History of Mathematics Project | Babylonian Reciprocals Tablet

History of
Mathematics

Project

around 1900–1600 BCE

Babylonian Reciprocals Tablet

Old Babylonian reciprocal table (with errors)

Tablet YPM BC 024321 from the Yale University Babylonian Collection has been dated to around 1900–1600 BCE. Both sides contain two columns of numbers written in the base-60 numeral system of the Babylonians. The obverse (front) of the tablet consists of the whole numbers from 56 through 71 with their reciprocals. The reverse side similarly contains the numbers 72 through 80 with their reciprocals. In both cases, the numbers sometimes wrap around onto the sides of the tablet.

Babylonian reciprocal tables are relatively common, clearly demonstrating the ability of Babylonian scribes to perform simple arithmetic calculations. This tablet is notable for giving approximate base-60 values for the reciprocals it contains that cannot be expressed finitely in base 60. It is also noteworthy that the tablet contains a number of computational (or transcription) errors. For example, in the eighth line of the front of the tablet, the approximate reciprocal 1/63 ≈ 0.015873 should be given as 57, 8, 34 (equal to 57/60² + 8/60³ + 34/60^4 ≈ 0.015873) but is instead written as 57, 8, 24 (equal to 57/60² + 8/60³ + 24/60^4 ≈ 0.0158722).

Artifact dimensions

4.7 cm × 7.0 cm × 1.7 cm

Original artifact location

Babylonia (historical name), Iraq (current name)

Current artifact location

Peabody Museum of Natural History, New Haven

Catalog number

YPM BC 024321 (current), YBC 10529 (historical)

Timeline

Interactive Content

Computational Explanation

Before exploring tablet YPM BC 024321, first consider a reciprocals table constructed in the familiar decimal (base-10) system starting with the whole numbers 1 through 10:

In[]:=

TextGridTable[{n,N[1/n,20]},{n,10}],



Out[]=

1	1.0000000000000000000
2	0.50000000000000000000
3	0.33333333333333333333
4	0.25000000000000000000
5	0.20000000000000000000
6	0.16666666666666666667
7	0.14285714285714285714
8	0.12500000000000000000
9	0.11111111111111111111
10	0.10000000000000000000

It is immediately clear that some reciprocals have terminating decimal expansions (meaning they contain only of zeros after a certain point), while the others have repeating groups of digits. For example:

1

0.5

0.25

0.2

0.125

0.1

… are all terminating decimals. The existence of finite expansions follows from the factorization of 10 as

102×5

, meaning any numbers whose factorizations contain only these factors (or 1 in the trivial case of 1/1  1):

In[]:=

FactorsOf10=FactorInteger[10][[All,1]]

Out[]=

{2,5}

… have reciprocals with a finite decimal representation. For example, the complete set of whole numbers up to 100 having this property in base 10 is:

In[]:=

FiniteBase10Reciprocals=Select[Range[100],ContainsOnly[FactorInteger[#][[All,1]],FactorsOf10]&]

Out[]=

{2,4,5,8,10,16,20,25,32,40,50,64,80,100}

… giving the reciprocals:

In[]:=

Table[Row[{r,"",N[r,Length@First@RealDigits[r]]},Spacer[1]],{r,1/FiniteBase10Reciprocals}]//Column

Out[]=



0.5



0.25



0.2



0.125



0.1



0.0625



0.05



0.04



0.03125



0.025



0.02



0.015625



0.0125

100



0.01

Another interesting mathematical tidbit, of which:

In[]:=

N

,20

Out[]=

0.11111111111111111111

is a special case, is the fact that the base-b expansion of

1/(b-1)

always consists of repeating 1's. This amusing fact follows immediately from the well-known sum:

In[]:=

Sum

,{n,∞}

Out[]=

-1+b

Therefore, in the preceding table,

1/90.

, where an overbar has been used to denote a repeating block (in this case of length 1) of digits.

Unlike our base-10 system for representing numbers, the Babylonian system of numeration consisted of base-60 numbers written in cuneiform script. The following table summarizes cuneiform representations of the whole numbers 1 to 40:

In[]:=

TextGridFlatten/@TransposePartitionTablen,

[◼]	AncientNumberRepresentation

[n,"Babylonian"],{n,40},10,DividersAll

Out[]=

1	11	21	31
2	12	22	32
3	13	23	33
4	14	24	34
5	15	25	35
6	16	26	36
7	17	27	37
8	18	28	38
9	19	29	39
10	20	30	40

Considering now tablet YPM BC 02432, the following transcription can be obtained (Jones 2010, following Neugebauer and Sachs 1945):

Out[]=

Including a similar transcription from the the reverse side (Neugebauer and Sachs 1945) and filling in digits that are missing or obscured but can be inferred gives the following set of numbers and base-60 reciprocal expansions:

In[]:=

Transcription={56{},57{},58{1,2,4},59{1,1,1},60{1},61{59,59},62{58,3,52},63{57,8,24},64{56,15},65{55,23,4,30},66{54,32,43,30},67{53,43,52},68{52,56,53,14},69{52,10,28},70{51,25,42},71{50,42,15},72{50},73{49,18,55},74{48,38,55},75{48},76{47,22,6},77{46,45,11},78{46,9,13,50},79{45,34,9,54},80{45}};

Display these in tabular form and append to them reciprocal expansions calculated from scratch using RealDigits in the Wolfram Language:

In[]:=

header={Style["n",Italic],"tablet","calculated"};

In[]:=

TextGridPrependReplacedata={#,#2,Take[DeleteCases[RealDigits[1/#,60,5][[1]],0],Length[#2]]}&@@@Transcription,

,header,

Rule[

]



Out[]=

n	tablet	calculated
56
57
58	1 2 4	1 2 4
59	1 1 1	1 1 1
60	1	1
61	59 59	59 59
62	58 3 52	58 3 52
63	57 8 24	57 8 34
64	56 15	56 15
65	55 23 4 30	55 23 4 36
66	54 32 43 30	54 32 43 38
67	53 43 52	53 43 52
68	52 56 53 14	52 56 28 14
69	52 10 28	52 10 26
70	51 25 42	51 25 42
71	50 42 15	50 42 15
72	50	50
73	49 18 55	49 18 54
74	48 38 55	48 38 55
75	48	48
76	47 22 6	47 22 6
77	46 45 11	46 45 11
78	46 9 13 50	46 9 13 50
79	45 34 9 54	45 34 10 37
80	45	45

A careful examination shows that while the tablet and calculated values generally agree, there are deviations. This can be seen more clearly by coloring the tablet values green if they agree with the correct value and red if not:

In[]:=

TextGridPrependReplace{#,Thread[Style[#2,Equal@@@Transpose[{#2,#3}]/.{TrueDarker@Green,FalseRed}]],#3}&@@@data,

,header,

Rule[

]



Out[]=

n	tablet	calculated
56
57
58	1 2 4	1 2 4
59	1 1 1	1 1 1
60	1	1
61	59 59	59 59
62	58 3 52	58 3 52
63	57 8 24	57 8 34
64	56 15	56 15
65	55 23 4 30	55 23 4 36
66	54 32 43 30	54 32 43 38
67	53 43 52	53 43 52
68	52 56 53 14	52 56 28 14
69	52 10 28	52 10 26
70	51 25 42	51 25 42
71	50 42 15	50 42 15
72	50	50
73	49 18 55	49 18 54
74	48 38 55	48 38 55
75	48	48
76	47 22 6	47 22 6
77	46 45 11	46 45 11
78	46 9 13 50	46 9 13 50
79	45 34 9 54	45 34 10 37
80	45	45

As can be seen, the tablet contains errors in 7 of its entries, though usually in the last digit only. For example, the approximate reciprocal of 63 is 57 8 34 …:

In[]:=

RealDigits[1/63,60,3][[1]]

Out[]=

{57,8,34}

… but this is incorrectly given on the table as 57 8 24:

In[]:=

[◼]	AncientNumberRepresentation

[{57,8,24},"Babylonian"]

Out[]=





Out[]=

Returning to some of the mathematical properties of the tablet, note that in base 60, only reciprocals whose factors include:

In[]:=

FactorsOf60=FactorInteger[60][[All,1]]

Out[]=

{2,3,5}

… have a finite base-60 expansions. Since the tablet lists reciprocals of the numbers 56 through 80, only its numbers:

In[]:=

TerminatingReciprocals=Select[Range[56,80],ContainsOnly[FactorInteger[#][[All,1]],FactorsOf60]&]

Out[]=

{60,64,72,75,80}

… have finite expansions. Since such numbers have short exact representations in base 60, this accounts for their brevity (1 or 2 digits compared to surrounding expansions which have 3 or 4) in the tablet:

In[]:=

TextGridReplacedata[[All,;;2]],

,

Rule[

]



Out[]=

56
57
58	1 2 4
59	1 1 1
60	1
61	59 59
62	58 3 52
63	57 8 24
64	56 15
65	55 23 4 30
66	54 32 43 30
67	53 43 52
68	52 56 53 14
69	52 10 28
70	51 25 42
71	50 42 15
72	50
73	49 18 55
74	48 38 55
75	48
76	47 22 6
77	46 45 11
78	46 9 13 50
79	45 34 9 54
80	45

Finally, as previously noted, the reciprocal of one less than the base of a numeration system always consists of a sequence of 1's when expressed in that base. As a result, it is amusing to note that had the scribe who wrote:

In[]:=

TextGridReplacedata[[2;;6,;;2]],

,

Rule[

]



Out[]=

57
58	1 2 4
59	1 1 1
60	1
61	59 59

… in cuneiform:

In[]:=

[◼]	AncientNumberRepresentation

N

,4,"Babylonian"

Out[]=

… wished, he could have continued writing as many cuneiform ones as space (and his patience) permitted:

In[]:=

[◼]	AncientNumberRepresentation

N

,20,"Babylonian"

Out[]=

Other Resources

History of Mathematics
Sumerian 7 Quotient and 10 Product Tablet

History of Mathematics
Babylonian Scribe School Multiplication Tablet

Wolfram Demonstration
Babylonian Numerals

Additional Reading

Bruins, E. M. "Tables of Reciprocals with Irregular Entries." Centaurus, Vol. 17, pp. 177–188, September 1973.
Jones, A. "Before Pythagoras: The Culture of Old Babylonian Mathematics: Numbers on Clay." Brochure accompanying the exhibition Before Pythagoras: The Culture of Old Babylonian Mathematics. On view at ISAW from November 12th–December 17th, 2010.
Neugebauer, O. and Sachs, A. (Ed.). "Reciprocals of Regular and Irregular Numbers: YBC 10529." In Mathematical Cuneiform Texts. New Haven, CT: American Oriental Society and the American Schools of Oriental Research, p. 16, 1945.
Ore, O. Fig 8-1 in Number Theory and Its History. New York: Dover, pp. 173–174, 1988.

Additional Links

Image Credits

Yale Peabody Museum of Natural History

1	11	21	31
2	12	22	32
3	13	23	33
4	14	24	34
5	15	25	35
6	16	26	36
7	17	27	37
8	18	28	38
9	19	29	39
10	20	30	40

1	11	21	31
2	12	22	32
3	13	23	33
4	14	24	34
5	15	25	35
6	16	26	36
7	17	27	37
8	18	28	38
9	19	29	39
10	20	30	40

Babylonian Reciprocals Tablet

Timeline

Interactive Content

Computational Explanation

Other Resources

History of Mathematics

History of Mathematics

Wolfram Demonstration

Additional Reading

Additional Links

Image Credits

1	11	21	31
2	12	22	32
3	13	23	33
4	14	24	34
5	15	25	35
6	16	26	36
7	17	27	37
8	18	28	38
9	19	29	39
10	20	30	40