The following is a general equation to express the relationship between two differing units of measure:
mA = nB
Where the symbol "A" could represent the concept ‘number of feet’ and in that it would bear the units "feet." The symbol "B" could represent the concept ‘number of miles’ and similarly would bear the unit "miles." The symbol "n" is the conversion factor and in the conversion between feet and miles it would be 5,280 and it would bear the units "feet/mile." The symbol "m" is understood to have the value of unity or 1 and it is dimensionless. It is by convention omitted from the actual expression of the equation.
A = nB
Which should be read in the case of the example as one mile equals 5,280 feet. The corollary of this relationship expresses "B" in terms of "A" or:
B = (m/n)A or as it is usually written:
B = A/n where the ‘m’ of value unity is eliminated.
Common conversion factors for geologists:
In each of the above the null value of one is equal to the null value of the other or:
In the case of converting between ‘a temperature in degrees Fahrenheit’ to ‘a temperature in degrees Celsius’ the relationship is more complex since a temperature of zero degrees Celsius (freezing point of water) equals a temperature of 32 degrees Fahrenheit. To make the conversion use:
oC = 0.555556(oF-32)
The origin of the 0.555556 conversion factor is illustrative of all conversion factors. Remember that water freezes at 32 oF and boils a 212 oF or it freezes at 0 oC and boils at 100 oC. Therefore for the same temperature range (between the freezing point of water and its boiling point there are 180 degree division of the Fahrenheit scale (212-32=180degrees) and 100 degree division of the Celsius scale (100-0=100degrees). The conversion factor is therefore: (100/180) = 0.555556
Note: there are a whole lot of different units of measure out there and many are real obscure. Something very useful for a student to do is learn where one can obtain conversion factors and better yet build your own personal list of conversions. Also keep in mind that some units will have differing quantitative values in different places, times, and situations. For example the nautical mile and the length we call a mile here on land in North Carolina or a US gallon and an Imperial gallon, not exactly the same thing.
The term percent, which is symbolized by %, is a contraction for the expression "parts per 100 parts". Lets say we have a class of 100 students and of those 5 will receive a grade of A for the course; or 5 students per 100 students will get an A. What percent (%) of the class gets an A?
5/100 = 0.05 = 5%
If there were 200 students in the class and 8 received A, what percent got an A?
8/200 = (8/2)/(200/2) = 4/100 = 0.04 = 4%
The division of the numerator, 8, and the denominator, 200, by 2 was an unnecessary step since dividing 8 by 200 give the same result, namely 0.04 which you read as 4 one hundredths. This' 0.04' is just a symbol that expresses that relationship. Below is a table or similar symbols which we run into all the time.
| 1 | one | 1/1 | one per one |
| 0.1 | one tenth | 1/10 | one per ten |
| 0.01 | one hundredth | 1/100 | one per one hundred or or one per cent or 1% * |
| 0.001 | one thousandth | 1/1000 | one per one thousand or 1 permil or 1 o/oo ** |
| 0.0001 | one ten thousandth | 1/10,000 | one per ten thousand or 0.1 permil or 0.1 o/oo |
| 0.00001 | one one hundred thousandth | 1/100,000 | one per one hundred thousand or 0.01 permil or 0.01 o/oo |
| 0.000001 | one millionth | 1/1,000,000 | one per million or 1ppm (ppm = parts per million parts) |
| 0.000000001 | one billionth | 1/1,000,000,000 | one per billion or 1ppb |
| 0.000000000001 | one trillionth | 1/1,000,000,000,000 | one per trillion or 1ppt |
* How many pennies in a dollar, 100 right??? What is another name for a penny? cent yes? Do you see the origin of the term percent?
** On New Years Eve 1999 we passed from one millennium to a new millennium. A millennium is 1000 years long. Do you see the connection to the roots of the term permil?
Back in the days of the slide rule it was very convenient to express a number in the form of something times 10 to the such-and-such power or:
34.5 became 3.45x101
The engineers expressed it in a similar manor but with slightly different notation:
34.5 became 3.45E1
This was done because in making multiplication and divisions on the slide rule you first dealt with the 3.45 and then you "slipped the stick" for the x101. A computer does the same thing but it is in the background and you never see it. Such notation is no longer absolutely necessary except that for very large and very small numbers where there is certainly a space advantage associated with these conventions. For example, which would you prefer to write:
0.000000000000000345 or 3.45x10-16
Therefore, it is still the custom in scientific writing to use symbols like 1.3x108 and in engineering reports to use symbols like 1.3E8. In both cases there is only one digit to the left of the decimal point and the power of 10 is always an integer (1, 2, 3, -1, -2, -3, etc.).
If your are trying to email data it is much easier to use the engineering form and much safer as some email programs may not be able to handle exponents.
Note that when we are dealing with the percent of something or the permil, or ppm, or ppb, etc we generally omit the scientific or engineering notation. For example, I have a sample that weighs 5.10E2 grams and 7.5E1 grams of that sample is iron. What is the concentration of iron in the sample?
(7.5E1 grams/5.10E2 grams) = 1.5E-1 or 0.15 which is expressed traditionally as 15 %.
Think before you leap because 1.5E-1 % does not equal 15%; it equals 0.15% !!!!!! The symbol 1.5E-1 is not equal to the symbol 1.5E-1% You can really screw up here if you do not understand this!!!
Furthermore, if something is reported as say 3,545.1 ppb it is better to say that it is 3.5451 ppm. As a rule, in values of this type no more than three digits should be found to the left of the decimal point.
When ever one number is operated on by another (multiplication and division) the resultant number is express with no more precision than that of the least precise number involved in the operation. In scientific notation the precision of a number is expressed by the number of digits to the right of the decimal point. For example 3.45x101 has two digits to the right of the decimal or is said to have 2 significant digits. The number 3.45 x107 also has 2 significant digits but number 1.000000 x101 had six significant digits. The number 0.00001 has zero significant digits since in scientific notation it becomes 1.x10-5 but the number 0.000010 becomes 1.0 x10-5 and has one significant digit. If one multiplies 4.001 x101 by 2.0 x101 the answer is 8.0 x102 not 8.002 x102.
In addition and subtraction the same also holds but we must look at it a little bit differently. If one were to add 1.0 x10-5 to 1.0 x100 the answer is 1.0 x100 and not 1.00001 x100 but if you added 1.0 x10-5 to 1.00000 x100 the answer would be 1.00001 x100 because both of the components being added together are expressing the same degree of measurement precision.
If in scientific notation a number is expressed without a decimal point it is considered to be an integer and it by definition has infinite precision. By custom it is also expressed generally without the x10n notation. This may be a bit confusing but then 3 times 1000, 3. x10o times 1. x103 , and 3.000 x100 times 1.000 x103 are not expressing the same thing.
When dealing with values expressed in percent (%) sometimes you need to look at the context of the report in order to determine the number of significant digits. Obviously 12% means 0.12 but if someone reports that ten percent of the student population is from Georgia did they mean 0.10 or 0.1??? a good guess would be the latter.
Examples:
| Street Notation | Scientific Notation | Engineering Notation |
| 100 | 1.x102 | 1.E2 |
| 100.1 | 1.001x102 | 1.001E2 |
| 0.001 | 1.x10-3 | 1.E-3 |
| 0.001001 | 1.001x10-3 | 1.001E-3 |
| integer 3 | 3 | 3 |
| integer 300 | 300 | 300 |
| 5.12% | 5.12% | 5.12% |
| 12.1% | 12.1% | 12.1% |
| 1.1 ppm | 1.1 ppm | 1.1 ppm |
| 115 ppm | 115 ppm | 115 ppm |
| 2755 ppb | 2.755 ppm | 2.755 ppm |
Solve the following problems and email your answers to me.
1) By the odometer on my bicycle it is 4.1 miles from the Coast Guard Station as the south end of Wrightsville Beach to the northern tip of Shell Island, how far is that in feet?
2) The porosity of a sandstone sample from an oil field reservoir was measured to be 11%. The field covers 640 acres and the reservoir has an uniform thickness of 28 feet. If all the pore space is filled with oil how many US stock tank barrels (standard unit of volume measure in the petroleum business of the U.S.A.) of oil are in the ground under the field?
3) An ore sample from a North Carolina gold mine was assayed at 5.1 ppm Au. The ore rock has a density of 2.68 grams/cm3. The vein has an average width of 1.1 feet. It has a surveyed length of 6783 feet. It has an explored depth of 125 feet. What is the amount of gold within the identified portion of the vein in Troy ounces?
4) The sand down at Wrightsville Beach is mostly the mineral quartz (density 2.65000 grams/cm3). Loose sand has a porosity of 36%. How many tons (Avoirdupois) of sand are contained in 1.000000 cubic yard?
5) You are to mix up 1.00 gallons of a 5.0% (by volume) solution of HCl. How much distilled water do you use and how much concentrated HCl do you use, and since all you have to work with in lab is a metric graduated cylinder you should express your answer in liters.