Math 409/409GHistory of Mathematics

Babylonian Numbering System

The Babylonians represented numbers in base 60 (the sexagesimal system) whereas we represent numbers in base 10 (the decimal system).

Base 10 and base 60 numbers

3

3

210

6012

1 0

0

a a

a a

1b b

b

c c

c

0 10 10 10

60 60 c 60 6

d

d d 0

d

b

002 1

6, , 60 60 6015 2 251 155 1

Converting from base 60

To convert a base 60 number to base 10, expand the number and do the arithmetic. Example:

10

10

2621 2115 15

54,0

5 5

300

, , 60 60

54,3

20

21

0 1

Converting to base 60

To convert a base 10 number to base 60, you start by repeatedly dividing by 60 until you get a quotient of 0.

Example: Converting 54,32110 to base 60

905 15 RRR60 54321 60 905 60 1

5150 152

• The “units digit” in the base 60 number is the remainder in the first division problem.

• The coefficient of the 601 term is the remainder in the second division problem.

• The coefficient of the 602 term is the remainder in the third division problem.

905 15 0 RRR60 54321 60 905 60 1

55

21 15

10 60154,321 ,5,215

Babylonian numbers

Babylonians represented their numbers in base 60, but the coefficients they used for the powers of 60 in this representation were in base 10, the same way we did in the last example.

10 60154,321 ,5,215

Babylonians numbers 1 to 59

Babylonians used an upright wedge for the number 1 and a sideways wedge for the number 10.

We will approximate these symbols by using:

1

10

Since the coefficients for the powers of 60 in the Babylonian system were in base 10, all they needed was a way to denote the numbers 1 through 59. So they used no more than 9 upright wedges for the units, and no more than 5 sideways wedges for the 10’s digit.

59

This positional notation usually caused no problem. But when it did, the Babylonians simply added spaces. Examples:

6025,1 1,5

6025,1 1,5

Babylonian problem with zero

Since the Babylonians had no symbol for zero, there was no way of knowing if their base 60 symbol for 1 represented 1 or 6010 or 360010. So they never achieved an accurate positional system.

In our notation, this is not a problem.

60 10 60 10 60 101 1 1,0 60 1,0,0 3600

Babylonian fractions

The Babylonians felt that all useful fractions could be approximated by finite terminating fractions in base 60.

In our base 60 notation, the fractional part of a number in separated from the whole number part by a semicolon. Example:

3012 605 5 ;305

Converting fractions to base 60

• Convert to a fraction over a power of 60.

• Convert the numerator to base 60.

1

2

3

60 60

60 3600

60 216,000

21

2

20

60

210

60

7 37 30

120 120 30 60

3 300;3,30

60 3

0 6

0

6 0

Converting fractions from base 60

2

3 45 112;3,45 12 12

60 60 16

This ends the lesson on the

Babylonian Numbering System