Basic Math Concepts
There are several math concepts that are not only essential to a career as an EH professional, but will also be important in order to you will need to master in order to successfully engage in EH717. The remainder of this page will cover these concepts.
Scientific Notation and the Metric System
In our everyday lives we normally express numbers in decimal notation. As EH professionals we often deal with very large numbers (e.g. numbers of molecules of TCE in a drinking water well) and very small numbers. For example, the number 6,320,000 can be written as 6.32 x 106. The following resources can help you learn or simply brush up on your skills of working with scientific notation:
In addition to scientific notation, it is essential that you be able to use the metric system for mass, length, volume and temperature. It is handy to remember the unit conversions detailed in the table below. Note that the symbol ≈ means approximately.
| Properties | Units of Conversion |
|---|---|
| Mass | 1 kg ≈ 2.2 lb |
| Length |
|
| Volume | 1 qt ≈ 0.95 L |
Conversion of temperature requires a formula and is not something that you need to memorize for this class.
The formula for converting degrees Fahrenheit (°F) to degrees Celsius (°C)
Commonly used prefixes for powers of 10 used with metric units are given below. You will see these very commonly in scientific papers.
| Power | Prefix | Abbreviation |
|---|---|---|
| 10-12 | pico | p |
| 10-9 | nano | n |
| 10-6 | micro | μ |
| 10-3 | milli | m |
| 10-2 | centi | c |
| 10-1 | deci | d |
| 103 | kilo | k |
| 106 | mega | M |
Examples: μg = micrograms, kg = kilograms
Unit conversion
It is essential that you are able to convert units. One way to do this is illustrated by the following example.
Suppose you want to convert 53 lb to kilograms:
The first term, 53 pounds, is what you want to convert. The second term is the conversion factor. Since 2.2 pounds equals 1 kg, we have not changed anything except the units. Most importantly, we have here written the conversion factor "upside down" so that the units that we want to change (lb) cancel, leaving the units that we want (kg). Once the equation is set up in this way, we can simply do the arithmetic on the numbers and cancel the units, producing the results on the right hand side. If the units do not cancel out the way you want, something has gone wrong! This general scheme can be expanded upon.
Suppose we want to convert 400 g/mL to units of kg/L:
As before the units cancel leaving the desired result. Here the conversion factors also cancel leaving the number unchanged. For more practice, you may review Math Skills Review - Dimensional Analysis.
Significant figures (digits)
Significant figures show the level of precision of a number. A measurement of 10 grams is different than a measurement of 10.040 grams. A measurement of 10.040 grams implies that we can accurately measure to the thousandths place and think that the measurement is different from 10.041 grams.
Significant figures include:
- All non-zero digits (1-9)
- Zeros within a number. Both 4308 and 40.05 contain four significant figures.
- Trailing zeros that aren't needed to hold the decimal point. For example, 4.00 has three significant figures.
Zeros that do nothing but set the decimal point are NOT significant. Thus, 450,000 has two significant figures and 0.000045 has two significant figures. For example, 450 expressed in scientific notation may have two (4.5 x 102) or three (4.50 x 102) significant figures. When doing the problems, remember to follow the rules of significant figures.
In some cases, it is not possible to determine the number of significant figures without information about how the number was obtained. For example, a measurement of 20 grams might have one significant figure or two; you would need to know more about the accuracy of the scale to determine if the zero is a place holder or an actual measured value.
Significant figures when performing calculations
The results of calculations performed with a computer or calculator may have many digits. The number should be rounded off to the appropriate number of significant figures.
- 12.5 + 14.47 + 98.3 = 125.3 (one decimal place)
- 1.090 x 0.002 = 0.002 (one significant figure)
Note: In the case of conversions, exact factors (such as 7 days per week or 1000 ml per l) are not used to determine the number of significant figures of the result.
If you are adding or subtracting, the result should be rounded to the same number of decimal places as the term with the least number of decimal places. If you are multiplying or dividing, the result should be rounded off to the same number of significant figures as the least precise factor. For example, if you are multiplying a number with 2 significant figures by a number with 4 significant figures, your result should have 2 sig figures.
The analytical results of a drinking water supply show the following levels of benzene: 29 ppb, 5.5 ppb, and 347 ppb. The arithmetic mean of these results is 127.16667. When using the appropriate significant figures, what should this result be rounded to?
When conducting a series of mathematical steps in a calculation, rounding off to the appropriate number of significant figures should only be done at the end of the calculation, not after each step. The final significant digit is rounded up if the first digit dropped is greater than 5 and rounded down if the first digit dropped is less than five. If the final digit dropped is equal to five, then the last digit is rounded up if the digit is odd.
For an on-line "quiz" of your ability to work with significant figures, see Widener University Sig Fig Tutorial.
Logarithms and Exponentiation
Environmental scientists often use logarithms to describe large and small numbers. For example, some important chemical properties such as pH are based on logarithms. For a refresher on logarithms—and their inverse function, exponentials, see the following website Math Skills Review - Logarithms.
Scientists typically use base 10 logarithms (often abbreviated log) or base e logarithms (often but not always abbreviated ln); the latter are also known as "natural logarithms." e is approximately 2.71828. It's important to know which logarithms are being used!
Logarithms are very easy to calculate with a calculator. For example, use your calculator to determine the results of log(125). What do you get?
The previous calculations used base 10. What happens when you use natural logarithms and plug ln(125) into your calculator?
If logarithms seem mysterious, consider the following tip using the example of log(125).
- Rewrite 125 in scientific notation: 1.25 x 102.
- Note that log(125) = 2.097
The results of Step 2 tell you how many powers of ten the number has as shown in the scientific notation in Step 1. Those numbers should always match.
It turns out that logarithms are often quite useful when we want to describe environmental data as the latter often have skewed (non-symmetrical) distributions, e.g., many values are low but some are quite high. You'll learn more about this later.
