=D/A
400 DISP "NUM= ";N
405 DISP "DEN= ";D
410 DISP "FRAC= ";N/D
415 STOP
The 300-325 section calculates the common greatest divisor, based off the "Greatest Common Divisor" program from 119 Practical Programs for the TRS-80 Pocket Computer (see source below).
Examples:
1/2 + 1/6
# TERMS = 2
Result: 2/3 (NUM = 2, DEN = 3)
1/7 + 1/8 + 1/2
# TERMS = 3
Result: 43/56 (NUM = 43, DEN = 56)
1/9 + 1/5 + 1/15 + 1/8
# TERMS = 4
Result: 181/360 (NUM = 181, DEN = 360)
Source
Craig, John Clark. 119 Practical Programs For The TRS-80 Pocket Computer TAB Books Inc. Blue Ridge Summit, PA. 1982 pg. 133
All original content copyright, © 2011-2022. 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.
