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. ,x‘ APPENDIX A
MATH AND SCIENCE BASICS INTRODUCTION Students enter this course from a wide variety of ﬁelds of study. Some are
mathematically sophisticated, while others use little math. These notes are intended to
assist the latter in handling the measurements and simple calculations used in this course.
Some instructors may wish to cover at least part of the material in class, while others may
assign it for selfstudy. The notes include the uso of an inexpensive pocket calculator to
perform certain operations that may be difﬁcult to do by hand, but are childishly simple
on the calculator. These sections are marked with a calculator icon. If you do not yet use
such a calculator, you are depriving yourself of a wonderful aid in everyday life! If you have done little or no math since 7th grade algebra, please do not be frightened
by this material. None of the calculations used in this course can by any stretch of the
imagination be called “advanced!” If at any point you hate difﬁculties, remember that
you always have at hand an instructor who will be happy to give you oneonone
assistance. We hope that an important byproduct of this course will be that in the future
you will be empowered to read newspaper and magazine articles on science and
technology with increased understanding (and perhaps even to detect the alltoofrequent
errors in such articles!). SCIENTIFIC NOTATION In astronomy, we often have to deal with, huge numbers. For instance, the age of
the Universe, the distance to a galaxy or the mass of a star, represent times, lengths and
masses, respectively, well bewnd our everyday experience. Astronomers have two ways
to cope with this inconvenience. The ﬁrst is to use an intelligent way” to represent
numbers, calied “scientiﬁc notation”. The second is to use different units of time, length,
era, that are more appropriate for astronomy—related problems (see UNITS Section). The scientiﬁc notation is a way to express very large (and very small) numbers, as
powers of ten: ' 10" = 1.0 _ 101 = 10.0 10" = 0.1 102 = 100.0 10'2 = 0.01 103 = 1,000.0 10'3 = 0.001
104 = 10,0000 _ 104 = 0.0001
105 = 100,0000 105 = 0.00001
10‘5 = 1,000,0000 10*5 = 0.000001
etc. etc. A quick way of changing a number from a decimal representation to a Scientiﬁc
notation representation is simply to count the number of position to the left or right of
the decimal point. The number of positions corresponds to the power of ten. In most
cases, scientiﬁc notation expresses numbers as the product of a number and a power of A.1 ten. For example, the mean distance from the Earth to the Sun is 149,600,0000 km. In _
scientiﬁc notation, we would express this as 1.496 x 108 km. The decimal point was
moved 8 positions to the left. In order to enter Scientiﬁc notation in your calculators, you should use the key
usually labeled EXP or E. For instance, if your calculator has the key EXP, to
type in 1.496 x 103, you must press 1.496 EXP s. ? Problems 1. Express the following numbers in scientiﬁc notation:
i. 394,000,000
ii. 0.0007985 52
iv. 0.65 MEASURING ANGLES About ﬁve thousand years ago, ancient Babylonians divided the circle into 360
degrees (or 360°), each degree into 60 are minutes (or 60’) and each arc minute into 60 are
seconds (or 60"). Although this system may have been consistent with philosophical
beliefs of the time, it sounds rather arbitrary today.  We will use the Babylonian system throughout this course: 1 degree = 60 arcminutes (1° = 60’)
1 arcminute = 603rcseconds (1’ =60”) ? Problems 2. This question is intended to get you used to manipulating angles:
ii. How many arcseconds are there in a degree? 7 How many degrees are there in an arcsecond? UNITS In the section about scientiﬁc notation, we saw how to write numbers in scientiﬁc
notation in order to conveniently express very big or very small numbers. A second way
out is to use appropriate units, so that we do not have to deal with cumbersome numbers
at all. For example, using miles to measure the distance to the Sun is much like using
inches to measure the distance from New York to Los Angeles. So, when we deal with distances of objects within our solar system, we use a
“yardstick” called the astrOnornical unit (or just AU). One astronomical unit is equal to
the average distance of the Sun from the Earth; ' A2 1 AU: 1.496x 108km=9.3>< 10" miles However, distances between the stars of our galaxy are huge compared to the
distance of the Earth'from the Sun, so the astronomical unit is not a convenient unit any
more. Therefore, we have to further stretch our Wardstick” and use as a unit of length the
distance a light beam travels in a year, which we call light year (or 1y). Thus, although “year” is a unit of time, a “light year” is a unit of length!
11y:6.324x10‘AU=9.46x10”km There is yet another unit to measure interstellar distances within galaxies. It is
called parsec (pc): 1 pc = 3.26 ly
One kiloparsec (kpc) is equal to one thousand parsecs:
1 kpc =‘ 103 pc
One megaparsec (Mpc) is equal to one million parsecs:
1 Mpc = 10‘ pc Although the aweinspiring size of the Universe makes necessary the use of such
long yardsticks, the measurement of distances is not the only case where we have to
introduce new units. For instance, masses of stars and galaxies "are so huge that instead of using kilograms (kg), we use our Sun’s mass itself” asa unit of mass. It is called a solar
mass (Me): 1 M9: 1.992 x103" kg Other common and useful units are: 1 kilogram (or kg) = 1000 grams (or g) 1 meter (or m) = 100 centimeters (or em) = 1000 millimeters (or m)
I Angster (or A) = 10‘“ meters (or m) 1 nanometer (or run) = 10" m = 10 A 1 inch (or in) = 2.54 centimeters (or cm) 1 foot (or ft) = 0.305 In 1 mile (or mi) = 1.609 kilometers (or km) A3 ? Problems 3. These questions are designed to give you a feeling for metric and astronomical
units as compared to English measures. Express: 1 cm in inches
iv. 1 kilometer in miles
v. 8.9 x 109 kilometers in AU SIGNIFICANT FIGURES The theory of signiﬁcant digits is concerned with the useful and n‘ustwonhy digits
in a number. Suppose you measure a child's height to the nearest centimeter as 63 cm.
This means that you are sure of the 6 and have determined that the 3 is better than 2 or 4;
therefore, both the 6 and the 3 are signiﬁcant digits. A signiﬁcant digit represents a
number whose value is supported by a reliable measurement. Exact numbers may be defined or counted and knoWn to be exactly correct For
example, you can count the number of pages in this appendix”. If there are 10 pages, there
is no doubt that this is exact and not 9.7 or 10.] pages. Deﬁned numbers are exact as well.
A kilometer is deﬁned to: be exactly 1000 meters. Exact numbers have an inﬁnite
number of signiﬁcant ﬁgures. ‘ Inexact numbers are what is yielded by most measurements, whose accuracy is
limited by the measuring device used. For example, the smallest unit on most metric
rulers is miliimeters. The ﬁrst estimated digit in length obtained from such a ruler would
be 01 millimeters. One can still measure the diameter of a penny to be 18.9 millimeters
using this” ruler. When measuring the penny you see that its size lies somewhere between
18 and 19 millimeters and is most likely 18.9 millimeters. The last is considered
signiﬁcant because it is the best estimate of the person who measured the penny. If only the results of a measurement are reported, the following conventions are
used to tell which digits are signiﬁcant: 1. All non—zero numbers are signiﬁcant. 2. Zeros used only to position the decimal point are not significant. Three statements cover all cases involving zeros:
a . Leading zeros (those to the left of the ﬁrst nonzero digit) are never
signiﬁcant.
b . Imbeddedzeros (those betvveen nonzero digits) are always signiﬁcant.
c. Trailing zeros (those to the right of nonzero digits) are signiﬁcant only if the decimal point is shown.
For example:
(a) 0.00334 three sig ﬁgs
(b) 1.0058 ﬁve sig ﬁgs
(c) 34.00 four sig ﬁgs
(d) 0.678 three sig ﬁgs
(e) 0.00004 one sig ﬁg A.4 r_,‘m_\ . (f) 1.38 x 105 three Sig figs
(g) 1.3800x105 ﬁve sig ﬁgs The rules used in this lab for determining the number of signiﬁcant digits in a
calculated ﬁnal answer are given below: (1.). When adding or subtracting numbers, the result should not be carried
beyond the ﬁrst column having a doubtful ﬁgure.
For example: , 8.16 + 74 = 82, not 82.16. (2). When multiplying or dividing, the result should have the same number of
digits as the least signiﬁcant number used in the calculation. For example: 83 x 1045 = 8700, not 8673.5. (3). To show the results of calculations to the proper number of significant
. ﬁgures, rounding must be used. In all labs be sure that all the measured and calculated numbers that you record have the
intended number of signiﬁcant digits. ? Problems 4. Find the answers to the following with the correct number of sig figs:
(iv) 4.256 + 0.008 (v) so :— 5.876
(vi) (4.6789 x 1024)  (9.267 x 1073) Be sure to use the correct number of sig ﬁgs for all the following problems! UNIT TRANSFORMATIONS With so many units around, it is not surprising that we often have to make unit
transformations. For instance, we may want to express a result in certain units, but we are
given data in different units. We will have to deal with unit transformations many times
during this courSe, so it is imperative that you feel comfortable with them. There is a very simple and general way to perform unit transformations, which
will be illustrated with the following two examples. Example 1: You measure the length of your desk to be 48.0 inches long. What is the
length of the desk in cm? Use the fact that l in = 2.54 cm. If we are given: 1 in = 2.54 cm A.5 We can divide both sides of the equation by 1 inch to get: 254
I = . (A)
1 m 7 Notice that equation (A) equals 1. Multiplying any side of an equaIiOn by i is allowed in
mathematics, so we have: 43.0311 = (48.0 in) x (1) Which is the same as: 48.0in = (48049) x (23m) = 122cm (B) 1in Now notice that by doing this, the unit of inches has canceled out, leaving just cm
as the ﬁnal unit for (B). When multiplying an equation by anothet equation that is equal
to onethe whalepaint istoputinthedenomitwtortheunitwe wanttogetridaﬁ so
that it eventually cancels out. If we knew that the length of the desk is 122 cm and
wanted to ﬁnd it in inches: 122cm=(12263§)x[ )=48.0in 254655 Example 2: A car is traveling at a speed of 60.0 miles per hour. How many meters per
second is this? 1 mile: 1.6km, 1 hr: 3,600 sec. We now have to perform several times what we did in the previous example: . (6001%)x 1.6km x 1,000m
600 _/.hr _ 603m: _ 1mi 1km __ 96,000111 __ 267 I
‘m‘ "' 1hr _ (1_ 3,6005ec ‘3,6005ec" ‘mm
X —.—
he 1%?
? Problems 5. The purpose of the question is to see if you can keep track of units and compute
answers to simple formulae. Calculate the mass of the Earth by using the formula; M: 21!: 9 ER 3 A6 , ’"Ww where M is the mass of the Earth (what we are trying to ﬁnd); p = 5.5 glam3 is the average density of the Earth; and Si = 6.4 x 108 cm is the radius of the
Earth. Express your results in scientiﬁc notation (i) in grams, (ii) in kilograms. 6. Suppose your car was modiﬁed so you could travel through space. At a velocity of
65 mi/hr, and driving nonstop, how long would it take for you to get to:
(i) Pluto (usually the outermost planet in our solar system), at a distance of
5.85 x 109 km? (ii) Proxima Centauri (the nearest star to the Sun), at a distance of 1.33 pc? ERRORS When we perform an experiment, we usually want to take measurements of
physical quantities, such as lengths, times, weights, etc. We may then substitute these
numbers into formulae and seek relations among them. In this process, there are two
sources of error involved: Human errors These include:
Reading errors, eg. misreading a ruler when measuring
Calculation errors, eg. punching the wrong number in the calculator Bias, Le. errors induced into data collection by the experimenter when helshe
knows the outcome of the experiment ' These errors can be avoided by being more careful, repeating the experiment,
asking a friend to verify our numerical results, and so on. Errors due to limitations imposed by nature For example: Measuring error, 3.3. when we measure the length of a line with a ruler, our
reading cannot by any better than the ruler’s smallest subdivision. We can of course use a better ruler or a different measurement tool, but this will
only decrease the error; it will noteliminate it altogether. Unlike human errors, these
errors are unavoidable. ' In the course of this lab, you will have to take many measurements. In most labs .
you will only he asked to explain what are some possible sources of error in your
results. It should be understood that you are asked to mention errors that occur due to the
second reason, i.e., due to limitations imposed by the laws of nature, and not due to
human error. Occasionally, you will be asked to ﬁnd a numerical answer to describe the errors of the experiment. The easiest way to do this is mathematically to compare your
observed result with the known result value in the following way: ' A.7 Resultm — Resth ‘7 ERROR =
a Re sultm x 100.0 This is called percentage error. ? Problems '7. Through an experiment you conducted you found that the Moon’s diameter is
3.550 x 103 Inn. The accepted value for the Moon’s diameter is 3.476 x 103 km
What is the percentage error of your result? SCALE FACTORS We will need to use scale factors when examining photographs and pictures These
pictures are representative of real life in every way except one. The distances between
objects in the pictures are obviously not the actual distances between actual objects. They
are sealed so as to ﬁt on the photograph. The best way to eXpla‘in the concept is with an
example. 7 Suppose you have an aerial photograph of the University of Florida and you want to
determine the size of the campus. On closer" inspection of the photograph, you ﬁnd The
Swamp is clearly visible. You know that the length of a regulation football ﬁeld is 120
yards (including the endzones). Knowing this, you can develop a scale for the photograph. Suppose you measure the length of thetﬁeid in the photograph to be S
centimeters. Then we say that: 120 ds
scale = mu??— 2 24yardslcm
5cm Now you can determine the true length of campus along University Avenue. Suppose you measure the length of University Avenue on the photograph, and you ﬁnd it to be 40
centimeters. Now multiply by the previously determined scale: real length = (scale) x (ruler measurement) = [24 x (40cm) 2 960 yards In other words, to ﬁnd the scale of a picture, we ﬁnd a feature whom length we
somehow" know. Then we measure its length on the picture with a ruler. The ratio of these two lengths is the scale, and we can use it to determine the sizes of other features on the
picture. All ? Problems 8. You measure the distance from Gainesville to Miami on a map. The measured distance is 14 cm. The map gives a scale of 40 km per cm. What is the actual
distance in kilometers? 9. In the picture above the distance between the cities Que Que and Harare is 100
miles. ,. (i) What is the scale of this picture in miles/mm? (ii) How long is Lake Kariba in miles? COMMON MATH FUNCTIONS '
This section is designed to refresh your memory on basic math functions you will see in this course, as well as to acquaint you with how to use the special function keys of
your calculator. ._' To perform logarithms with your calculator, locate the key marked LOG or log on
your calculator. If you want to calculate log 5 and your calculator is a graphing calculator,
you most likely have to press LOG then 5. Sometimes you have to press = to get the
answer.__ Otherwise, you ﬁrst press 5 then LOG. Try it! logs = 0.698970004 Ifyou have the equation: x = log y and you have to solve for y, you must put both sides in the exponent of 10 (in this course, logarithms are base 10 unless otherwise
noted). It will look like this: 10 " = 10 “g Y
But a basic identity of logarithms is 101°“ 2 y. So we get: 10"=y A.9 BE ‘ To perform 10 7'9 on your calculator (don’t confuse this with scientiﬁc notation),
ﬁrst locate the button labeled 10 x or 10 3' on yOur calculator. Type 7.9 and then the 10 ‘
(or 10 y). Sometimes you must press = to get the answer. Try it! 107'9 = 7943282147 There is yet another important exponentiai key to know about. What if the base of
the exponential isn’t 10? Occasionally, you must perform calculations like 5 3 or 3.2 7'89. To perform a calculation such as 5.6 9, ﬁrst ﬁnd the button marked x’ or y’. Most
of the time you get the result by typing 5.6 then x’ (or y‘) then 9 then = . Try it! 5.69 =
54l6169.448 Often we have an answer already displayed on our calculator that we would like to
divide into 1. Rather than writing the number dowu or storing it, clearing the calculator and doing the operation, many calculators are equipped with a special button just for this
task. I: E To perform an operation like 3%; ﬁrst locate the button labeled 11x. Type in
2.867 then press 11x. Try it! 3.3;? = 0348795651 ? Problems . 10. When solving these problems use the calculator techninIes above: ' 6.67 10”
(i) Lugs“) =? (ii) You have the following equation: 3 = log:
I). Let s = 32.4 and solve for t. c. max. 7 d. Now ﬁnd 5.4’. M96 Dee. 7997 A.10 _. mm} ...
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