Babylonian Numbers

In an October blog “Decimals,” I showed you how we use the decimal (base-10) number system to generate the natural numbers we use today in math. In the blog “Mayan Numbers,” I showed you how the ancient Mayan civilization created numbers using a vigesimal (base-20) number system. In today’s blog, we’ll visit another ancient culture and learn how the Babylonian civilization used a sexegesimal (base-60) number system to do math.

Babylonia existed from about 18th c. BC until the 6th c. BC. This nation existed in the southern region of Mesopotamia between the Tigris and Euphrates River. Today, this area of the world is now part of Iraq.

The Babylonians used one of the earliest forms of written expression: the cuneiform script. They wrote in soft clay tablets using a wedge-shaped reed stylus. When the clay was hardened in the sun, the writing permanently became a part of the tablet.
They used just two symbols to define the 59 digits as follows:

The Babylonians wrote their numbers from right to left. The rightmost slot in any number has a multiple of 1. The multiplier for the next slots to the left of this column increased by a multiple of 60. The Babylonians defined their numbers by using one or more of the 59 digits from above. Although the Babylonians understood the concept of nothingness, they did not have a symbol for the digit or value of zero. They would leave a space or use a non-numeric placeholder to indicate that no digit was defined for a position within a number.

Now let’s use this knowledge to translate a Babylonian number (A).

We first translate the Babylonian digits into a Hindu-Arabic number and we multiply it by the multiplier associated with its position (B). We add the values of each position and come up with the value for the Babylonian number (C).

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Decimals

Today, we are going to take a closer look at the elements that make up the set of natural numbers. If you remember from a previous blog, this set represents all positive integers within the set of real numbers.

All real numbers (which include natural numbers) are made up of one or more digits. The digits we use in our modern numbering system are 0,1,2,3,4,5,6,7,8,9. Note that there are ten numbers that are made up of one digit (0-9). There are ninety two-digit numbers (10-99), and there are one thousand three-digit numbers (100-999). All real numbers can be expressed in this decimal notation, even fractions. This will be looked into in a future blog.

The position of the digits within a numeral is as important as the digits themselves. The numeral “476” is not the same as the numeral “674” even though both numerals use the same three digits: 4, 7, and 6. In order to understand why this is the case, we need to understand the decimal number system. I’ll show the structure of this system by illustrating the number “six hundred and seventy-four” in three different ways.

In this system, each position represents a multiplier used to determine the magnitude of the digit in its slot (A). The rightmost slot in any number has a multiplier of 1. The slot to its left represents a multiplier of 10. The third slot from the right represents a multiplier of 100. Theoretically, this can continue indefinitely.  Very large (or very small) multipliers can also be expressed using scientific notation.Each slot represents a different multiplier that is ten times more than the slot to the right.  Since all the multipliers in this system are multiples of ten, the decimal notation system is also referred to as a base-10 system. The arithmetic expression (B) is a way of expressing the relationship between the multipliers and its digits. The number (C) is a shorthand way of expressing this arithmetic expression.  There are other numerical systems that use a multiplier other than ten.  We’ll explore these systems and their practical applications in a future blog.