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# Units and Prefixes Study Guide

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Updated on Sep 26, 2011

## Introduction

Measurements and observations form the base of our scientific knowledge of the world. Some quantities we measure can be calculated in terms of others, while some cannot; those that cannot are the most fundamental quantities. The International System of Units (SI) is based on seven fundamental units from which we can express all other units. Sometimes, we deal in practice with quantities much larger or much smaller than the defined unit. In such cases, it is convenient to use prefixes. Last but not least, we look at significant figures and guidelines for their use in computations and the scientific notation.

## Fundamental and Derived Units

Physics is exciting when it can predict how nature will behave in one situation on the basis of data obtained in another situation. Therefore, a great deal of effort goes into making measurements as accurate and reproducible as possible in a given situation.

For example, the atomic mass unit u has a value of one-twelfth (1/12) the mass of the carbon isotope C12. As long as we stay within the boundaries of atomic physics, the picture is nice and clear: The hydrogen atom has (approximately) a one-unit mass, 1 u; the oxygen atom has a mass of 16 u; and so forth. And these data can be obtained from nice atomic collision measurements. The ugly part (due to our somewhat arbitrary definition of u) comes when we want to link this to another mass standard, such as the kilogram. We find the following:

1 u = 1.66067 · 110–27 kg

and all our nice atomic data transform into strings carrying four or five decimals. It can make us wonder why the kilogram is the only one of the seven fundamental units still defined in terms of a physical platinum-iridium cylinder artifact kept in Paris.

Often, it is possible to split the units of measurement into simpler units. For example, a unit of area, say, a square foot, may be actually the area of a square that is 1 foot long by 1 foot wide. In other words, we can reduce the area measurement to a length measurement.

But there are times when this trick is not possible anymore. For example, to measure the length, we need to actually perform a direct comparison to a convenient unit of length. We call length a fundamental quantity and the unit of length a fundamental unit. Other well-known fundamental units are the ones measuring time and mass. We should think of these base units as some of the building blocks that we use in our dimensional puzzle to get units for other important physical quantities such as speed, force, or energy.

### Choosing Units

Choosing a unit for a particular quantity is a highly subjective exercise; choosing a good one will allow accurate measurements that are easily accessible and reproducible so that everybody understands when we document our observations.

## International System of Units (SI)

Revolutionary times call for revolutionary units! The main ideas for an international system of units were born out of the French Revolution, when the metric system and the kilogram were introduced to measure lengths and weights. Another revolution, and a New World, had already adopted a decimal system—you guessed it, in the United States, where a dime has ten cents, a dollar has ten dimes, and so forth. But the United States fell short of putting Thomas Jefferson's decimal idea in practice outside the financial system.

Eventually, these revolutionary ideas prevailed and took over the world, when in 1960, the International System of Units (S1) was adopted by the 11th General Conference on Weights and Measurements.

This system defines seven fundamental quantities and their associated units: length and meter, mass and kilogram, time and second, electric current and ampere, temperature and degree Kelvin, luminous intensity and candela, and, last but not least, the amount of substance and the mole. All other units can be consistently derived from these seven fundamental units.

The metric system had a strong influence on the scientific community. Therefore, eventually the first coherent system of units and a precursor to SI showed up in the 1860s as the CGS system. It defined the centimeter, gram, and second (or CGS) as fundamental units for length, mass, and time. Another hundred years had to pass for us to fully realize we needed another four more fundamental units to get the whole picture, that is, so that we could derive any other measurement in terms of these seven fundamental quantities.

A coherent system of units enforces a certain set of rules for manipulating the results of the actual measurements. To get sensible results, only quantities with the same unit can be added or subtracted.

When listing data and performing calculations, do the following:

• Write down the units explicitly, convert everything to the same unit, and use the same prefixes throughout a calculation.
• Treat all units as algebraic quantities. In particular, when identical units are divided, they cancel each other out.
• Use conversion factors between different unit systems. For instance, the conversion factor of 1 lb = 0.4536 kg might be used to go back and forth between 51 and British units of mass.
• Check to see that your calculations are correct by verifying that the units combine algebraically to give the desired unit for the answer. Remember, only quantities with the same units can be added or subtracted.

Some usual practical conversion factors between different units in the SI and the u.s. or British system are as follows:

1 inch = 2.54 centimeters

1 yard = 3 feet = 36 inches = 91.44 centimeters

1 mile = 1.609 kilometers

1 pound (lb) = 0.4536 kilograms

1 quart = 946 milliliters = 0.946 liters

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