## Babylonian Irregular Reciprocal V4

### Preface

gold Here is TCL calculations for Babylonian irregular reciprocal algorithm in calculator shell.

### Introduction

In the cuneiform math problems and coefficient lists on clay tablets, there are coefficient numbers which were used in determining the amount of materials and the daily work rates of the workers. In most cases, the math problem is how the coefficient was used in estimating materials, work rates, and math problems. One difficulty is determining the effective magnitude or power of the number coefficient in the base 60 notation. In cuneiform, numbers in base 60 are written using a relative notation. For example, 20 could represent either 20*3600,20,20/60, 20/3600, or even 1/20. The basic dimensions and final tallies were presented in the cuneiform accounts on clay tablets, but some calculations, some units, and some problem answers (aw shucks!) were left off the tablet. Successive or iterated math solutions are called algorithms and the Babylonian methods are some of the earliest algorithms documented circa 1600 BCE. The TCL procedures are descendants of this idea. The Babylonians did not use algebra notation, decimal notation, or modern units, so the reader will have to bear some anachronisms in the TCL code. At least one approach for the modern reader and using modern terminology is to develop the implied algebraic equations and decimal equivalents from the cuneiform numbers. Then the TCL calculator can be run over a number of testcases to validate the algebraic equations.

## Pseudocode Section

```           # using pseudocode for Babylonian irregular reciprocal algorithm.
# possible problem instances include, given irregular n , find 1/n
# irregular defined as not in standard B. table of reciprocals
target_number= supplied value
#   decompose target number
c = a + b
a+b is not unique, test a for prime, even, odd?
find a as factorable into 2**x, 3**y, 5**z, or n/<(2**x)*(3**y)*(5**z)>???
take (1/a)
table lookup (1/a)
take (1/a)*(b)
take 1/(1+(1/a)*(b))
take  (1/a)* (1/(1+(1/a)*(b)))
check_answer new area =? desired goal ,  n*(1/n) =? 1  , (yes/no logic)
set answers and printout with resulting values```

### Testcases Section

In planning any software, it is advisable to gather a number of testcases to check the results of the program. The math for the testcases can be checked by pasting statements in the TCL console. Aside from the TCL calculator display, when one presses the report button on the calculator, one will have console show access to the capacity functions (subroutines).

#### Testcase 1

table 1printed in tcl wiki format
quantity value comment, if any
1:testcase_number
10.0 :target number (c=a+b) meters
2.0 :decomposed a meters
8.0 :decomposed b meters
1. :optional
1. :optional
1.0 :check product c*(1/c) =? 1
0.100 :irregular reciprocal meters

#### Testcase 2

table 2printed in tcl wiki format
quantity value comment, if any
2:testcase_number
20.0 :target number (c=a+b) meters
4.0 :decomposed a meters
16.0 :decomposed b meters
1. :optional
1. :optional
1.0 :check product c*(1/c) =? 1
0.050 :irregular reciprocal meters

#### Testcase 3

table 3printed in tcl wiki format
quantity value comment, if any
3:testcase_number
5.0 :target number (c=a+b) meters
2.0 :decomposed a meters
3.0 :decomposed b meters
1. :optional
1. :optional
1.0 :check product c*(1/c) =? 1
0.200 :irregular reciprocal meters

## Appendix Code

### appendix TCL programs and scripts

```        # pretty print from autoindent and ased editor
# Babylonian Irregular Reciprocal Algorithm calculator
# written on Windows XP on eTCL
# working under TCL version 8.5.6 and 1.0.1
# gold on TCL WIKI, 25jan2017
package require Tk
package require math::numtheory
namespace path {::tcl::mathop ::tcl::mathfunc math::numtheory }
set tcl_precision 17
frame .frame -relief flat -bg aquamarine4
pack .frame -side top -fill y -anchor center
set names {{} { target number (c=a+b) meters :} }
lappend names { decomposed a meters :}
lappend names { decomposed b meters : }
lappend names { answers: optional : }
lappend names { optional :}
lappend names { optional: }
lappend names { check product c*(1/c) =? 1 : }
lappend names { irregular reciprocal 1/meters :}
foreach i {1 2 3 4 5 6 7 8} {
label .frame.label\$i -text [lindex \$names \$i] -anchor e
entry .frame.entry\$i -width 35 -textvariable side\$i
grid .frame.label\$i .frame.entry\$i -sticky ew -pady 2 -padx 1 }
proc about {} {
set msg "Calculator for Babylonian Irregular Reciprocal Algorithm
from TCL WIKI,
written on eTCL "
tk_messageBox -title "About" -message \$msg }
proc calculate {     } {
global side1 side2 side3 side4 side5
global side6 side7 side8
global testcase_number
incr testcase_number
set side1 [* \$side1 1. ]
set side2 [* \$side2 1. ]
set side3 [* \$side3 1. ]
set side4 [* \$side4 1. ]
set side5 [* \$side5 1. ]
set side6 [* \$side6 1. ]
set side7 [* \$side7 1. ]
set side8 [* \$side8 1. ]
set target_number \$side1
set decom_a \$side2
set decom_b \$side3
set term1 1
set term2 1
# initialize placeholder answer
set reciprocal 1.
catch {set term1 [* [/ 1. \$decom_a ] \$decom_b ]}
set term2 [/ 1. [+ 1. \$term1 ]]
set reciprocal [* [/ 1. \$decom_a ] \$term2 ]
set check_answer_product [* \$target_number \$reciprocal ]
# check for lazy entries of zero, revert to modern way of reciprocals
if { \$side2 < .00001 } { set reciprocal [/ 1. \$target_number ] }
if { \$side3 < .00001 } { set reciprocal [/ 1. \$target_number ] }
if { \$side2 < .00001 } { set check_answer_product [* \$target_number \$reciprocal ] }
if { \$side3 < .00001 } { set check_answer_product [* \$target_number \$reciprocal ] }
set side5 1.
set side6 1.
set side8 \$reciprocal
}
proc fillup {aa bb cc dd ee ff gg hh} {
.frame.entry1 insert 0 "\$aa"
.frame.entry2 insert 0 "\$bb"
.frame.entry3 insert 0 "\$cc"
.frame.entry4 insert 0 "\$dd"
.frame.entry5 insert 0 "\$ee"
.frame.entry6 insert 0 "\$ff"
.frame.entry7 insert 0 "\$gg"
.frame.entry8 insert 0 "\$hh"
}
proc clearx {} {
foreach i {1 2 3 4 5 6 7 8 } {
.frame.entry\$i delete 0 end } }
proc reportx {} {
global side1 side2 side3 side4 side5
global side6 side7 side8
global testcase_number reference_factor flag
console show;
puts "%|table \$testcase_number|printed in| tcl wiki format|% "
puts "&| quantity| value| comment, if any|& "
puts "&| \$testcase_number:|testcase_number | |& "
puts "&| \$side1 :|target number (c=a+b) meters |   |&"
puts "&| \$side2 :|decomposed a meters | |& "
puts "&| \$side3 :|decomposed b meters | |& "
puts "&| \$side4 :|answers: optional| |&"
puts "&| \$side5 :|optional  | |&"
puts "&| \$side6 :|optional |  |&"
puts "&| \$side7 :|check product c*(1/c) =? 1  |  |&"
puts "&| \$side8 :|irregular reciprocal meters |  |&"
}
frame .buttons -bg aquamarine4
::ttk::button .calculator -text "Solve" -command { calculate   }
::ttk::button .test2 -text "Testcase1" -command {clearx;fillup 10.  2.  8.0 1.  1.  1. 1. 0.1}
::ttk::button .test3 -text "Testcase2" -command {clearx;fillup 20.  4.0  16.  1.  1.  1. 1. .05 }
::ttk::button .test4 -text "Testcase3" -command {clearx;fillup 5.   2.0  3.0 1.  1.  1. 1. .2 }
::ttk::button .clearallx -text clear -command {clearx }
::ttk::button .cons -text report -command { reportx }
::ttk::button .exit -text exit -command {exit}
pack .calculator  -in .buttons -side top -padx 10 -pady 5
pack  .clearallx .cons .about .exit .test4 .test3 .test2   -side bottom -in .buttons
grid .frame .buttons -sticky ns -pady {0 10}
. configure -background aquamarine4 -highlightcolor brown -relief raised -border 30
wm title . "Babylonian Irregular Reciprocal Algorithm Calculator"   ```

### Pushbutton Operation

For the push buttons, the recommended procedure is push testcase and fill frame, change first three entries etc, push solve, and then push report. Report allows copy and paste from console.

For testcases in a computer session, the eTCL calculator increments a new testcase number internally, eg. TC(1), TC(2) , TC(3) , TC(N). The testcase number is internal to the calculator and will not be printed until the report button is pushed for the current result numbers. The current result numbers will be cleared on the next solve button. The command { calculate; reportx } or { calculate ; reportx; clearx } can be added or changed to report automatically. Another wrinkle would be to print out the current text, delimiters, and numbers in a TCL wiki style table as

```  puts " %| testcase \$testcase_number | value| units |comment |%"
puts " &| volume| \$volume| cubic meters |based on length \$side1 and width \$side2   |&"  ```