# 1.3: Measurements

Scientists present their evidence in the form of a measurement. People make many statements in everyday life. Some are qualitative and some are quantitative. A friend might say "This music is great!" or "It was colder yesterday than today," or, "There are only four forces of nature." There is no reliable way to quantify the first statement, even though your friend may feel it is true. However, the next two statements can be quantified, and they can be subjected to actual measurements that will either support or refute the assertions. In science, we try to deal only with statements that can be quantified. Otherwise, we would have no way to compare results.

Scientists use measurements to arrive at precise and quantitative statements. A measurement must have two components: a number and a unit, which specifies the type of the quantity that is being counted. Notice that numbers alone are not that useful. Units must be in a system that is well defined. The statement "It was cold yesterday" can be quantified by saying that "It was 15 degrees yesterday," but you still need to know whether this is on the Celsius or Fahrenheit system of units. The statement "The stock market fell 50 points" is quantified, but to understand it, you need to know what a point on the Dow Jones average actually represents.

Astronomers need to handle very large and very small numbers. Scientific notation is a useful shorthand for writing numbers of any size (it is also often called exponential notation). For example, instead of saying that the nearest star is about 40,000,000,000,000 kilometers away, we can say it is 4 x 10^{13} kilometers away. (The exponent "13" represents the number of zeros following the significant figure "4.")

The system of units in physical science is amazingly simple. All the diverse measurements we can make in the physical world — speed andforce and temperature and electric charge and energy and so on — are derived from only three fundamental properties: mass, length, and time. To these properties we attach the familiar units in the metric system: kilograms to measure mass, meters to measure length, and seconds to measure time. Almost every other type of measurement is just some combination of these units. For example, area is just lengthmultiplied by length, and momentum is just mass multiplied by velocity — which is the same as mass multiplied by length divided by time. Even concepts such as energy or temperature can be expressed as combinations of the same three quantities. A simple system of units helps scientists make sense of a complicated world.

Why do scientists generally use units in the metric system? The metric system was first put into widespread use after the French Revolution of 1789. The architects of the French Revolution wanted to make a break with the culture defined by royalty and hereditary power and usher in an Age of Reason. As part of this sweeping set of social changes, they introduced a set of units based on a decimal counting system. The metric system was designed to replace the English system where a gallon is 8 pints, a foot is 12 inches, a pound weight is 16 ounces, a pound Sterling is 20 shillings, and so on. These units have their origins in medieval European history! At a time when few people were literate or numerate, it was easier to have measurements that could be easily subdivided. The numbers 8, 12, 16, and 20 all have three or more factors, while 10 only has two factors.

However, the metric system is much simpler. For example, a meter (which is roughly a yard) is 100 centimeters, and a kilometer is 1000 meters. Thomas Jefferson was very impressed by the metric system and pushed for it to be adopted in the United States. He would be very disappointed if he knew that nearly 230 years later his country was the world's last holdout against adopting the metric system. It is ironic that the world’s most advanced technological society still clings to inches, miles, acres, gallons, pints, pounds, and even horsepower ratings! These anachronistic units are awkward to deal with and the result is a subtle but profound disconnect between scientists and the general public.

Astronomy deals with temperature extremes from the coldness of intergalactic space at 3 degrees above absolute cold to the shock wave of asupernova at a billion (10^{9}) degrees. It spans objects that range in size from microscopic cosmic dust grains (10^{-6} meters) to the distance that light has traveled in the age of the universe (10^{23} kilometers). Discussions range from the sparseness of the space between galaxies at a density of 10^{-19} kilograms per cubic meter, to the density inside a black hole at 10^{18} kilograms per cubic meter. The difference between these last measurements is a factor of 10^{37}, or 10 followed by 37 zeros! With such factors, it is not surprising that astronomers use scientific notation.

Notice the number of significant figures, or non-zero digits, in a scientific measurement. For example, look at the numbers 13,000 kilometers and 12,756 kilometers. In scientific notation, we would write these numbers 1.3 x 10^{4} kilometers and 1.2756 x 10^{4} kilometers. The first number implies quite a rough measurement; changing the least significant figure would give you 12,000 or 14,000, which is a difference of 15 percent. But the second number implies a very fine measurement; changing the least significant figure would give you 12,755 or 12,757, which is a difference of a tiny fraction of a percent. So the number of significant figures is a measure of precision. Note that a calculator will often return a large number of significant figures, but that doesn't mean you should infer a high precision to the number. The precision of a number in science depends on how the measurement was made.

However, we should point out that precise numbers are not always required in science and are sometimes not even possible to measure. Some measurements of atomic phenomena are precise to twelve significant figures (one part in a trillion) while some numbers in cosmologyare only precise to one significant figure. You can understand most of the material in astronomy without having to deal with very precise numbers. In the example just given, it is more important to remember that the Earth is roughly 13,000 kilometers across than to try to memorize that it is exactly 12,756 kilometers across. Few numbers in astronomy are known with a precision of more than three significant figures. Sometimes the uncertainty is a factor of ten, which is called an order of magnitude. Knowing an approximate value allows you to estimate many effects without extensive calculations or looking things up in books. This basic scientific skill is called estimation. Estimation saves time and allows scientists to distinguish promising hypotheses from foolish ones.