# Numeration Systems

## 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 number's 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 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 blinding 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:

 decimal binary 0 = 0 1 = 1 2 = 10 3 = 11 4 = 100 5 = 101 6 = 110 7 = 111 8 = 1000 9 = 1001 10 = 1010

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. But 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 rightmost 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.

Barrow, John D. Pi in the Sky: Counting, Thinking, and Being. Oxford: Oxford University Press, 1992.

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

Swetz, Frank J. Capitalism & Arithmetic: The New Math of the 15th Century. LaSalle, IL: Open Court Press, 1987.

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

Weisstein, Eric W. The CRC Concise Encyclopedia of Mathematics. New York: CRC Press, 1998.

Patrick Moore

## 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 which 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.