Math 104 - Cooley Math For Elementary Teachers I OCC

Activity #22 – The Babylonian Numeration System

California State Content Standard – Mathematical Reasoning – Grade Six
2.5 Express the solution clearly and logically by using the appropriate mathematical notation and terms and clear language;
support solutions with evidence in both verbal and symbolic work.

The Babylonian numeration system (circa. 2000 BCE) was developed about the same time as the Egyptian system, yet much earlier than the Roman system. The Babylonian system was an additive positional system, and

because of this they were the first to introduce the notion of place value, in which the position of a symbol in a

numeral determined its value. With the introduction of a positional system comes the need for the concept of zero

to denote an empty space corresponding to a particular place value. The Babylonians never developed this

necessary concept of zero, and as a result often the numeral represented by the symbols could be interpreted

several ways.

The additive positional system of the Babylonians made it much easier to write numerals that represented large

numbers by using very few symbols. This system only contained two symbols:

= ten = one

The place value is not a decimal, base-10 system, like the current system used today. The Babylonian system uses

base-60, or sexagesimal, number system.

J Example:

a) The following Babylonian numeral represents the number 23, because

2·(10) + 3·(1) = 23.

J Example:

b) The following Babylonian numeral represents the number 764, because

12·(601) + 44·(600) = 764.

J Example:

c) The following Babylonian numeral represents

the number 88347, because 24·(602) + 32·(601) + 27·(600) = 88347.

J Example:

d) The following Babylonian numeral represents the number 43223, because

12·(602) +0·(601) + 23·(600) = 43223.

Recall that the Babylonians never developed the concept of zero, so whenever a place value was left empty, they

left a space in the symbol sequence. This was often confusing since the spacing wasn’t uniform. Consider the

symbol , which because of the blank space to the right of it could represent the number 1, the number 60, the

number 3600, etc. Later, the symbol was used to denote an empty place value when used between other symbols. But, it was never used as a zero like we use zero, so 60 = and was NEVER written as .

This Babylonian divider was an ingenious addition, yet there are inherent problems, so this system is somewhat flawed.

J Exercises:

Write the following Babylonian numerals in Hindu-Arabic form.

1)

2)

3)

Express the following numbers in Babylonian numerals.

4) 831

5) 41681

6) 57652

7) Try writing the Babylonian numeral for 67. Do you see the confusion in this system?

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