As previously discussed, a vital part of scientific endeavor involves making quantitative observations or measurements. Every measurement has two aspects: a number and a unit. Once measurements are made, they are typically used to express, explain, or exemplify scientific ideas. In addition, once made they are often mathematically manipulated to better describe the concept under investigation. Therefore, knowing the measurement system typically used, applying a straightforward approach in handling the numbers (which could be very large or very small), and understanding mathematical machinations that may be required are all critical for success.
It is important that scientists around the world use the same units when communicating information. For this reason, scientists use the modernized metric system, designated in 1960 by the General Conference on Weights and Measures as the International System of Units. This is commonly known as SI, an abbreviation for the French name Le Système International d’Unités. It is now the most common system of measurement in the world. There are minor differences between the SI and metric systems. For the most part, the quantities are interchangeable.
The reason SI is so widely accepted is due to its straightforward, simple approach. The system includes only seven base units that can be inflated or deflated by the use of prefixes related by a decimal. Quantities other than those described by the base units can be derived by mathematical manipulation of the base units or other derived units.
The seven base units that can be used to express the fundamental quantities of measurement are specifically defined within the system. For example, the meter is defined as the length of the path traveled by light in a vacuum during a time interval of 1/299,792,458 seconds. These base units and their symbols are shown in Table 1.1.
Table 1.1 SI Base Units
| Quantity | Unit | Abbreviation |
| mass | kilogram | kg |
| length | meter | m |
| time | second | s |
| electric current | ampere | A |
| temperature | kelvin | K |
| amount of substance | mole | mol |
| luminous intensity | candela* | cd |
If the magnitude of an SI unit is not appropriate for the object being measured, prefixes can be combined with the unit to adjust the unit's size. The prefixes represent multiples or fractions of 10. Table 1.2 gives some prefixes commonly used in the SI system.
Table 1.2 Prefixes Used with SI Units
For an example of how a prefix works in conjunction with the base unit, consider the term kilometer. The prefix kilo- means “multiply the base unit by 1,000,” so a kilometer is 1,000 meters. By the same reasoning, a millimeter is 1/1,000 of a meter. Because of the prefix system, units can be easily related by some factor of 10. Below is a list of some SI unit equivalents for various fundamental quantities.
A unit of length, used especially in expressing the length of light waves, is the nanometer, abbreviated as nm and equal to 10−9 meter. A unit of mass, used especially in measuring the mass of samples in the lab, is the milligram, abbreviated as mg and equal to 0.001 grams.
The metric system standards were chosen as natural standards. The meter was once described as 1/10,000,000 of the distance between the equator and the North Pole but now, defined by SI, is the length of the path traveled by light in a vacuum during a time interval of 1/2.99792458 × 108 second.
There are some interesting relationships between volume and mass units in the SI system. Because water is most dense at 4°C, the gram was intended to be 1 cubic centimeter of water at this temperature. This means, then, that:
The most commonly used temperature scale in scientific work is the Celsius scale. It gets its name from the Swedish astronomer Anders Celsius and dates back to 1742. For a long time it was called the centigrade scale because it is based on the concept of dividing the distance on a thermometer between the freezing point of water and its boiling point into 100 equal markings or degrees.
The SI temperature scale and unit are related to the Celsius scale and unit. The SI temperature scale is based on the lowest theoretical temperature (called absolute zero). This temperature has never actually been reached, but scientists in laboratories have reached temperatures within about a 100 trillionths of a degree above absolute zero. Sir William Thomson, also known as Lord Kelvin, proposed this scale on which a unit is the same size as a Celsius degree but where the zero mark has been displaced lower. Consequently, it is referred to as the Kelvin temperature scale. In the Kelvin scale, the magnitude of the temperature value is directly proportional to the average kinetic energy of the sample. That's why the Kelvin scale is often called the absolute temperature scale. The direct relationship between temperature and energy is not the case when using the Celsius scale, in which negative numbers and zero are commonly measured values. For that reason, it is common in calculations involving temperature that values measured in Celsius be converted to the Kelvin scale.
Through experiments and calculations, it has been determined that absolute zero is 273.15 degrees below zero on the Celsius scale. This figure is usually rounded off to −273°C.
Figure 1.7 displays the graphic and algebraic relationships among three temperature scales: the Celsius and Kelvin, commonly used in chemistry, and the Fahrenheit.
Note: In Kelvin notation, the degree sign is omitted: 283 K. The unit is the kelvin, abbreviated as K.
Since there are only seven SI base units but many more than seven quantities that need to be described by a unit, most SI units are constructed through mathematical manipulation of related units. These units are said to be derived. A derived unit is comprised of base or other derived units put together in some mathematical operation that describes the considered quantity. Often, the combination of units is given a name to simplify the expression of the unit. It is common for the name to have been given in honor of a famous scientist who worked in the particular branch of science for which the derived unit is commonly used. Table 1.3 contains some commonly derived quantities, how their units are derived, as well as (perhaps) the name given to the unit.
Table 1.3 Derived Quantities and Units
| Quantity | Derivation by Quantity |
Derivation(s) by Unit |
Name |
| area | length × length |
m2, cm2, or dm2 |
--- |
| volume | length × length × length |
m3, cm3, or dm3 |
liter (L) = dm3 |
| density | mass/volume |
g/cm3, g/mL, or g/L |
--- |
| velocity | distance/time |
m/s |
--- |
| acceleration | distance/time2 |
m/s2 |
--- |
| force | mass × acceleration |
kg · m/s2 |
newton (N) |
| energy | force x distance |
N x m or kg · m2/s2 |
joule (J) |
Often the magnitude of a measurement is very large or very small. So the numbers used to express the value can be awkward to describe or to compare with other values. Scientific notation alleviates both of these problems. It uses a standard way to express numbers that easily allows the numbers to be compared. A number written in scientific notation has two factors multiplied by each other. The first factor is a number with only one digit to the left of the decimal. This mandates the number be greater than or equal to 1 and less than 10. In this way, the magnitude of the first factor in any scientific notation will always be somewhat similar in value. The second factor is always expressed as some power of 10. Since the first factor is always similar in magnitude, when comparing scientific notation numbers the exponent on the base 10 can easily be used to compare the sizes of the numbers expressed.
With large numbers, such as 3,400,000, first move the decimal point to the left until only one digit remains to the left of it (3.400000). Then indicate the number of moves of the decimal point as the exponent of 10 (3.4 × 106). With a very small number such as 0.0000034, first move the decimal point to the right until only one non-zero digit is to the left of it (0000003.4). Then indicate the number of moves as the negative exponent of 10 (3.4 × 10−6).
With numbers expressed in this exponential form, you can now use your knowledge of exponents in mathematical operations. An important fact to remember is that in multiplication you add the exponents of 10, and in division you subtract the exponents. Addition and subtraction of two numbers expressed in scientific notation can be performed only if the numbers have the same exponent.
Two other factors to consider in measurement are precision and accuracy. Precision indicates the reliability or reproducibility of a measurement. Accuracy indicates how close a measurement is to its known or accepted value.
For example, suppose you were taking a reading of the boiling point of pure water at sea level on a normal day. Using the same thermometer in three trials, you record 96.8, 96.9, and 97.0 degrees Celsius. Since these figures show a high reproducibility, you can say that they are precise. However, the values are considerably off from the accepted value of 100 degrees Celsius, so you say they are not accurate. In this example, you would probably suspect that the inaccuracy was the fault of the calibration of the thermometer.
Regardless of precision and accuracy, all measurements have a degree of uncertainty. This is usually dependent on one or both of two factors—the limitation of the measuring instrument and the skill of the person making the measurement. Uncertainty can best be shown by example.
The graduated cylinder in Figure 1.8 contains a quantity of water to be measured. It is obvious that the quantity is between 30 and 40 milliliters because the meniscus lies between these two marked quantities. Now, checking to see where the bottom of the meniscus lies with reference to the ten intervening subdivisions, we see that it is between the fourth and fifth. This means that the volume lies between 34 and 35 milliliters. The next step introduces the uncertainty. We have to guess how far the reading is between these two markings. We can make an approximate guess, or estimate, that the level is more than 0.2 but less than 0.4 of the distance. We therefore report the volume as 34.3 milliliters. The last digit in any measurement is an estimate of this kind and is uncertain. It is, however, better to make this guess than not. If we reported the value to be 34 milliliters, we know that would be too low. If we reported the value to be 35 milliliters, we know that would be too high. The general rule is that you should report the value of your measurement to be 1 place past the measurement about which you are absolutely sure.
Any time a measurement is recorded, it includes all the digits that are certain plus one uncertain digit. These certain digits plus the one uncertain digit are referred to as significant figures. These digits are all reasonably certain because you absolutely know all but the last and have made a guess based on reason for that last one. The more digits you are reasonably able to record in a measurement, the less relative uncertainty there is in the measurement. If a measurement is made and a nonzero value is reported in a place, it had to have been reasonably measured or a value would not be there. The value of 0 may be interpreted in different ways, however. Sometimes it is a significant figure (meaning it is in a measured place), but other times it is not a significant figure (meaning it is simply a placeholder giving proper magnitude to a number). Table 1.4 summarizes the rules of significant figures.
Table 1.4 Zero Rules for Significant Figures
| Rule | Example | Number of Significant Figures |
| All digits other than zeros are significant. | 25 g | 2 |
| 5.471 g | 4 | |
| Zeros between nonzero digits are significant. (They had to have been in absolutely measured places.) | 309 g | 3 |
| 40.06 g | 4 | |
| Final zeros to the right of the decimal point are significant. (The don't need to be there to express the magnitude of the number; therefore, they must have been measured.) | 6.00 mL | 3 |
| 2.350 mL | 4 | |
| In numbers smaller than 1, zeros to the left or directly to the right of the decimal point are not significant. | 0.05 cm |
1 The zeros merely mark the position of the decimal point. |
| 0.060 cm |
2 The first two zeros mark the position of the decimal point. The final zero is significant. |
One last rule deals with final zeros in a whole number. These zeros may or may not be significant, depending on the measuring instrument. For instance, if an instrument that measures to the nearest mile is used, the number 3,000 miles needs to have four significant figures. If, however, the instrument in question records miles to the nearest thousands, there should be only one significant figure. The number of significant figures in 3,000 could be one, two, three, or four, depending on the limitation of the measuring device.
This problem can be avoided by using scientific notation. By convention, the first factor in any scientific notation number used to express a measurement contains only significant figures. For this example, the following notations all indicate different numbers of significant figures:
| 3 × 103 3.0 × 103 3.00 × 103 3.000 × 103 |
one significant figure two significant figures three significant figures four significant figures |
Another common way to make zeros significant, when they would not normally be considered so, is to place a decimal point at the end of the whole number when it would not normally be shown. For example:
| 3000 | (one significant figure) |
| 3000. | (four significant figures) |
When you do calculations involving numbers that do not have the same number of significant figures in each, keep the following two rules in mind.
First, in multiplication and division, the number of significant figures in a product or a quotient of measured quantities is the same as the number of significant figures in the quantity having a smaller number of significant figures.
Second, when adding or subtracting measured quantities, the sum or difference should be rounded to the same number of decimal places as the quantity having the fewest decimal places.
Dimensional analysis is a mathematical technique often used by scientists to convert a quantity given in one unit to the same quantity expressed in a different unit. This may eliminate the need to express the number in scientific notation if it is too big or too small to work with easily, thereby expressing the measurement in an appropriately sized unit. For example, you may alter the measured mass of a large sample of iron given in milligrams to kilograms so that it may be more easily compared to other large samples of metals for which you know their mass in kilograms.
To use dimensional analysis, an equation needs to be written that mathematically changes the units of the given value to that of the unknown value. The first part of the dimensional analysis equation begins with the given quantity itself. Next, a fraction that equals 1 is multiplied by the given quantity. In the fraction, the desired units are in the numerator and the units of the given quantity are in the denominator. Since the fraction is equal to 1, the given quantity itself stays the same. The example that follows shows how 155,500 milligrams can be expressed in kilograms. This may be a more appropriate expression of the given quantity based on the quantity's size.
Analysis of the equation shows that the mg unit in the given quantity is canceled out by the mg unit in the denominator of the fraction. The unit that does not cancel is then associated with the answer. The mathematical operation is then applied to the numbers to produce the value in front of that unit.