This web page provides a review of the mathematical concepts you will apply in this course.
Fractions are sometimes expressed as a percent. To express a fraction as a percentage, find the decimal form and multiply by 100. The percent symbol ``%'' simply means ``divide by 100.'' For example 1/2 = 0.5 = 0.5× 100% = 50%; 3/5 = 0.6 = 0.6× 100% = 60%. Some examples on converting percentages to fractions or decimals: 5.8% = 5.8/100 = 0.058; 0.02% = 0.02/100 = 0.0002; the Sun is 90% Hydrogen means that 90 out of every 100 atoms in the Sun is Hydrogen.
The use of the phrase ``factor of'' is very similar to the use of ``times''. For example, 1 pint is a factor of 4 smaller than 1 gallon, or 1 gallon is a factor of 4 bigger than 1 pint.
a | = | a1 |
a × a | = | a2 (not 2 × a!) |
a× a × a | = | a3 (not 3 × a!) |
a× a× a× a | = | a4 |
a× a× a× a× a | = | a5 |
Some special rules apply when you divide or multiply numbers raised to some power. When you have an multiplied by am, the result is a raised to a power that is the sum of the exponents:
When you have an raised to a power m, you multiply the exponents:
Scientific calculators have a ``yx'' key or a ``xy'' that takes care of raising numbers to some exponent. Some fancy calculators have a ``^'' key that does the same thing. Some calculators have ``x2'' and ``x3'' keys to take care of those frequent squaring or cubing of numbers. Check your calculator's manual or your instructor. The Basic Skills Computer Lab has some excellent software that can improve your skills with exponents. Try it out!
A square root of a number less than 1, gives a number larger than the number itself: Sqrt[0.01] = 0.1 because 0.1 × 0.1 = 0.01 and Sqrt[.36] = 0.6 because 0.6 × 0.6 = 0.36.
The cube root of a quantity is a number that when multiplied by itself two times, the product is the original quantity:
Scientific
calculators have ``'' and sometimes ``
'' keys to take
care of the common square roots or cube roots. An expression
means the
nth root of a. How can you use your calculator for something like
that? You use the fact that the nth root of a is a raised
to a fractional exponent of 1/n. So we have:
![]() |
= | a1/n |
Sqrt[a] = ![]() |
= | a1/2 |
Cube-root[a] = ![]() |
= | a1/3 |
100 = 1. | = | 1. with the decimal point moved 0 places |
101 = 10. | = | 1. with the decimal point moved 1 place to the right |
102 = 100. | = | 1. with the decimal point moved 2 places to the right |
106 = 1000000. | = | 1. with the decimal point moved 6 places to the right |
and | ||
10-1 = 0.1 | = | 1. with the decimal point moved 1 place to the left |
10-2 = 0.01 | = | 1. with the decimal point moved 2 places to the left |
10-6 = 0.000001 | = | 1. with the decimal point moved 6 places to the left. |
The exponent of 10 tells you how many places to move the decimal point to the right for positive exponents or left for negative exponents. These rules come in especially handy for writing very large or very small numbers.
Since we'll be working with very large and very small numbers, we'll be using scientific notation to cut down on all of the zeroes we need to write. Proper scientific notation specifies a value as a number between 1 and 10 (called the mantissa below) multiplied by some power of ten, as in mantissa×10exponent. The power of ten tell you which way to move the decimal point and by how many places. As a quick review:
10 = 1 × 101, 253 = 2.53 × 100 = 2.53 × 102 and
15,000,000,000 = 1.5 × 1010 which you'll see sometimes written as 15
× 109 even though this is not proper scientific notation. For small
numbers we have: = 1 × 10-1,
× 10-2 or about 0.395 × 10-2 = 3.95 ×
10-3.
When you divide two values given in scientific notation, divide the mantissa
numbers and subtract the exponents in the power of ten. Then adjust the
mantissa and exponent so that the mantissa is between 1 and 10 with the
appropriate exponent in the power of ten. For example: × 1010-23 = 0.5 × 10-13 = 5 × 10-14.
Notice what happened to the decimal point and exponent in the examples. You subtract one from the exponent for every space you move the decimal to the right. You add one to the exponent for every space you move the decimal to the left.
Most scientific calculators work with scientific notation. Your calculator
will have either an ``EE'' key or an ``EXP'' key. That is for
entering scientific notation. To enter 253 (2.53 × 102), you would
punch 2 .
5
3
EE or
EXP
2. To enter 3.95 × 10-3, you would punch 3
.
9
5
EE or EXP
3
[
key]. Note that if the calculator displays ``3.53 -14'' (a space
between the 3.53 and -14), it means 3.53 × 10-14 NOT
3.53-14! The value of 3.53-14 = 0.00000002144 =
2.144×10-8 which is vastly different than the number
3.53×10-14. Also if you have the number 4 × 103 and you
enter 4
×
1
0
EE or
EXP
3, the calculator will interpret that as 4 × 10 ×
103 = 4 × 104 or ten times the number you really want!
One other word of warning: the EE or EXP key is used only for scientific
notation and NOT for raising some number to a power. To raise a number to some
exponent use the ``yx'' or ``xy'' key
depending on the calculator. For example, to raise 3 to the 4th power as in
34 enter 3 yx or
xy
4. If you instead entered it using the
EE or EXP key as in 3
EE or EXP
4, the calculator would
interpret that as 3×104 which is much different than 34 =
81.
prefix | meaning | example |
---|---|---|
pico | 10-12 = 1/1012 | 1 picosecond = 10-12 second |
nano | 10-9 = 1/109 | 1 nanometer = 10-9 meter |
micro | 10-6 = 1/106 | 1 microgram = 10-6 gram |
milli | 10-3 = 1/1000 | 1 millikelvin = 10-3 kelvin |
centi | 10-2 = 1/100 | 1 centimeter = 1/100th meter |
kilo | 1000 | 1 kilogram = 1000 grams |
mega | 106 | 1 megasecond = 106 seconds |
giga | 109 | 1 gigakelvin = 109 kelvins |
tera | 1012 | 1 terameter = 1012 meters |
1 meter (m) | = 39.37 inches | = 1.094 yards (about one big step) |
1 kilometer (km) | = 1000 meters | 0.62137 mile |
1 centimeter (cm) | = 1/100th meter | 0.3937 inch (1/2.54 inch, about width of pinky)) |
1 gram (g) | = 0.0353 ounce | produces 0.0022046 pounds of weight on the Earth |
1 kilogram (kg) | = 1000 grams | produces 2.205 pounds of weight on the Earth |
1 metric ton | = 1000 kilograms | produces 2,205 pounds of weight on the Earth |
To convert fahrenheit to celsius use: ° C = (5/9)(° F - 32).
To convert fahrenheit to kelvin use: K = (5/9)(° F - 32) + 273.
To convert celsius to fahrenheit use: ° F = (9/5)° C + 32.
To convert kelvin to fahrenheit use: ° F = (9/5)(K - 273) + 32.
See the light chapter for further discussion of the kelvin, celsius, and fahrenheit scales.
last updated 11 January 1998