30 in Babylonian sexagesimal
× 1
Three ten-wedges. The side of YBC 7289 is marked 30; its scale is determined by context.
ANCIENT · BASE 60
Babylonian sexagesimal
Base-60 place values, with wedge signs for units and tens.
Convert numbers- Place
- Mesopotamia · present-day Iraq
- Period / variant
- Old Babylonian · c. 1800–1600 BCE
EXAMPLES
42 · Base 60
Four tens and two ones in a single sexagesimal place.
Worked example
Write 42 in Babylonian sexagesimal
Read from left to right. The rightmost place counts ones; each place to its left is worth 60 times as much.
42 is less than 60, so it fits in one group in the ones place.
- × 1
4 ten-wedges (corner wedges) and 2 one-wedges (vertical wedges) give 4 × 10 + 2 = 42. At the ones place, it contributes 42.
Check
4 × 10 + 2 = 42
Sources and notation notes
The separated cells fix the integer place values used in this example.
Bill Casselman, Analysis of YBC 7289 · University of British Columbia · Bill Casselman, Mathematical commentary on YBC 7289
Meet the symbols
Within one place, repeat the one wedge for units and the ten wedge for tens. Here are all the combinations from 1 to 59, plus an empty place labeled 0 in this teaching grid.
- 0
- 1
- 2
- 3
- 4
- 5
- 6
- 7
- 8
- 9
- 10
- 11
- 12
- 13
- 14
- 15
- 16
- 17
- 18
- 19
- 20
- 21
- 22
- 23
- 24
- 25
- 26
- 27
- 28
- 29
- 30
- 31
- 32
- 33
- 34
- 35
- 36
- 37
- 38
- 39
- 40
- 41
- 42
- 43
- 44
- 45
- 46
- 47
- 48
- 49
- 50
- 51
- 52
- 53
- 54
- 55
- 56
- 57
- 58
- 59
How the symbols fit together
Count the wedges inside each group first. Then read the groups using positions worth sixty times one another.
- Inside one group, a vertical wedge means 1 and a corner wedge means 10. Four ten-wedges and two one-wedges make 40 + 2 = 42.
- One position can hold a value from 0 to 59. Once you reach sixty, you need another position.
- For 61, the left group has one wedge in the sixties position. That means one group of sixty: 60.
- The right group has one wedge in the ones position. Add 60 + 1 = 61. Our separated cells and labels show which place is which; historical readings also depended on context.
42 in Babylonian sexagesimal
× 1
Four tens and two ones in a single sexagesimal place.
61 in Babylonian sexagesimal
× 60× 1
1 × 60 + 1. The same one-wedge occupies two places.
This is an educational transcription of Old Babylonian notation. Empty places are blank cells, not a historical zero sign. The cells and integer scale labels are modern aids: historical notation could leave the scale ambiguous. Later placeholder conventions are not mixed into this earlier variant.
History
Old Babylonian scribes pressed reed styluses into clay. Their mathematical tablets combine a base-sixty place system with additive groups of tens and ones. The square-root tablet YBC 7289 is an extraordinary surviving example.
How to read it
Count the wedges inside each group first. Then read the groups using positions worth sixty times one another.
- Inside one group, a vertical wedge means 1 and a corner wedge means 10. Four ten-wedges and two one-wedges make 40 + 2 = 42.
- One position can hold a value from 0 to 59. Once you reach sixty, you need another position.
- For 61, the left group has one wedge in the sixties position. That means one group of sixty: 60.
- The right group has one wedge in the ones position. Add 60 + 1 = 61. Our separated cells and labels show which place is which; historical readings also depended on context.
Limits
This is an educational transcription of Old Babylonian notation. Empty places are blank cells, not a historical zero sign. The cells and integer scale labels are modern aids: historical notation could leave the scale ambiguous. Later placeholder conventions are not mixed into this earlier variant.
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