AMA1D01C by Dr. Joseph Lee

๐’€ญ Division by Regular Numbers ๐’€ญ

Babylonian Reciprocal Multiplication Method
๐’น รท ๐’Œ‹๐’ˆซ = ๐’น ร— ๐’ผ
Regular Numbers (2แตƒ ร— 3แต‡ ร— 5แถœ) and Their Reciprocals

Click any number to use it as the divisor. Regular numbers have exact finite reciprocals in base-60.

Division Calculator
Dividend (ๅฏฆ):
60ยฒ
;
60ยน
;
60โฐ
รท
Divisor (ๆณ•):
โœ“ Regular (2ยฒร—3)
Division Result (ๅ•†)
Babylonian Method: Multiply by Reciprocal
Reciprocal (IGI) of
Cuneiform Representation:
Step-by-Step Calculation

๐Ÿ“œ Regular Numbers in Babylonian Mathematics

The ancient Babylonians (c. 2000-300 BCE) developed a sophisticated base-60 (sexagesimal) number system. A key concept was regular numbers โ€” numbers of the form 2a ร— 3b ร— 5c โ€” which have exact finite reciprocals in base-60.

Division was performed by multiplication by the reciprocal:

a รท n = a ร— (1/n) = a ร— IGI(n)

The term IGI (๐’…†) meant "reciprocal" in Sumerian. Babylonian scribes memorized extensive tables of reciprocals for regular numbers.

๐’€ญ Why Base 60?

  • 60 = 2ยฒ ร— 3 ร— 5 (highly composite)
  • Divisible by 1,2,3,4,5,6,10,12,15,20,30,60
  • Many fractions have exact representations
  • Still used: time (60 sec/min) and angles (60'/ยฐ)

๐’น Regular Numbers Properties

  • Only prime factors: 2, 3, and 5
  • Have exact finite base-60 reciprocals
  • First few: 1,2,3,4,5,6,8,9,10,12,15,16...
  • Non-regular: 7,11,13,14,17,19,21,22,23...

๐Ÿ“‹ Standard Reciprocal Table

20;30
30;20
40;15
50;12
60;10
80;7,30
90;6,40
100;6
120;5
150;4
160;3,45
180;3,20
200;3
240;2,30
250;2,24
270;2,13,20
300;2
320;1,52,30
360;1,40
400;1,30
450;1,20
480;1,15
500;1,12
540;1,6,40

๐Ÿงฎ Algorithm

  • Check if divisor is regular (factors: 2,3,5 only)
  • Compute reciprocal: IGI(n) in base-60
  • Multiply: dividend ร— IGI(n)
  • Result is exact (no remainder)