12 Number Systems
Our number system is a western adaptation of the Hindu-Arabic numeral system developed somewhere between the first and fourth centuries AD. However, numbers have been recorded with tally marks throughout history. The Ishango Bone[1] from Africa is about 25,000 years old. It’s the lower leg bone from a baboon, and contains tally marks. We know the marks were used for counting because they appear in distinct groups.
This reindeer antler[2] from France is about 15,000 years old, and also shows clearly grouped tally marks.
Of course, we still use tally marks today![3]
Base ten numbers (the ones you have probably been using your whole life), and base b numbers (the ones you’ve been learning about in this chapter) are both positional number systems.
Definition
A positional number system is one way of writing numbers. It has unique symbols for 1 through b – 1, where b is the base of the system. Modern positional number systems also include a symbol for 0.
The positional value of each symbol depends on its position in the number:
- The positional value of a symbol in the first position is just its face value.
- The positional value of a symbol in the second position is b times its value.
- The positional value of a symbol in the third position is times its value.
- And so on.
The value of a number is the sum of the positional values of its digits.
Definition
In an additive number system, the value of a written number is the sum of the face values of the symbols that make up the number. The only symbol necessary for an additive number system is a symbol for 1, however many additive number systems contain other symbols.
History: Roman numerals
The ancient Romans used a version of an additive number systems. The Romans represented numbers this way:
| number | Roman Numeral |
| 1 | I |
| 5 | V |
| 10 | X |
| 50 | L |
| 100 | C |
| 500 | D |
| 1,000 | M |
So the number 2013 would be represented as MMXIII. This is read as 2,000 (two M’s), one ten (one X), and three ones (three I’s).
For any additive number system very large numbers become impractical to write. To represent the number one million in Roman numerals it would take one thousand M’s!
However, the Roman numerals did have one efficiency advantage: The order of the symbols mattered. If a symbol to the left was smaller than the symbol to the right, it would be subtracted instead of added. So for example nine is represented as IX rather than VIIII.
Think / Pair / Share
If you don’t already know how to use Roman numerals, research it a little bit. Then answer these questions.
- Write the numbers 1–20 in Roman numerals.
- What is the maximum number of symbols needed to write any number between 1 and 1,000 in Roman Numerals? Justify your answer.
The earliest positional number systems are attributed to the Babylonians (base 60) and the Mayans (base 20). These positional systems were both developed before they had a symbol or a clear concept for zero. Instead of using 0, a blank space was us