Number system: a system for representing numbers.

In his short story "Funes the Memorious", Jorge Louis Borges describes a teenager who had assigned individual names to all numbers up to 24,000.

"In place of seven thousand thirteen, he would say (for example), Maximo Perez; in place of seven thousand fourteen, The Train; other numbers were 'Luis Melian Lafinur', 'Olimar', 'Brimstone', 'Clubs', 'The Whale', 'Gas', 'The Cauldron', 'Napoleon', 'Agustín de Vedia' (...) I attempted to explain that this rhapsody of unconnected terms was precisely the contrary of a system of enumeration. I said that to say 'Three hundred and sixty-five' was to say 'three hundreds', 'six tens', 'five units': an analysis which does not exist in such numbers as Negro Timoteo or The Flesh Blanket. Funes did not understand me, or did not wish to understand me. "

To give numbers individual names we would certainly need an unlimited memory just like the teenager in Borges's story. The idea of creating numerals as a conglomerate of some few isolated digits is very old. The simplest system is a one-digit system consisting of a plain series of strokes |, ||, |||, |||| etc., still found in the Roman symbols for the numbers 1 to 3.

The use of more digits enables us to represent numbers in a shorter and more elegant manner. In the decimal system, which we inherited from Arabic mathematicians, numbers are composed of 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Other numbering systems use a different number of digits. Any number can be represented in any numbering system, though the representations will, of course, differ.

The advantage of modern numbering systems consists in the fact that the value of a number does not just depend on the digits used for its representation, but also on the position of the digits within the respective number symbol. In the decimal system, for example, each position from the left to the right has ten times the value of the previous position, that is, for example

365.25 = 100 ∙ 3 + 10 ∙ 6 + 1 ∙ 5  + 1/10 ∙ 2 + 1/100 ∙ 5

In this way we can represent any large or small numbers, which was not possible prior to this device. One important numbering system is the binary system developed by Leibniz, which still plays an important part in computer technology and is discussed in other places in this dictionary such as the issue of communication with ►extraterrestrials. The binary system recognizes only two digits, 0 and 1, and each position within a number symbol has twice the value of the previous position. Thus, for example, the number 1100.11 in the binary system is equivalent to the number 12.75 in the decimal system:

1100.11 = 8 ∙ 1 + 4 ∙ 1 + 2 ∙ 0 + 1 ∙ 0 + 1/2 ∙ 1 + 1/4 ∙ 1 = 12.75

We can accordingly define number systems with three, four, or any number of different digits. In computer technology the hexadecimal system, which uses 16 digits, is also of importance. ►Babylonian astronomers used a system with 60 different digits, though without the number zero. We still use this system today, in combination with normal decimal numbers, to count minutes and seconds.


© Johann Christian Lotter   ■  Infinity  ■  Links  ■  Forum