This web page provides a review of the mathematical concepts you will apply in this course.
Some programs they have are Academic Development (AC DV) classes, Basic Skills Computer Lab, High Tech Center, Math Lab, Study Skills classes, Program for Learning Disabled Students, and Peer Tutoring. The Basic Skills Computer Lab has a lot of computers with extremely user-friendly software that can help you improve your arithmetic, pre-algebra, and elementary algebra skills. You go at your own pace and it is free! The Math Lab is a regular class and is not free. If you need to use the Peer Tutoring, you may not find an ``astronomy'' tutor. However, most astronomy students have difficulties with the math, so ask for a MATH tutor. They may have a PHYSICS tutor that could help.
There are also the AC DV 70a-f mini-courses for those at the Reading Level 1 skill level that will help you make your study time a more efficient and productive process. Each mini-course is 5 sessions long. Course titles: Time Management, Test Taking, Textbook Reading, Note Taking, Memory, Vocabulary. See the schedule in the class calendar.
The staff at the Learning Center is about the best in the country and the programs are excellent in quality and quantity. This is what students say, not some know-it-all instructor!
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 |
)m =
(a1/n)m
= a1/n × m = am/n.
| 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: ƒ C = (5/9)(ƒ F - 32).
To convert fahrenheit to kelvin use: K = (5/9)(ƒ F - 32) + 273.
To convert celsius to fahrenheit use: ƒ F = (9/5)ƒ C + 32.
To convert kelvin to fahrenheit use: ƒ 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
(206) 543-1979
University of Washington
Astronomy
Box 351580
Seattle, WA 98195-1580