Numeration systems

Numeration systems are methods for representing quantities. As a simple example, suppose you have a basket of oranges. You might want to keep track of the number of oranges in the basket. Or you might want to sell the oranges to someone else. Or you might simply want to give the basket a numerical code that could be used to tell when and where the oranges came from. In order to perform any of these simple mathematical operations, you would have to begin with some kind of numeration system.

Why numeration systems exist

This example illustrates the three primary reasons that numeration systems exist. First, it is often necessary to tell the number of items contained in a collection or set of those items. To do that, you have to have some method for counting the items. The total number of items is represented by a number known as a cardinal number. If the basket mentioned above contained 30 oranges, then 30 would be a cardinal number since it tells how many of an item there are.

Numbers can also be used to express the rank or sequence or order of items. For example, the individual oranges in the basket could be numbered according to the sequence in which they were picked. Orange #1 would be the first orange picked; orange #2, the second picked; orange #3, the third picked; and so on. Numbers used in this way are known as ordinal numbers.

Finally, numbers can be used for purposes of identification. Some method must be devised to keep checking and savings accounts, credit card accounts, drivers' licenses, and other kinds of records for different people separated from each other. Conceivably, one could give a name to such records (John T. Jones's checking account at Old Kent Bank), but the number of options using words is insufficient to make such a system work. The use of numbers (account #338-4498-1949) makes it possible to create an unlimited number of separate and individualized records.

History

No one knows exactly when the first numeration system was invented. A notched baboon bone dating back 35,000 years was found in Africa and was apparently used for counting. In the 1930s, a wolf bone was found in Czechoslovakia with 57 notches in several patterns of regular intervals. The bone was dated as being 30,000 years old and is assumed to be a hunter's record of his kills.

The earliest recorded numbering systems go back at least to 3000 B.C. , when Sumerians in Mesopotamia were using a numbering system for recording business transactions. People in Egypt and India were using numbering systems at about the same time. The decimal or base-10 numbering system goes back to around 1800 B.C. , and decimal systems were common in European and Indian cultures from at least 1000 B.C.

One of the most important inventions in western culture was the development of the Hindu-Arabic notation system (1, 2, 3, … 9). That system eventually became the international standard for numeration. The Hindu-Arabic system had been around for at least 2,000 years before the Europeans heard about it, and it included many important innovations. One of these was the placeholding concept of zero. Although the concept of zero as a placeholder had appeared in many cultures in different forms, the first actual written zero as we know it today appeared in India in A.D. 876. The Hindu-Arabic system was brought into Europe in the tenth century with Gerbert of Aurillac (c. 945–1003), a French scholar who studied at Muslim schools in Spain before being named pope (Sylvester II). The system slowly and steadily replaced the numeration system based on Roman numerals (I, II, III, IV, etc.) in Europe, especially in business transactions and mathematics. By the sixteenth century, Europe had largely adopted the far simpler and more economical Hindu-Arabic system of notation, although Roman numerals were still used at times and are even used today.

Numeration systems continue to be invented to this day, especially when companies develop systems of serial numbers to identify new products. The binary (base-2), octal (base-8), and hexadecimal (base-16) numbering systems used in computers were developed in the late 1950s for processing electronic signals in computers.

The bases of numeration systems

Every numeration system is founded on some number as its base. The base of a system can be thought of as the highest number to which one can count without repeating any previous number. In the decimal system used in most parts of the world today, the base is 10. Counting in the decimal system involves the use of ten different digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. To count beyond 9, one uses the same digits over again—but in different combinations: a 1 with a 0, a 1 with a 1, a 1 with a 2, and so on.

The base chosen for a numeration system often reflects actual methods of counting used by humans. For example, the decimal system may have developed because most humans have ten fingers. An easy way to create numbers, then, is to count off one's ten fingers, one at a time.

Place value

Most numeration systems make use of a concept known as place value. That term means that the numerical value of a digit depends on its location in a number. For example, the number one hundred eleven consists of three 1s: 111. Yet each of the 1s in the number has a different meaning because of its location in the number. The first 1, 1 11, means 100 because it stands in the third position from the right in the number, the hundreds place. (Note that position placement from the right is based on the decimal as a starting point.) The second 1, 1 1 1, means ten because it stands in the second position from the right, the tens place. The third 1, 11 1, means one because it stands in the first position from the right, the units place.

One way to think of the place value of a digit is as an exponent (or power) of the base. Starting from the right of the number, each digit has a value one exponent larger. The digit farthest to the right, then, has its value multiplied by 10 ⁰ (or 1). The digit next to it on the left has its value multiplied by 10 ¹ (or 10). The digit next on the left has its value multiplied by 10 ² (or 100). And so forth.

The Roman numeration system is an example of a system without place value. The number III in the Roman system stands for three. Each of the Is has exactly the same value (one), no matter where it occurs in the number. One disadvantage of the Roman system is the much greater difficulty of performing mathematical operations, such as addition, subtraction, multiplication, and division.

Examples of nondecimal numeration systems

Throughout history, numeration systems with many bases have been used. Besides the base 10-system with which we are most familiar, the two most common are those with base 2 and base 60.

Base 2. The base 2- (or binary) numeration system makes use of only two digits: 0 and 1. Counting in this system proceeds as follows: 0; 1; 10; 11; 100; 101; 110; etc. In order to understand the decimal value of these numbers, think of the base 2-system in terms of exponents of base 2. The value of any number in the binary system depends on its place, as shown below:

2 ³ (=8)

2 ² (=4)

2 ¹ (=2)

2 ⁰ (=1)

The value of a number in the binary system can be determined in the same way as in the decimal system.

Anyone who has been brought up with the decimal system might wonder what the point of using the binary system is. At first glance, it seems extremely complicated. One major application of the binary system is in electrical and electronic systems in which a switch can be turned on or off. When you press a button on a handheld calculator, for example, you send an electric current through chips in the calculator. The current turns some switches on and some switches off. If an on position is represented by the number 1 and an off position by the number 0, calculations can be performed in the binary system.

Base 60. How the base-60 numeration system was developed is unknown. But we do know that the system has been widely used throughout human history. It first appeared in the Sumerian civilization in Mesopotamia in about 3000 B.C. Remnants of the system remain today. For example, we use it in telling time. Each hour is divided into 60 minutes and, in turn, each minute into 60 seconds. In counting time, we do not count from 1 to 10 and start over again, but from 1 to 60 before starting over. Navigational systems also use a base-60 system. Each degree of arc on Earth's surface (longitude and latitude) is divided into 60 minutes of arc. Each minute, in turn, is divided into 60 seconds of arc.