Workshop: Mathematics and Measurements
“Anything worth measuring is worth measuring well.”
Source unknown
To determine if a runner broke the world’s record for a marathon, you must carefully measure the time that passed between the start and finish of the race and compare it to the record time for that marathon. Since time can be measured and expressed as an amount, it is called a quantity. Ten seconds, two minutes, and five hours are examples of quantities of time. Other familiar quantities that are important in chemistry include mass (similar to the more familiar weight), length, volume, and density.
The International System of Units
In 1960, a group of scientists from many fields and many countries agreed upon a set of metric units that would serve as a standard for scientific communication. This standard set of units is known as the International System of Units and is abbreviated SI (the abbreviation is derived from the French spelling le Systeme International d’ Unites). Seven quantities are the foundation for SI, and each has a base unit in which that quantity is expressed. Table 1 lists the base units for length, mass, volume, temperature, time and chemical amount, along with their abbreviations and their relationships to common United States units.
Table 1: Seven fundamental units in the SI system. All other units can be decomposed to these units
Quantity
 U.S.
 SI Base Unit
 Chemistry

Mass (weight)  Pound (lb)  Kilogram (kg)  “Gram” (g, mg) 
Volume  Gallon (gal)  Liter (L)  “Liter” (mL, L) 
Temperature  Fahrenheit (oF)  Kelvin (K)  K & Celsius (oC) 
Length  Mile (mi), feet(ft), Inches (in)  Meter (m)  “Meter” (cm, mm, nm) 
Time  Second (s)  Second (s)  Second (s) 


 Mole (mol) 
Table 2: SI Base Units Equivalents
Quantity
 Base Unit
 Abbreviation
 U.S. Equivalent

Mass  kilogram  kg  2.205 pounds 
Volume  liter  L  0.946 quarts 
Length  meter  m  39.37 inches 
The three SI base units for mass, volume and length in Table 1 were chosen because they correspond to magnitudes which are convenient for everyday measurement. They are well suited for measurements on a scale that we can directly relate to. However, chemists often work with tiny quantities such as those used to express the diameter of a hydrogen atom or huge quantities such as the number of particles in a kilogram of carbon. These numbers are beyond the range of our senses and cannot be conveniently expressed in standard notation in SI units. Thus, the system of scientific notation is used to express very small and very large quantities.
Scientific notation is a method of expressing numbers as a product of two factors. The first factor is a number that is greater than or equal to 1 but less than 10. The second factor is 10 raised to a power. The power of 10, or exponent, is positive for numbers greater than 10 and negative for numbers less than 1. Table 2 gives examples showing both the ordinary decimal form and the exponential form for some quantities. Notice how scientific notation eliminates the need to write a long list of zeros in very small and very large numbers.
Table 2. Examples of Quantities Expressed in Scientific Notation
Quantity  Ordinary Decimal Form  Scientific Notation 
Diameter of a hydrogen atom  0.000 000 000 074 m  7.4 x 10–11 m 
Mass of a hydrogen atom  0.000 000 000 000 000 000 000 000 11 kg  1.1 x 10–25 kg 
Number of molecules in 2.0 g of hydrogen  600 000 000 000 000 000 000 000  6.0 x 1023 
Metric Prefixes
To further simplify the expression of measured quantities, scientists use prefixes with metric units to represent powers of ten. The following Tables list metric prefixes with a range of 25 orders of magnitude and those frequently used in chemistry. Notice that each prefix has an abbreviation and an equivalent power of ten. You should know those prefixes in bold face italics and their associated powers of ten listed in Table 3. You may need to memorize them if you can't remember them.
Table 3. Commonly used prefixes and Equivalent Powers of Ten. Know the bold italics.
Prefix  Abbreviation  Power of Ten  Example 
pico  p  10–12  picogram, pg 
nano
 n  10–9  nanometer, nm 
micro  µ  10–6  microsecond, µs 
milli
 m  10–3  milliliter, mL 
centi
 c  10–2  centimeter, cm 
kilo
 k  103  kilometer, km 
mega
 M  106  megahertz, MHz 
giga
 G  109  gigabyte, GB 
Mathematical Operations in Scientific Notation
Addition and Subtraction
To add or subtract numbers that are expressed in scientific notation, use the “enter exponent” key on your calculator to express the power of ten. It is usually labeled “EE” or “EXP.” Do not use the 10 x or y x key for scientific notation exponents. The calculator usually will express the result in scientific notation automatically. For example:
3.25 x 10^{3} + 4.66 x 10^{4 } =
Calculator key sequence: 3 . 2 5 EE 3 + 4 . 6 6 EE 4 =
Result: 4.985 x 10^{4}
Multiplication
To multiply numbers that are expressed in scientific notation, use your calculator. For example:
(3.1 x 10^{2}) x (5.2 x 10^{4}) =
Calculator key sequence: 3 . 1 EE 2 ¥ 5 . 2 EE 4 =
Result: 1.612 x 10^{7}
Division
To divide numbers that are expressed in scientific notation, again, let your calculator do the work. For example:
(7.5 x 10^{2}) / (5.9 x 10^{–4}) =
Calculator key sequence: 7 . 5 EE 2 ÷ 5 . 9 EE 4 +/– =
Result: 1.271... x 10^{6}
Powers
To raise a number expressed in scientific notation to a power, use the yx key on your calculator. For example:
(2.5 x 10^{2})^{3} =
Calculator key sequence: 2 . 5 EE 2 yx 3 =
Result: 1.5625 x 10^{7}
Measured Quantities and Significant Figures
There is a degree of uncertainty in every measurement, and this uncertainty, by convention, is reflected in the last recorded digit of any measured quantity. If several people measure the same distance with a ruler, their measurements will probably differ in the last digit. However, these measurements will cluster around the true value. Some will be equal to the true value, some will be higher, and some will be lower. The average of a series of measurements is generally considered to be the most accurate value.
The number of significant figures in a measurement is the total of the number of digits known with certainty plus the one uncertain digit. The digits known with certainty are those that can be read precisely from the measuring instrument. The last digit recorded in any measured quantity—the uncertain digit—is estimated.
When a measurement is written in scientific notation, the first factor represents the significant digits in the measured quantity. The second factor, the power of ten, represents the magnitude of the measurement, or the number of decimal places, and therefore has nothing to do with the number of significant figures in the quantity. If the threesignificantfigure measured quantity 134 pounds is written in scientific notation as 1.34 x 10^{2} pounds, the number of significant figures cannot change, so this must also have three significant figures.
The following rules summarize the conventions used by chemists when working with measured quantities:
 The last digit expressed in a measured quantity is the uncertain, or estimated, digit.
 Zeros that serve as place holders in ordinary decimal notation are not significant. For example, the measured quantity 0.001 m has one significant figure. This becomes more apparent when we write the quantity in scientific notation: 1 x 10^{–3} m.
 When adding and subtracting numbers representing measured quantities, the number of decimal places in the sum or difference are limited by the number of decimal places in the measured quantity that has the least number of decimal places. For example: 0.152 g + 0.26 g = 0.41 g.
 When multiplying and/or dividing numbers representing measured quantities, the number of significant figures in the result is the number of significant figures in the factor with the least number of significant figures. For example: (1.5 x 10^{2}) x (1.350 x 10^{3}) = 2.0 x 10^{5}.