History of Mathematics Project | Babylonian School Multiplication Tablet

History of
Mathematics

Project

around 1900–1600 BCE

Babylonian School Multiplication Tablet

Ancient multiplication exercises—with teacher and student errors

Reciprocal and multiplication tables were commonly composed in scribal schools of Sumer and Babylonia. They had relatively standard forms in which student scribes copied an original series of lines multiple times to reinforce "penmanship" as well as mastery and memorization of the facts. This exemplar originated in the ancient Sumerian city of Nippur around 1900–1600 BCE and was reassembled from fragments found by several separate archaeological expeditions.

The front of the tablet contains a teacher's (on the left) and student's (on the right) copy of the 45 times table. The reverse sides contains a list of divisors of 12,960,000 from 1 1/2 to 81 with quotients and a 50 times table written alternately three times. Both scribe's and student's copies contain a number of errors.

Artifact dimensions

17.3 in. × 11 in. × 3.3 in. (central portion, possibly omitting lower-right piece attached post-Hilprecht)

Original artifact location

Nippur (historical name), Afak, Al-Qadisiyah, Iraq (current name)

Current artifact location

Penn Museum, Philadelphia

Catalog number

CBS 11340 + CBS 11402 + UM 29-16-752

Timeline

Interactive Content

Computational Explanation

Multiplication tables are relatively common among Babylonian mathematical tablets. The tablet CBS 11340 + CBS 11402 + UM 29-16-752 is composed of three separately labeled pieces all originating from the ancient Sumerian city of Nippur and dated to around 1900–1600 BCE. It contains teacher's and student's copies of the 45 times table on the obverse (front).

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

To replicate the multiplication table, create a grid showing the numbers and multiplications involved written in both Hindu-Arabic and Babylonian cuneiform numerals. Comparing with a schematic drawing of the tablet published by V. Hilprecht in 1906 shows clear agreement:

StyleRowTextGridTablen,

[◼]	AncientNumberRepresentation

[n,"Babylonian"],Row[{n,"×",45,"",45n}," "],

[◼]	AncientNumberRepresentation

[45n,"Babylonian"],{n,8,15},

ImageTake[

]

,

Rule[

]

//Rasterize

Out[]=

Interestingly, the second line on the front of the tablet, corresponding to 2 × 45, contains an error by the teacher:

Out[]=

Here, the teacher included an extra wedge so that what should be:

Out[]=

2×4590

… is actually written as:

Out[]=

2×45150

The student also has his share of errors. For example, lines 9–10 on the student side of the tablet contain two errors:

Out[]=

In the second of these on line 10:

Out[]=

10×45270

… has been written instead of the correct:

Out[]=

10×45450

The reverse side of the tablet contains a list of divisors of 12,960,000 from 1 1/2 to 81 with quotients and a 50 times table written alternately three times. As was the case on the front of the tablet, the computations on the back also contain a number of errors.

Other Resources

History of Mathematics
Sumerian 7 Quotient and 10 Product Tablet

History of Mathematics
Babylonian Reciprocals Tablet

Wolfram Demonstration
Babylonian Numerals

Wolfram Demonstration
Multiplication Table

Wolfram Demonstration
Exploring the Times Table

Additional Reading

Hilprecht, H. V. Object 20 in Mathematical, Metrological and Chronological Tablets from the Temple Library of Nippur. Philadelphia, PA: Department of Archaeology, University of Pennsylvania, p. 60 and Pl. 10, 1906.
Neugebauer, O. Mathematische Keilschrift-Texte. Springer, pp. 12, 54 and 62, 1935.
Robson, E. Fig 1.4 in "Tables and tabular formatting in Sumer, Babylonia, and Assyria, 2500 BCE–50 CE." Ch. 1 in The History of Mathematical Tables: From Sumer to Spreadsheets (Ed. M Campbell-Kelly, M. Croarken, R. Flood and E. Robson). Oxford, England: Oxford University Press, p. 32, 2007. [misidentifies the tablet as CBS 2142]

Additional Links

Cuneiform Digital Library Initiative: P230019

Image Credits

Penn Museum

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 School Multiplication Tablet

Timeline

Interactive Content

Computational Explanation

Other Resources

History of Mathematics

History of Mathematics

Wolfram Demonstration

Wolfram Demonstration

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