NUMBER SYSTEMS

Intrinsically, engineering involves the analysis, evaluation, and expression of quantities of materials and forces of nature. It requires common symbols for counting objects and a numbering by which the magnitudes of length, mass, time, and other physical properties can be expressed.

Most engineering work employs arabic numerals and is based on the deci-mal numeration system. With this system, any number, regardless of size, can be expressed with 10 basic numbers or digits. The decimal system represents numbers in groups of 10. Ten is the scale or base of the decimal system.

The value of each digit in the decimal system depends on the symbol and its position in the numeral. Thus the numeral 321 stands for 3 hundreds, 2 tens, plus one. The numeral 123 contains the same digits but represents a dif-ferent number because the digits are in difdif-ferent positions.

In the decimal system, the position value can also be expressed by powers of 10, the exponent indicating the number of times the number 10 (the base) is to be multiplied by itself:

Place Name Exponent Meaning

Ones 0 10⁰ 1

Tens 1 10¹ 10

Hundreds 2 10² 100

Thousands 3 10³ 1000, and so forth

There are, of course, other numeration systems that can be used to solve engineering problems, for example, the binary system that forms the basis for the operation of electronic computers. The binary system is based on only two digits, 0 and 1. In computers, this corresponds to the positions of electronic switches: on or off.

The binary system groups numbers by powers of 2. The position of the digit determines the value in terms of powers of two. The value of the rightmost digit is 2⁰or 1; the value of the digit to its immediate left is 2¹; the value of the next digit to the left is 2²; and so on. Binary numbers can be converted to dec-imal numbers by summing the place values of the digits in terms of decdec-imal

7.2 NUMBER SYSTEMS

177

numbers. The meanings of several binary numbers and equivalent decimal numbers are shown in Table 7.1.

7.3 DIMENSIONS

To solve engineering problems, it is necessary that engineers describe or char-acterize the material world in terms of dimensions. We commonly think of dimensions in terms of spatial extent or size, that is, length, width, and height.

But many other dimensions are used to describe the physical properties of the objects and materials of engineering. There are fundamental dimensions such as length, time, and mass, and combinations of such dimensions called derived dimensions. For example, velocity, the ratio of length and time, is a derived dimension. Other examples of fundamental and derived dimensions are shown in Table 7.2.

Some quantities used in engineering calculations have no dimensions.

Examples of dimensionless quantities are ratios of quantities having the same dimension, such as π, the ratio of the circumference of a circle to its diameter.

TABLE 7.1 Examples of Binary Numbers and Equivalent Decimal Numbers

Binary Number Meaning Equivalent Decimal

Number

1 1  2⁰ 1

10 (1  2¹) (0  2⁰) 2

11 (1  2¹) (1  2⁰) 3

100 (1  2²) (0  2¹) (0  2⁰) 4 101 (1  2²) (0  2¹) (1  2⁰) 5 110 (1  2²) (1  2¹) (0  2⁰) 6

TABLE 7.2 Examples of Fundamental and Derived Dimensions Fundamental Dimensions Derived Dimensions

Length, L Area, L²

Time, T Volume, L³

Mass, M Velocity, L/T

Electric current, I Acceleration, L/T²

Temperature Mass density, M/L³

Amount of substance, (mole) Force, ML/T²

Luminance intensity Energy, ML²/T²

7.4 UNITS

The word unit is defined as a precisely stated quantity in terms of which other quantities of the same kind can be stated. For each dimension, there is a need for one or more reference amounts in order to describe quantitatively the phys-ical properties of some object or material. For example, the dimension of length has been measured in units of miles, metres, feet, the distance from a person’s nose to fingertip, and many others. We measure time in units of seconds, min-utes, hours, months, and so on.

Since we deal with a large number of dimensions, a system of units is needed for reliable and reproducible measurements and for good communica-tion. Technological developments in transportation and communications and increasing international trade have emphasized the need for a common lan-guage of measurement, a system capable of measuring any physical quantity with units that are clearly and precisely defined and that possess a logical rela-tionship between units to facilitate calculations. The Système Internationale d’Unitès (SI) is widely recognized as such a system. Graham¹has described the history of the development of the SI system and its essential features in an arti-cle published by the American Society for Testing and Materials. The artiarti-cle is reprinted in the following pages with permission of the author and ASTM.

There are many systems of measurement in use throughout the world, but primarily two that have been in use for years—the English system and the met-ric system. These systems suffer from many variations and problems.

The English system, which is common in the United States, has grown piece by piece over at least 3000 years, with little relationship between units. The element of precision has been provided by the National Bureau of Standards in the United States and the National Physical Laboratory in England since the beginning of the twentieth century, and individual units are adequate to any measuring task. As a system it is poor, and the many differences in its detail between English-speaking countries presents a problem.

The old metric system (speaking of today’s common European system) also has problems. In contrast with the English system, which grew in a haphazard fashion, it was commissioned by the French government nearly 200 years ago and was designed to be an integrated, universal measurement system. The United States and the British Commonwealth nations refused to join in its use but Germany, France, Italy, and others proceeded to develop their industries and industrial standards around it. Many variations in the system have devel-oped, however, because no controls were set up to unify use, and the common metric system is as awkward and varied as the English system.

In the late nineteenth century these variations of units of measure, particu-larly the minor differences that had grown up between countries using the met-ric system, were an obstacle to the rapidly increasing sophistication of science and industry.

7.4 UNITS

179

1J. D. Graham, “The International System of Units (SI),” Metrification—Managing the Industrial Transition, ASTM STP 574, American Society for Testing and Materials, 1975, pp. 42–56.

In 1875 five years of international discussion on measurement culminated in the signing by 17 countries (including the United States) of the Metre Convention. An international organization was established to provide a sound basis of precise uniform world measurement units. This organization consisted of an international working committee to provide the technical base (CIPM), an international bureau for laboratory work (BIPM), and the treaty body, the General Conference on Weights and Measures (CGPM), scheduled to meet in Paris every six years. The initials are those of the French names for these soci-eties, and are universally used to designate them.

This organization has developed a basis for the standardization of the world’s measurement units by issuing in 1960 the International System of Units, abbreviated SI (from its French name) in all languages. This is a com-plete, coherent system, is the basis of all official measurement in the world, and is the system that the United States is now beginning to adopt. It is a major obligation of all the world’s technical people to understand it, respect it, and use it properly.