Computation:
Scientists use computation extensively because results stated mathematically are really more valuable. In the previous chapter, "Scientific Method," we discussed that you should identify measurable quantities and their units. The famous scientist Lord Kelvin once said something to the effect that "If you cannot assign a number to a thing, you can hardly claim to truly understand it." This may seem to exaggerate the situation, but it is a basic principle of modern science, especially in the so called "hard" sciences astronomy, physics, and chemistry.
Mathematics is a powerful language for describing scientific, economic, and other numeric data. Using math in doing science will not only make the science useful, but more interesting. Use your imagination in applying your mathematics skills to your science problems. While a calculator is an essential tool for "number crunching," a computer spreadsheet program can make your data come to life. If you have access to a computer, learn to use a spreadsheet program like Excel, Quattro Pro, or Lotus 1-2-3. Play with your data, graphing it, computing averages, and standard deviations.
The rest of this section discusses the extremely important technique variously called "unit factor analysis" or "dimensional analysis." This technique is based on the fact that all numeric quantities have units or dimensions. Consider the number 44. In astronomy it might refer to an angle, so the unit would be degrees from a reference direction. In physics, 44 might be 44 meters per second, a speed, or 44 Amperes, an electrical current. In business, 44 might refer to 44 widgets or 44 dollars or 44 percent. The point is that numbers are generally meaningful ONLY if there is a unit, a dimensional quantity, associated with the number.
Dimensioned numbers follow a few simple rules.
Rule 1: The unit of a dimensioned number may be converted to another unit of the same kind. For example, hours may be converted to seconds, minutes, or years; Acres may be converted to square feet, square miles, or hectares; and pounds (force) may be converted to newtons, ounces, dynes, or tons. The method to convert a quantity’s unit is to multiply by unity, dimensionless one. Unity may take on a huge variety of forms. The form of this unity is easily constructed from any conversion. For example, 1 mile = 5280 feet = 1609 meters, so unity equals
Rule 2: If two dimensioned numbers have identical units, they may be added or subtracted. For example, $21 plus 5 miles does not make sense, but $21 + $5 = $26 and 33 miles - 5 miles = 28 miles do make sense.
Note that Rule 1 and Rule 2 may be used together. Suppose you wanted to add 528 feet to 2 miles. If you wanted everything in feet you would first convert 2 miles to feet,
.
Next, add in the extra 528 feet to get the total distance
.
If you wanted everything in miles, first convert 528 ft to miles,
.
Next, add the 2 miles to the converted 528 ft to get
.
Rule 3: If two quantities have different units, they can only be multiplied or divided by one another. The new units are computed by the standard rules of algebra on the numerators and denominators. For example, 3 feet per second (recall that "per" generally means division so 3 ft/sec) and 12 square feet are written
Multiplying these together gives
in the numerator and seconds in the denominator, so the product is
which is the volume per time, a volume flow rate. This is the amount of water flowing down a pipe or channel at 3 feet per second if the pipe or channel cross sectional area is 12 square feet. To convert to gallons per second, use Rule 1 with the conversion 7.49 gallons = 1 ft3
The second computation that one could legally perform is to divide 3 feet per second by 12 square feet to get
or to divide 12 square feet by 3 feet per second to get
neither of which seem to make any sense. Often the units themselves will indicate how to interpret the quantity.
As a fun illustration, consider figuring something like the amount of water lost from a dripping faucet. Suppose the faucet drips so that it fills a tablespoon every minute. The rate is then 1 Tablespoon per minute. Recall (or look up in a cookbook or other handbook) that 4 Tablespoons = ¼ cup, 2 cups = 1 pint, 2 pints = 1 quart, 4 quarts = 1 gallon, 60 minutes = 1 hour, 24 hours = 1 day, and 365 days = 1 year. The water waste rate is
This number should be reported as 3000 gallons per year because the initial number, 1 Tablespoon per minute, is only good to one "significant digit." If you are quite sure that the original measurement is 1.0 Tablespoon per minute and not 1.1 or 0.9 Tablespoon per minute then you have two significant digits and the calculation should be reported as 2700 gallons per year.