Calculating in floating sexagesimal place value notation, 4000 years - PowerPoint PPT Presentation

Calculating in floating sexagesimal place value notation, 4000 years ago Christine Proust Laboratoire SPHère (CNRS & Université Paris-Diderot) ARITH 22 22 nd Symposium on Computer Arithmetic Lyon June 22-24, 2015

François Thureau-Dangin (1872-1944) 1938 Textes Mathématiques Babyloniens . Otto Neugebauer (1899-1990) 1935-1937 Mathematische Keilschrifttexte I Neugebauer & Sachs, 1945 Mathematical Cuneiform Texts .

School tablet from Nippur , Old Babylonian period (HS 217a, University of Jena)

Multiplication table by 9 1 9 15 2:15 2 18 16 2:24 3 27 17 2:33 4 36 18 2:42 5 45 20-1 2:51 6 54 20 3 7 1:3 30 4:30 8 1:12 40 6 9 1:21 50 7:30 10 1:30 HS 217a 11 1:39 8.20 a-ra 2 1 8.20 12 1:48 13 1:57 14 2:6 20x9 = 180 = 3x60 In base 60 : 3:0 But the scribes wrote: 3 This is puzzling. 20x9 is written 3 20x9 = 3 The square of 30 is written 15 30x30 = 15 The square root of 15 is written 30

(Sachs 1947) Is the lack of graphical systems to determine the place of the unit in the number an imperfection of the cuneiform script?

yes The algorithm for reciprocals according to Abraham Sachs (1947)

no

Scribal schools in Old Babylonian period (ca. 2000-1600 BCE) Scribal school in Nippur

The curriculum at Nippur Level Content Elementary Metrological lists : capacities, weights, surfaces, lengths Metrological tables : capacities, weights, surfaces, lengths, heights Numerical tables : reciprocals, multiplications, squares Tables of square roots and cube roots Intermediate Exercises : multiplications, reciprocals, surface and volume calculations

A list of proverbs 1. Someone who cannot produce "a-a”, from where will he achieve fluent speech? 2. A scribe who does not know Sumerian -- from where will he produce a translation? 3. The scribe skilled in counting is deficient in writing. The scribe skilled in writing is deficient in counting. 4. A chattering scribe. Its guilt is great. 5. A junior scribe is too concerned with feeding his hunger; he does not pay attention to the scribal art. 6. A disgraced scribe becomes a priest. School tablet from Nippur , Old Babylonian period … (Ist Ni 5376, Istanbul)

Metrological lists capacity weight School tablet from Nippur, surface Old Babylonian period (HS 249, University of Jena) length

Metrological lists: measurements of capacity 1/3 sila 1/2 sila 2/3 sila 1 sila 5/6 sila ca. 1 liter 1 sila 1 1/3 sila 1 ban 1 1/2 sila 1 2/3 sila ca. 10 liters 1 5/6 sila 2 sila 3 sila 4 sila 5 sila 6 sila 7 sila 8 sila 9 sila 1 ban še 1 ban 1 sila 1 ban 2 sila School tablet from Nippur, Old Babylonian period 1 ban 3 sila 3 (HS 1703, University of Jena)

1 šusi 10 2 šusi 20 3 šusi 30 4 šusi 40 5 šusi 50 6 šusi 1 1 šusi = 1 finger, 7 šusi 1:10 ca. 1.6 cm 8 šusi 1:20 9 šusi 1:30 1/3 kuš 1:40 1 kuš 1/2 kuš 2:30 = 1 cubit, 2/3 kuš 3:20 ca. 50 cm 5/6 kuš 4:10 1 kuš 5 1 1/3 kuš 6:40 1 1/2 kuš 7:30 1 2/3 kuš 8:20 2 kuš 10 School tablet from Nippur, Old Babylonian period (HS 241, University of Jena)

Reciprocals Multiplication tables by 50 45 44:26:40 7:30 40 7:12 36 7 30 6:40 25 6 School tablet from Nippur, 24 5 Old Babylonian period 22:30 (Ist Ni 2733, Istanbul 4:30 Museum) 20 4 18 3:45 16:40 3:20 16 3 Numerical tables 15 2:30 12:30 2:24 12 2 10 1:40 9 1:30 8:20 1:20 8 1:15 Table of squares

2 30 3 20 4 15 5 12 Table of reciprocals 6 10 8 7:30 9 6:40 10 6 12 5 15 4 16 3:45 18 3:20 20 3 24 2:30 25 2:24 27 2:13:20 30 2 32 1:52:30 36 1:40 40 1:30 45 1:20 School tablet from unknown provenance, 48 1:15 Old Babylonian period 50 1:12 (MS 3874, Schøyen collection, 54 1:6:40 copy Friberg) 1 1 1:4 56:15 1:21 44:26:40

2 30 3 20 4 15 5 12 The division by a number was performed by 6 10 mean of the multiplication by the reciprocal 8 7:30 9 6:40 of this number. 10 6 12 5 15 4 5 ÷ 30 = 5 × 2 = 10 16 3:45 18 3:20 20 3 24 2:30 2 ÷ 44:26:40 = 2 × 1:21 = 2:42 25 2:24 27 2:13:20 30 2 32 1:52:30 36 1:40 40 1:30 45 1:20 48 1:15 50 1:12 54 1:6:40 1 1 1:4 56:15 1:21 44:26:40

Multiplying 4 50 4 50 ----------------------- 41 40 3 20 3 20 16 ----------------------- 23 21 40

Reciprocal 17:46:40 Its reciprocal 3:22:30 ==================== 17:46:40 [9] 2:40 22:[30] 3:22:[30] A chattering scribe, his guilt is great. ----------------------------------- A chattering scribe, his guilt is great ====================== 17:46:40 9 1:30*

Calculation of surface SPVN 2.10 2.10 4.26 ! .40 1/3 kuš 3 3 šu-si its side --------------------- Its surface what ? --------------------- Its surface 13 še igi-4 ! gal 2 še ============== Metrological notations Ni 18 School tablet from Nippur Istanbul Museum

Multiplication table (SPVN) Table of surfaces Table of lengths SPVN → s urface measure length measures → SPVN

School tablets from Nippur Extract of metrological table Extract of m etrological table Extract of table of squares for lengths for surfaces 2 times 2 4 6 šu-si 1 1 / 3 sar 20 7 šu-si 1.10 3 times 3 9 1 / 2 sar 30 8 šu-si 1.20 4 times 4 16 2 / 3 sar 40 9 šu-si 1.30 5 times 5 25 5 / 6 sar 50 1 / 3 kuš 3 1.40 6 times 6 36 1 sar 1 1 / 3 kuš 3 1 šu-si 7 times 7 49 1.50 1 1 / 3 sar 1.20 1 / 3 kuš 3 2 šu-si 8 times 8 1.4 2 1 1 / 2 sar 1.30 1 / 3 kuš 3 3 šu-si 2.10 9 times 9 1.21 1 2 / 3 sar 1.40 1 / 3 kuš 3 4 šu-si 2.20 10 times 10 1.40 1 5 / 6 sar 1.50 11 times 11 2.1 12 times 12 2.24 SPVN SPVN Metrological notations SPVN Metrological notations

The algorithm for reciprocal Obverse 4:26:40 Its reciprocal 13:30 ============== reverse 4:26:40 9 40* 1:30 13:30 *error of the scribe: he wrote 41 instead of 40

2 30 3 20 4:26:40 9 4 15 40 1:30 5 12 6 10 13:30 8 7:30 9 6:40 10 6 12 5 4:26:40 ends with the regular number 6:40, so 4:26:40 is "divisible" by 15 4 16 3:45 6:40 . 18 3:20 To divide 4:26:40 by 6:40, we must multiply 4:26:40 by the reciprocal of 20 3 6:40. 24 2:30 25 2:24 The reciprocal of 6:40 is 9 . 27 2:13:20 The number 9 is placed in the right hand column. 30 2 4:26:40 multiplied by 9 is 40, thus, 40 is the quotient of 4:26:40 by 32 1:52:30 6:40. 36 1:40 40 1:30 This quotient is placed in the left hand column. 45 1:20 The reciprocal of 40 is 1:30. 48 1:15 The number 1:30 is placed in the right hand olumn. 50 1:12 54 1:6:40 To find the reciprocal of 4:26:40, one only has to multiply the 1 1 reciprocals of the factors of 4:26:40, that is to say, the numbers 9 and 1:4 56:15 1:30 placed in the right hand column. 1:21 44:26:40 This product is 13:30, the reciprocal sought.

2 30 3 20 4:26:40 9 4 15 40 1:30 5 12 6 10 13:30 8 7:30 9 6:40 10 6 12 5 15 4 16 3:45 18 3:20 20 3 24 2:30 Left hand column: 25 2:24 27 2:13:20 4:26:40 = 6:40 × 40 30 2 32 1:52:30 36 1:40 40 1:30 45 1:20 Right hand column 48 1:15 50 1:12 9 × 1:30 = 13:30 54 1:6:40 1 1 1:4 56:15 1:21 44:26:40

Obverse: 3 columns, # 1-16 CBS 1215 Reverse: 3 columns, #16-21 Provenance: unknown Datation: OB period (ca. 1800 BCE) University of Pennsylvania, Philadelphia Publication: Sachs 1947, Babylonian Mathematical Texts 1 Copy Robson 2000: 14 2.5 4.10 8.20 16.40 33.20 Entries 1.6.40 2.13.20 4.26.40 … 10.6.48.53.20