This page is under development. Comments are welcome, but please load any comments in the comments section at the bottom of the page. Please include your wiki MONIKER and date in your comment with the same courtesy that I will give you. Its very hard to reply reasonably without some background of the correspondent on their Wiki bio page. Thanks,gold
gold Here are some calculations for Babylonian Brothers Inheritance problems using TCL expressions.
There are differences between the TCL calculator and the Babylonian solutions. The TCL calculator has room for only 5 slots of brothers on the existing displays. Solutions for beyond 5 brothers will have to loaded on a printout as overflow. The TCL calculator seeks base_10 solutions with modern decimal fractions. It is supposed that th calculator could force an integer solution in base_10, but that might be another constraint on the solution, if solvable. The Babylonians were seeking integer solutions or integer & fraction solutions that were most compatible with base_60 fractions. The Neugebauer solution for dividing 60 among 5 brothers is the sequence of integers 4,8,12,16,20 with constant delta of integer 4. As an original translator, Neugebauer probably had the best sense of the typical solution in base_60, that the Babylonians were shooting for. However, the Neugebauer solution is believed an arithmetic sequence, rather than a geometric solution found in most of the other problems, that we know of. The relevant TCL expressions are expr {4+8+12+16+20} = 60 and expr {4./60+8./60+12./60+16./60+20./60} = 60/60 or 1.
Lets try an alternate solution to the brothers inheritance problem with modern algebra with modern decimal and percent notation. The Babylonian judge has three brothers in an estate of 60 shekels. This alternate solution starts with the elder brother. In setting up the problem, the share of the first brother is called X1 and the share of the second brother would be (1-1/N) or (1-1/3). The share of the second brother would be (1-1/3), expr { 2./3} , or factor 0.666*X1. The share of the third brother would be (1-1/N)*(1-1/N), (1-1/3)*(1-1/3), expr { (2./3)*(2./3)} , or factor 0.4444*X1. Adding the shares in X1, the equation is X1+0.6666*X1+0.4444*X1 = 60 shekels. Reduction is expr { 1+.6666+.4444} = 2.11099 , 2.11099*X = 60. So X1 = expr { 60/2.11099 } and X1 = 28.4226 for first brother share. Second brother share would be expr { 28.4226*0.6666 } or 18.9461. The third brother share would be expr { 28.4226*0.4444} or 12.6307. The percents of brothers 1,2,&3 was 47, 32, & 21 percent. The check answer was expr { 28.4226+18.9461+12.6307 } or 59.9994, rounding to 60. The average share for three brothers was expr { 60./3 } or 20, somewhat close to second brother share. The TCL calculator solution for brothers 1,2,&3 sharing 60 was entered as +60/+5/+5. Starting with the elder brother, the resulting shares were returned {list 28.4211 18.9474 12.6316}, reasonably close to the hand calculation.
An alternate solution starting with the younger brother is possible. Once again, the Babylonian judge has three brothers in an estate of 60 shekels. The younger brother share is called X3. The second brothers factor is expr { (1+1./3)}. The second brothers share is factor 1.3333*X3. The elder brothers factor is expr { (1+1./3)*(1+1./3)}. The second brothers share is factor 1.7777*X3. Adding the shares in X3, the equation is (1.+1.3333+1.7777)*X3 = 60 shekels. From expr { 1+1.3333+1.7777 } equals 4.11099, the youngest brother share is X3 = expr { 60/4.1109} or 14.5953. The second brother share is expr { 14.5953*1.3333 } equals 19.4595. The elder brother share is expr { 14.5953*1.7777 } equals 25.9465. The check answer is expr { 14.5953+19.4595+ 25.9465 } equals 60.0013. The TCL calculator solution for brothers 1,2,&3 sharing 60 was entered as +60/+5/-5. The TCL shares were returned {list 14.5945 19.4594 25.9459 }, reasonably close to the hand calculations.
For an alternate inheritance solution, lets try the double portion inheritance method for the elder brother, mentioned in the Bible. The patriarch wishes to portion out 60 shekels to 3 brothers with the elder brother receiving a double portion. The generic formula for a portion or single share would be 1/(N+1), where N is the number of brothers. As understood here, the elder brother would receive 2/(N+1) and the younger brothers would receive 1/(N+1). Also, the formula works for one brother as the only and eldest brother would receive 2/(1+1). (2/2) reduces to one and the whole inheritance. For the three brothers, one would divide 60 by N+1, expr { 60./4 } = 15 shekels for a single share. Elder brother gets a portion of 2X or expr { 2*15 } or 30. So brothers 1,2,&3 get shares of 30,15,&15. The check answer is expr { 30+15+15} = 60. The shares of the brothers 1,2&3 are 50,25,&25 percent. In opinion, while elder brother still gets the lion's share, the Hebrew solution is more fair to the younger brothers. In the Babylonian inheritance rules, the youngest brother of a large family would get next to nothing even if the youngest brother could figure out the math!
As a trial solution, looking at using the TCL calculator for developing an estate or field division solution could be expected in a sequence of brother inheritance shares in real estate. In rounded numbers, the sequence for seven brothers was { list W = 235, 210,175,140,105,70,35} as field widths dividing a triangular field using a set of constant lengths { L = kay} and dividing the front of an inheritance, ref Friberg. In other words, the seven brothers would each receive a trapezoidal or other triangular surface area bounded by W(N),W(N-1), and L. The elder and middle brothers would receive trapezoidal fields bounded by W(N),W(N-1), and L. The youngest brother would receive a triangular field bounded by W(N-6) and L. The sum of widths approximates expr { 235+210+175+140+105+70+35} = 970 units. But the total field area was not a hard number given in the original tablet, but probably 900-1200 units and compatible with common area unit of one bur.
Getting into the weeds of Babylonian units, a bur was 1800 square nindas or 64800 square cubits. A ninda was 12 cubits or 6 meters. In Babylonian math problems, the classic rectangle was 4:3 side ratio and would hold two classic triangles, which used the 3:4:5 rule for sides. For the conceptual study, a rectangular field of 2 burs could be divided into 2 triangular fields of 1 bur or 64800 square cubits. A triangular field of one bur using the classic triangle rule of 3:4:5 would be approximated by width of 312 cubits and length of 416 cubits. The initial or total estate would be triangular area equals expr { .5*312*416 } square cubits.
Continuing with the problem set up, the length or front portion of each brother would be expr { 416./7}, 59.4, or rounding to 60 cubits. Maybe a mismatch, but the calculator solution for 900 units is { list Width = 208.06 173.38, 144.49, 120.409, 100.341, 83.61, 69.68 }. Another computer run for expr {64800./60 } or 1080 units was { list Width = 233.7, 200.3, 171.7, 147.19, 126.1, 108.1, 92.69} . Remember the last width is for a right triangle (Area= 0.5*L*W) and not quite figured out how to treat a triangle in the false shares algorithm. But the Babylonian solution on the tablet is maybe 0.5*L?
The formulas in the brothers inheritance problems have some similarity to the formulas in Babylonian interest rate problems. In other words, the Babylonian formulas and TCL code could be set or morphed to generate an increase crease curve or interest rate table. Usually, the Nth share in decreasing shares is { elder brother share } * (1-1/b)**(N-1). With a different constant c, the Nth share is { elder brother share } * (1-1/c)**(N-1). For example, starting with the youngest brothers share and calculating the increase for the successive brothers in a stairstep, the Nth share in increasing shares is { youngest brother share } * (1+1/c)**(N-1). Resetting the variables for an interest rate problem, the Principal and Interest (P&I) would increase every year in the formula P&I =(1+1/c)**(N+1), where 1/c is interest factor and N is the year. For example, the Babylonian rule that silver would double in 5 years works out as P&I = expr { (1+1./5)**4 } or 2.0735, as P&I past the fourth year.
The TCL calculator solution in base_10 is about +- 3 percent absolute error. That is, 3 percent difference could be expected from the available Babylonian solutions in base_60 on the clay tablets. On one Babylonian inheritance problem with a method and answers in clear cuneiform, the TCL calculator showed about 0.03 percent error from the textbook solution. So far, the majority of the problems do not show the worked methods or Babylonian answers. But the internal algorithm in the TCL calculator is on the right track. In both base_10 and base_60, there is going to be some calculation slop and rounding on multiple brothers. Remember the student is generally working either from eldest brother down or youngest brother up, so any error propagation should be in a consistent directions. From the problem setups and available problem method, the Babylonian mathematicians and students are working out solutions to four base_60 relative places. The TCL calculator solution is kept in { TCL precision 17 } to study the Babylonian algorithms, not to imply accuracy beyond 3 decimal places or accuracy beyond the numbers on the clay tablets.
Modern mathematicians recognize three types of sequences as 1) arithmetic sequences of terms with a common and constant difference (a+N, b+N,c+N,d+N...), 2) geometric progression with any two terms in a constant ratio (a,a*r,a*r**2,a*r**3,a*r**4...), and 3) other sequences defined in simple or complex formulas (t(N)=(N**2)+1 or s(N)=(N**3)-1). Some math sequences are neither arithmetic sequences nor geometric progression in strict definition, but the available cuneiform tablets suggest the Late Babylonian mathematicians were familiar with all three sequence types. The brother inheritance problems are believed to represent geometric progression to solve for silver pieces or land parcels. Some terminology in the early brother inheritance problems suggest that geometric progression methods was first and early applied to land inheritances and graphical survey of triangular fields and geometric figures. Also, there are some geometry problems where brothers or soldiers were given equal portions or ladders of land from triangles, trapezoids, or other geometric figures. Possibly, the later arithmetic problems were developed or demonstrate solutions for silver pieces and unitless numbers. The exact relationship of the brother inheritance problems to Babylonian law is uncertain, but the available brother inheritance problems are recognized as more a tour de force in sequence mathematics than the law courts.
more than one formula for 1) tables and 2) calculator shell
gold This page is copyrighted under the TCL/TK license terms, this license .
Disclaimers: This page, screenshots, and TCL source code is copyrighted under the TCL/TK license terms. Editorial rights and disclaimers retained under the TCL/TK license terms and will be defended as necessary in court.
Please place any comments here, Thanks.