# 10 Alternative Numeral and Number Systems

The distinction between numbers and numerals is subtle but important. A number is an idea that expresses how much or how many. A numeral is a written symbol that expresses a number. Here are ten ways to represent numbers that differ from the Hindu-Arabic (decimal) system.

## Tally marks

Numbers are abstractions that stand for real things. The first known numbers came into being with the rise of trading and commerce — people needed to keep track of commodities such as animals, harvested crops, or tools. At first, traders used clay or stone tokens to help simplify the job of counting. Over time, tally marks scratched either in bone or on clay took the place of tokens.

## Bundled tally marks

As early humans grew more comfortable letting tally marks stand for real-world objects, the next development in numbers was probably tally marks scratched in *bundles* of 5 (fingers on one hand), 10 (fingers on both hands), or 20 (fingers and toes). Bundling provided a simple way to count larger numbers more easily.

Of course, this system is much easier to read than non-bundled scratches — you can easily multiply or count by fives to get the total. Even today, people keep track of points in games using bundles such as these.

## Egyptian numerals

Ancient Egyptian numerals are among the oldest number systems still in use today. Egyptian numerals use seven symbols.

Number | Symbol |
---|---|

1 | Stroke |

10 | Yoke |

100 | Coil of rope |

1,000 | Lotus |

10,000 | Finger |

100,000 | Frog |

1,000,000 | Man with raised hands |

Numbers are formed by accumulating enough of the symbols that you need. For example,

7 = 7 strokes

24 = 2 yokes, 4 strokes

1,536 = 1 lotus, 5 coils of rope, 3 yokes, 6 strokes

## Babylonian numerals

Babylonian numerals, which came into being about 4,000 years ago, use two symbols:

1 = Y

10 = <

For numbers less than 60, numbers are formed by accumulating enough of the symbols you need. For example,

6 = YYYYYY

34 = <<<YYYY

For numbers 60 and beyond, Babylonian numerals use place value based on the number 60.

61 = Y Y | (one 60 and one 1) |

124 = YY YYYY | (two 60s and four 1s) |

611 = < <Y | (ten 60s and eleven 1s) |

## Ancient Greek numerals

Ancient Greek numerals were based on the Greek letters. The numbers from 1 to 999 were formed using the symbols shown:

## Roman numerals

Although Roman numerals are over 2,000 years old, people still use them today, either decoratively (for example, on clocks, cornerstones, and Super Bowl memorabilia) or when numerals distinct from decimal numbers are needed (for example, in outlines). Roman numerals use seven symbols, all of which are capital letters in the Latin alphabet (which pretty much happens to be the English alphabet as well):

I = 1 | V = 5 | X = 10 | L = 50 |

C = 100 | D = 500 | M = 1,000 |

## Mayan numerals

Mayan numerals developed in South America during roughly the same period that Roman numerals developed in Europe. Mayan numerals use two symbols: dots and horizontal bars. A bar is equal to 5, and a dot is equal to 1. Numbers from 1 to 19 are formed by accumulating dots and bars. For example,

3 = 3 dots

7 = 2 dots over 1 bar

19 = 4 dots over 3 bars

Numbers from 20 to 399 are formed using these same combinations, but raised up to indicate place value. For example,

21 = raised 1 dot, 1 dot (one 20 + one 1)

399 = raised 4 dots over 3 bars, 4 dots over 3 bars (nineteen 20s + three 5s + four 1s)

## Base-2 (binary) numbers

Binary numbers use only two symbols: 0 and 1. This simplicity makes binary numbers useful as the number system that computers use for data storage and computation.

Like the decimal system you're most familiar with, binary numbers use place value. Unlike the decimal system, binary place value is based not on powers of ten (1, 10, 100, 1,000, and so forth) but on powers of two (20, 21, 22, 23, 24, 25, 26, 27, 28, 29, and so on), as seen here:

512s | 256s | 128s | 64s | 32s | 16s | 8s | 4s | 2s | 1s |

## Base-16 (hexadecimal) numbers

The computer's first language is binary numbers. But in practice, humans find binary numbers of any significant length virtually undecipherable. Hexadecimal numbers, however, are readable to humans and still easily translated into binary numbers, so computer programmers use hexadecimal numbers as a sort of common language when interfacing with computers at the deepest level, the level of hardware and software design.

The hexadecimal number system uses all ten digits 0 through 9 from the decimal system. Additionally, it uses six more symbols:

A = 10 | B = 11 | C = 12 |

D = 13 | E = 14 | F = 15 |

Hexadecimal is a place-value system based on powers of 16.

1,048,576s | 65,536s | 4,096s | 256s | 16s | 1s |

As you can see, each number in the table is exactly 16 times the number to its immediate right.

## Prime-based numbers

One wacky way to represent numbers unlike any of the others is prime-based numbers. Prime-based numbers are similar to decimal, binary, and hexadecimal numbers in that they use place value to determine the value of digits. But unlike these other number systems, prime-based numbers are based not on addition but on multiplication.

31s | 29s | 23s | 19s | 17s | 13s | 11s | 7s | 5s | 3s | 2s |

You can use the table to find the decimal value of a prime-based number.