Scientific Notation and Units of Measure

The numbers in this book (in scientific writing of all sorts, really) may look odd to readers unfamiliar with mathematics, but in fact the notation is straightforward and the abbreviations fairly easy to learn. Scientific notation is simply a convenient way of writing very large or very small numbers. It expresses numbers as a multiple of a "conveniently written" number and a power of ten. For example, we could write 230 as the product of 2.3 x 100. However, since 100 is the same as ten squared, or 102, we could also write 230 as 2.3 x 102. Expressing this particular number in scientific notation does not really offer any advantage to us, but imagine having to write 2,300,000,000 (two billion, three hundred million). We could express this as the product of 2.3 and one billion. Because one billion is the same as ten multiplied by itself nine times (try it on a hand calculator), or 109, we could write the number much more compactly as 2.3 x 109. Expressing very small numbers is also easily done. For example, 0.023 is the same as 2.3 x 0.01. However, 0.01 is the same as ten divided by itself twice, and mathematically, this can be written as 10-2. Thus, we can conveniently write the very small number 0.0000000023 as 2.3 x 10-9. Computers have an even more compact way of writing large or small numbers: they simply express the power of ten as a number preceded by an e (for exponent). Therefore, 0.0000000023, or 2.3 x 10-9, may appear on a computer printout or on the display of a hand calculator as 2.3e-9.

Whenever one deals with real quantities, rather than pure numbers, one must be able to describe what those numbers quantify. For example, a number like n is a pure number. It can be calculated from the ratio of two lengths (the circumference and radius of a circle), but the number itself does not signify a length. A number that does indicate a length, however, must include a descriptor of some sort, in other words, a unit of measure. Units usually are written after the number. So, we would describe the radius of a circle as x meters, or feet, or whatever.

Units that describe fundamental quantities, like mass, length, time, or temperature, have simple units. One often must deal with compound units, however, combining two or more simple units. A speed, for example, is a ratio of a length or distance traveled and a time required to travel it. One unit of speed could therefore be meters per second, which could be written in abbreviated form as m/s. Scientific convention dictates another form, though, as the multiple of distance and the inverse of time. So, the compound unit m/s could also be written as m x (1/s). Since an inverse of a quantity is equivalent to the quantity raised to the power of -1, we can write the unit of velocity as m s-1.

This might seem the ultimate in obfuscation: what's wrong with a simple ratio, like m/s? There is a good reason for using this style, though. Sometimes units become very complicated. Weight, for example, is actually the product of a mass and an acceleration imparted to the mass by the gravitational attraction of the Earth. The unit for mass is straightforward, the kilogram (kg). An acceleration, on the other hand, is a change of speed with time, or (meters per second) per second (no, I did not inadvertently repeat "per second"). The unit in simple ratio form is m/s/s. Or should it be (m/s)/s? Or should it be m/(s2)? And when we multiply acceleration by mass to get a weight (properly designated with the newton, N), the potential for confusion abounds. Is it kg x m/s/s? kg x (m/s/s)? kg x ((m/s)/s)?

Or we could simply write it as kg m s-2. By writing the units with the inverses as negative powers, the unit can be specified unambiguously and without the need to decode a multitude of parentheses.

You will also see many of the unit designators with prefixes, such as millimeters (mm) and kilojoules (kJ). These represent multipliers to the basic units. For example, the standard unit of length is the meter (m). Sometimes it is inconvenient to express a length in meters, though. The distance between, say, New York City and San Francisco is 4,713,600 meters, but it is more common to express as at 4,714 kilometers (km), where kilostands for "1,000 times."

Certain prefixes are used quite frequently, especially those that specify multiples of 103 or 10-3, as listed in the accompanying table. Sometimes a unit of measure is used even though the number could be "simplified" by using another. So, for example, we express the distance between San Francisco and New York as 4,714 km, even though we could more easily (and properly) write it as about 4.7 Mmâ€”kilometers are simply more conventional than megameters. Common numbers like distances would never be expressed in megameters or nanomilesâ€” it would bring us too close to the Dilbertian world of pocket protectors and calculator holsters.

Prefix |
How it is said |
Multiplier |

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