# 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. 9451003), 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 againbut 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, 11 1, means ten because it stands in the second position from the right, the tens place. The third 1, 111, 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 100 (or 1). The digit next to it on the left has its value multiplied by 101 (or 10). The digit next on the left has its value multiplied by 102 (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:

23 (=8)

22 (=4)

21 (=2)

20 (=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.

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# Numeration Systems

Why numeration systems exist

History

The bases of numeration systems

Base 2

Base 10 or decimal

Base 60

Place-value systems

Resources

Numerals are symbols or groups of symbols that represents a number. For example, the symbols 12, twelve, and XII are different numerals that all represent the same number. Numeration systems are structured methods or procedures for counting in order to determine the total units in a collection. Numeration systems consist of counting bases (base 2, base 5, base 10, base 20, etc.) and some form of representation. This representation might be as primitive as the hand signals used in aborigine cultures and in the trading pits of stock exchanges, or it could be written on paper or inscribed magnetically in an electronic medium like a computer hard-drive.

## Why numeration systems exist

Numeration systems exist for three reasons: to identify, to order, and to tally.

Numeration systems are used to identify people and property, because they preserve confidentiality, increase security, and minimize errors caused when there are many people with the same name or many identical objects in the same production run in a factory assembly

### Base ten numerals

tens = 10

hundred = 100

thousands = 1,000

ten thousands = 10,000

hundred thousands =100,000

millions = 1,000,000

ten millions = 10,000,000

hundred millions 100,0000,0000

billions = 1,000,000,000

line. There are thousands of people named John Jones, and even if John Jones uses his middle initial, he can still be confused with another John Jones with the same initial. Thus, numeration systems are developed for credit cards, social security cards, bank accounts, serial numbers for products, and other reasons. These identification numbers might be very long to defeat a criminal who is randomly guessing at numbers in order to steal from someones bank account or credit card account.

Numeration systems also define a person or units order in a series, for example, to determine who crosses a finish line in a race in first, second, or third place. Numbers that define order are known as the ordinal numbers (first, second, etc.) and contrast with the cardinal numbers (one, two, three, etc.) which express a tally or total of units.

Finally, numeration systems are used to tally or total; to find out how many items or units are involved in a calculation involving addition, subtraction, multiplication, or division.

## History

No one knows exactly when ordered numeration systems began, but counting has been around for tens of thousands of years. 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 30,000 years old and is assumed to be a hunters record of his kills.

The earliest recorded numbering systems go back at least 3000 BC, when Sumerians in Mesopotamia were using a numbering system for recording business transactions, and Egyptians and people in ancient India were also using numbering systems around the same time. The decimal, or base 10, numbering system goes back to at least 1800 BC, and decimal systems were common in European and Indian cultures from at least 1000 BC. One of the most important innovations in western culture was the development of the Hindu-Arabic notation system (1, 2, 3, . . . 9), which is the international standard today. The Hindu-Arabic system had been around for at least 2,000 years before the Europeans heard about it, and it has many important innovations. One of these was the place-holding concept of zero. Although the concept of zero as a null place holder had appeared in many cultures in different forms, the first actual written zero as is known today appeared in India in 876 AD. The Hindu-Arabic system was brought into Europe in the tenth century with Gerbert of Aurillac (9451003), and it slowly and steadily began to replace Roman numerals (I, II, III, IV, . . .) in Europe, especially in business transactions and mathematics. By the sixteenth century, Europe was well versed in the far simpler and more economical Hindu-Arabic system of notation, though Roman Numerals were still used, 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 (or base 16) numbering systems used in computers were extensively developed in the late 1950s for processing electronic signals in computers.

## The bases of numeration systems

The base of a numeration system is its frame of reference or the starting point on which it grounds its counting method. Although any numeration system must be abstract, the basic concept of number makes more sense to people if it has some obvious, immediate reference point in human experience. For that reason, many bases of numeration systems are founded upon the most obvious and immediate things in a persons visual field: a persons arms, hands, fingers, and toes. Common bases of numeration systems are the two arms of a person (base 2 system), the fingers of one hand (base 5 system), the fingers of both hands (base 10 system), or the total of all a persons fingers and toes (base 20 system). There are many other bases for numeration systems (base 4, base 7, base 8, base 16, etc.), but only a few will be discussed here.

## Base 2

Although most base 2 numeration systems have now been replaced by decimal (or base 10) systems, the base 2 system was one of the most common numeration systems in ancient times. In a base 2 system, to indicate a number like three or four, the person says two-and-one or two-and-two. The number 10 is indicated with two-and-two-and-two-and-two-and-two. However, as a person counts to higher and higher numbers in a base 2 system, it becomes harder and harder to remember ones place in the long string of twos. Thus, as cultures grew more complex and needed to count to higher numbers, base 2 systems became obsolete.

## Base 10 or decimal

The base 10 or decimal system has now spread throughout the world and is the most commonly used numeration system today. The digits to the left and right of the decimal point are named according to their distance from the decimal. The first ten numbers, in their order of distance from the left of the decimal point are:

These numbers continue indefinitely. To the right of the decimal point the numbers are one tenth, one hundredth, one thousandth, one ten-thousandth, one hundred-thousandth, one millionth, and so on.

## Base 60

The base 60 system seems very strange to Western readers. From long habit, Westerners are accustomed to the decimal system, and it is easy to understand numbering systems based on two (arms), five (fingers), ten (fingers on both hands), and so on. However, the base 60 system survives in the time-measuring system of 60 seconds to a minute and 60 minutes to an hour. It also survives in angle measurement and in navigational systems that measure longitude and latitude: 60 seconds equal one minute of arc, 60 minutes equal one degree of arc, and 360 degrees of arc equal an entire circle.

The base 60 system began with the Sumerians in Mesopotamia around 3000 BC. No one knows how it got started, though scholars speculate that it had something to do with the 60 to 1 ratio between the weights of the Sumerian measurement system. Others speculate that it was the result of the combining of a base 6 and base 10 numbering system. A rational explanation for using 60 as a base is that 60 can be divided evenly by 2, 3, 4, 5, and 6, which simplifies many computations.

## Place-value systems

A place-value system assigns a certain value to the spatial location of a number in a series. For example, in the decimal system, a numbers position relative to others in a series defines its category as being in the tens, hundreds, thousands, ten-thousands, and so on. In the number 1,234, the 4 occupies the slot representing zero through 9, the 3 occupies the slot representing 10 through 99, the 2 occupies the slot representing 100 through 999, and the 1 occupies the slot representing 1000 through 9999.

Place value systems are important because they make common arithmetic functions much more efficient. If people are to manipulate spatial symbols readily, they need a method that is simple, consistent, and symmetrical so that numbers can be lined up visually, and can be quickly grouped at a glance according to their value. Without the place values of the decimal system, simple arithmetic functions of addition, subtraction, multiplication, and division are enormously difficult because they are intimidating, time-consuming, overly complicated, and prone to error.

The Roman numeral system (I, II, III, IV, . . .) lacks an efficient way to represent place, and it makes simple arithmetic functions very difficult to perform for most people. Compare below the simple process of adding 17, 38, and 3 in Roman numerals and Hindu-Arabic numerals.

 XVII 17 XXXVIII 38 III 3 LVIII 58

Most people who are familiar with Hindu-Arabic numbers find that adding the Roman numerals on the left is baffling.

Although place-value systems make it easier for people to do arithmetic, they also help computers perform electronic computations at very fast speeds. A common place-value system used in computers is the binary number system, which is a base 2 system. The binary system has two values: 0 and 1. These values correspond with the signals high and low in the electronic circuits of computers. Because these numbers are so simple, computers can process them electronically up to a trillion times per second, depending on the speed of the computer.

In the binary system, each place from right to left is valued at 2 times the place to its right. Thus, the first place can be zero or one, the second place to the left is valued at two, the third place to the left is valued at four, the fourth place to the left is valued at eight, and so on. The following list indicates the binary values of the first ten numbers of a decimal system:

### KEY TERMS

Arc The continuous path described by a curved line.

Base The foundation or reference point upon which a counting system is built.

Hindu-Arabic numbers Although commonly called Arabic numerals, the numbering system represented as 1, 2, 3, 4, . . ., 9 represents a combination of innovations from Arabic and Hindu (or Indian) cultures.

Latitude The lines that run east and west on a map that are used to measure the distance north and south of the equator.

Longitude The lines on a map that run perpendicular to the equator, which are used to measure distances east and west.

Mesopotamia The area in the ancient Middle East between the Tigris and Euphrates rivers, which is now in Iraq.

Place-value The location of a number relative to others in a sequence. In the decimal system the number 3 in the series 2,300 occupies the hundreds place.

Roman numerals The numbering system developed during the Roman Empire: I, II, III, IV, V, and so on.

For example, the decimal number 3 above has two 1s in its binary format. The 1 on the right in the binary format is equal to 1, because its place value can only be 1 or 0. However, the 1 on the left in the binary format (for the decimal number 3) occupies the place that is valued at 2 in the binary system. Consider another example: look at the decimal number 10 as it is formatted in the binary system: 1010. The fourth number (1) from the right occupies the place valued at 8; the 0 in the third place means it is valued at zero; the 1 in the second place from the right means it is valued at 2; and the 0 in the right-most place means zero. Thus, in the binary place-value system, 8 + 0 + 2 + 0 = 10.

Although this system seems cumbersome to people who are used to the decimal notation system, it is perfectly suited for the ways that computers manipulate electric currents to process large quantities of data at very fast rates.

## Resources

### BOOKS

Ball, W.W. Rouse. A Short Account of the History of Mathematics. London: Sterling Publications, 2002.

Burton, David M. Elementary Number Theory. Boston, MA: McGraw-Hill Higher Education, 2007.

Clawson, Calvin C. The Mathematical Traveler: Exploring the Grand History of Numbers Cambridge, MA: Perseus Publishing, 2003.

Reid, Constance. From Zero to Infinity: What Makes Numbers Interesting. Wellesley, MA: A.K. Peters, 2006.

Schroeder, Manfred Robert. Number Theory in Science and Communications: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-similarity. Berlin, Germany, and New York: Springer, 2006.

Vinogradov, Ivan Matveevich. Elements of Number Theory. Dover Publications, 2003.

Weisstein, Eric W. The CRC Concise Encyclopedia of Mathematics. Boca Raton, FL: Chapman & Hall/CRC Press, 2003.

Patrick Moore