# MeasurementThe Metric System: Metric and Scientific Notation

by Anthony Carpi, Ph.D.

By the 18^{th} century, dozens of different units of measurement were commonly used throughout the world. Length, for example, could be measured in feet, inches, miles, spans, cubits, hands, furlongs, palms, rods, chains, leagues, and more. The lack of common standards led to a lot of confusion and significant inefficiencies in trade between countries. At the end of the century, the French government sought to alleviate this problem by devising a system of measurement that could be used throughout the world. In 1790, the French National Assembly commissioned the Academy of Science to design a simple decimal-based system of units; the system they devised is known as the metric system. In 1960, the metric system was officially named the Système International d'Unités (or SI for short) and is now used in nearly every country in the world except the United States. The metric system is almost always used in scientific measurement.

## Metric system basics

The simplicity of the metric system stems from the fact that there is only one unit of measurement (or base unit) for each type of quantity measured (length, mass, etc.). The three most common base units in the metric system are the meter, gram, and liter. The meter is a unit of length equal to 3.28 feet; the gram is a unit of mass equal to approximately 0.0022 pounds (about the mass of a paper clip); and the liter is a unit of volume equal to 1.05 quarts. So length, for example, is always measured in meters in the metric system; regardless of whether you are measuring the length of your finger or the length of the Nile River.

To simplify things, very large and very small objects are expressed as multiples of ten of the base unit. For example, rather than saying that the Nile River is 6,650,000 meters long, we can say that it is 6,650 thousand-meters long. This would be done by adding the prefix "kilo" (meaning 1,000) to the base unit "meter" to give us 6,650 kilometers for the length of the Nile River. This is much simpler than the American system of measurement, in which we have to remember inches, feet, miles, and many more units of measurement. Metric prefixes can be used with any base unit. For example, a kilometer is 1,000 meters, a kilogram is 1,000 grams, and a kiloliter is 1,000 liters. Six common prefixes used in the metric system are listed below.

Common Metric Prefixes | Unit Multiples |
---|---|

Table 1: Common metric prefixes. | |

kilo | 1,000 |

hecto | 100 |

deca | 10 |

(base unit) | - |

deci | 0.1 |

centi | 0.01 |

milli | 0.001 |

The subunits are used when measuring very large or very small things. It wouldn't make sense to measure your weight in grams for the same reason that you wouldn't measure it in ounces - the unit is too small. You would express your weight in kilograms (each kilogram is equal to 1,000 grams or about 2.2 pounds).

## Changing metric units through decimal places

The metric system is a called a decimal-based system because it is based on multiples of ten. Any measurement given in one metric unit (e.g., kilogram) can be converted to another metric unit (e.g., gram) simply by moving the decimal place. For example, let's say a friend told you that he weighed 72,500.0 grams (159.5 lbs). You can convert this to kilograms by moving the decimal three places to the left. In other words, your friend weighs 72.5 kilograms.

Because the metric system is based on multiples of ten, converting within the system is simple. Here's a shortcut: If you are converting from a smaller unit to a larger unit (moving upward in the table shown above), move the decimal place to the left in the number you are converting. If you are converting from a larger unit to a smaller unit (moving down in the table), move the decimal to the right. The number of places you move the decimal corresponds to the number of rows you are crossing in the table. For example, let's say someone told you that you had to walk 8,939.0 millimeters to get to the grocery store. That sounds like a long walk, but let's convert the number into meters to see how long it really is. The base unit, meter, is three rows above the millimeter, so the decimal should be moved three places to the left.

It's less than 9 meters to the grocery store - or about 30 feet. Metric units can be abbreviated for simplicity. Abbreviations for the base units are the first letter of the unit name: m = meter, g = gram, and l = liter. Subunits can be abbreviated using the first letter of the prefix and the first letter of the base unit (all lowercase): mm = millimeter, kg = kilogram, etc.

## Scientific notation

In science, it is common to work with very large and very small numbers. For example, the diameter of a red blood cell is 0.0065 cm, the distance from the Earth to the sun is 150,000,000 km, and the number of molecules in 1 g of water is 33,400,000,000,000,000,000,000. It gets cumbersome to work with such long numbers, so measurements such as these are often written using a shorthand called scientific notation.

Each zero in the numbers above represents a multiple of 10. For example, the number 100 represents 2 multiples of 10 (10 x 10 = 100). In scientific notation, 100 can be written as 1 times 2 multiples of 10:

100 = 1 x 10 x 10
= 1 x 10^{2} (in
scientific notation)

Scientific notation is a simple way to represent large numbers because the 10's exponent (2 in the example above) tells you how many places to move the decimal of the coefficient (1 in the example above) to obtain the original number. In our example, the exponent 2 tells us to move the decimal to the right two places to generate the original number:

Scientific notation can be used even when the coefficient is a number other than 1. For example:

This shorthand can also be used with very small numbers. When scientific notation is used with numbers less than one, the exponent on the 10 is negative, and the decimal is moved to the left, rather than the right. For example:

Therefore, using scientific notation, the diameter of a red blood cell is 6.5 x 10^{-3} cm, the distance from the Earth to the sun is 1.5 x 10^{8} km, and the number of molecules in 1 g of water is 3.34 x 10^{22}.

Also note that in scientific notation, the base numeral is always represented as a single digit followed by decimals if necessary. Therefore, the number 0.0065 is always represented as 6.5 x 10^{-3}, never as 0.65 x 10^{-2} or 65 x 10^{-4}.