Until now, most of the numbers we've used have been scalar (unitless). However, in the real world, most numbers have units associated with them. There are special rules which apply when doing math on numbers with units.
13.1.1 Addition and subtraction
The following rules apply:
• If the units are the same, add or subtract normally.
• If the units are different, then:
• If they can be converted to the same units, then addition and subtraction can occur.
• If they can't be converted to the same units, then addition and subtraction are not possible.
Examples
4 apples + 3 apples = 7 apples 4 apples - 3 apples = 1 apple
In the following case, we try adding apples and oranges, but encoutner a roadblock:
4 apples + 3 oranges = ?
These two items cannot be directly added together. However, depending on the context, it may be possible to convert them into "pieces of fruit" and add them. The result would look like this:
4 pieces of fruit + 3 pieces of fruit = 7 pieces of fruit
While the same may apply to subtraction, it is highly dependent on context (for example, the problem may be solvable if you treat it as receiving 4 pieces of fruit and giving 3 pieces of fruit in exchange.).
More commonly, you will encounter situations where you use standard unit conversion (e.g. 1 foot = 12 inches):
Unit math
1 foot + 3 inches = 12 inches + 3 inches = 15 inches
A complex example where unit conversion is required:
60 MPH - 30 fps
= 60 miles/hr - 30 ft/sec
= (60 miles × 5280 ft/mile)/(1 hr × 3600 sec/hr) - 30 ft/sec
= 316,800 ft / 3600 sec - 30 ft/sec
= 88 ft/sec - 30 ft/sec
= 58 ft/sec
This method is called unit cancellation, and is quite useful is science and engineering.
Here is an example where it isn't possible to convert to the same units:
4 apples + 3 degrees = ?
13.1.2 Multiplication and division
When multiplying or dividing numbers with units, the units are also multiplied or divided.
Examples
Here's are examples using the same base units:
2 feet × 3 feet = 6 square feet = 6 ft2 6 square feet / 2 feet = 3 feet
2 meters × 3 meters × 4 meters = 24 cubic meters = 24 m3 24 m3 / 2 m = 12 m2
Here are examples using different base units:
2 feet × 3 pounds = 6 foot pounds (a unit of torque) 100 miles / 2 hours = 50 miles/hr = 50 MPH
60 pounds / 4 square inches = 15 pounds/square inch = 15 PSI
200 miles / 10 MPH = 200 miles / (10 miles/hr) = 200 miles × (1 hr / 10 miles) = 20 hrs
It is also possible to multiply or divide a number with units by a scalar (a unitless number):
3 apples × 2 = 6 apples 3 apples / 2 = 1.5 apples
So, two groups of 3 apples each gives us 6 apples, while 3 apples divided into 2 equal groups gives us 1.5 apples per group.
Unit math
13.1.3 Powers, roots, and exponents
Just like multiplication or division, the operation also applies to the units.
Examples
(2 ft)3 = 23 ft3 = 8 ft3 = 8 cubic ft
(100 miles2)12 = 10 miles
So, in the first case we calculated that a cube 2 feet long on each side has a volume of 8 cubic feet, while in the second case we calculated that a square plot of 100 square miles of land would be 10 miles long on each side.
To distinguish between feet, square feet, and cubic feet, the term "linear feet" is sometimes used to clarify that we are talking about regular feet, not square or cubic feet.
13.1.4 Unit conversion
Sometimes it is necessary to do a unit conversion by itself. The unit cancellation method shown previously can be applied to convert from any unit to any other compatible units.
Examples
Convert 10 miles per day to inches per second:
(5280 ft/mile) × 10 miles/day = 52,800 ft/day (12 inches/ft) × 52800 ft/day = 633,600 inch/day 633,600 inch/day × (1 day/24 hr) = 26,400 inch/hr 26,400 inch/hr × (1 hr/60 min) = 440 inch/min 440 inch/min × (1 min/60 sec) = 7.33333 inch/sec
14 Introduction to significant digits
Significant digits are one crude way to denote a range of values. They apply to measured or estimated numbers where the quantity can vary from the provided value.
14.0.5 Examples
By default, significant digits are measured by
2000 means 2000 ± 500 = 1500 to 2500 200 means 200 ± 50 = 150 to 250 20 means 20 ± 5 = 15 to 25 2 means 2 ± 0.5 = 1.5 to 2.5 0.2 means 0.2 ± 0.05 = 0.15 to 0.25 0.02 means 0.02 ± 0.005 = 0.015 to 0.025
(Note: The first three examples may have more significant digits depending on context.) A terminal decimal point can be added to change the meaning of the first three cases:
2000. means 2000 ± 0.5 = 1999.5 to 2000.5 200. means 200 ± 0.5 = 199.5 to 200.5
20. means 20 ± 0.5 = 19.5 to 20.5
An additional zero after the decimal point also changes the meaning:
2000.0 means 2000 ± 0.05 = 1999.95 to 2000.05 200.0 means 200 ± 0.05 = 199.95 to 200.05
20.0 means 20 ± 0.05 = 19.95 to 20.05 2.0 means 2 ± 0.05 = 1.95 to 2.05
Here we've added another zero after the decimal point:
2000.00 means 2000 ± 0.005 = 1999.995 to 2000.005 200.00 means 200 ± 0.005 = 199.995 to 200.005
20.00 means 20 ± 0.005 = 19.995 to 20.005 2.00 means 2 ± 0.005 = 1.995 to 2.005 0.20 means 0.2 ± 0.005 = 0.195 to 0.205
Introduction to significant digits
14.0.6 Limitations
• Significant digits only allow for ranges where the amount that can be added to the base value and the amount that can be subtracted are the same.
• The value added and subtracted must also be 5 times or divided by a factor of 10.
• Exact values have no error or variation. Thus, $8472.35 will only mean $8472.35.
14.0.7 Use with scientific and engineering notation
To represent certain ranges, like 2000 ± 50, it's necessary to use either scientific or engineering notation:
0.20 × 104 Mistake: this is neither proper scientific notation nor proper engineering notation:
Proper scientific notation must have a nonzero digit to the left of the decimal point. Proper engineering notation must use multiples of 3 (e.g., 5 x 103, 603 x 106, 42.3 x 109, ...)
2.0 × 103
14.0.8 Beyond significant digits
More complex ranges can be represented using the ± sign:
2000 ± 38.3
If the plus and minus values are different, the actual range can be listed, or separate plus and minus values can be listed. The range shown to the right of the plus and minus symbol is called an error. In this example, there are two significant digits, and one doubtful digit.
Note that the ± sign shouldn't be confused with the ± operator, which can indicate either an exact addition or subtraction. This will be covered in Algebra1.
14.0.9 Accumulating error
As you perform mathematical operations with numbers, the error also increases.
• When you add or subtract numbers, the ranges are added together.
• When multiplying numbers, the error is converted into a percent of the main value (e.g. 20 ± 1 becomes 20 ± 5%). The percentages the two measurements added together, and the percentage is converted back to reflect the error of the original number.