|
Astronomy
10, Spring 2000 |
||||
Working With Numbers
The ancients were intimately involved with the sky. It was a familiar
part of their environment, like the earth beneath their feet, the air, the trees
around them. Many of us today spend much of our time indoors, within cities,
and rarely see or pay any attention to the natural environment. We cannot
urge you strongly enough to go outside and observe these phenomena for yourself.
Although much of modern astronomical knowledge is obtained with large telescopes
and sophisticated equipment, there are many things which can be learned by direct
observation with the naked eye. If you put a little effort in this direction,
your study of astronomy will likely become more meaningful and personal.
Now is a good time to familiarize yourself with some
important concepts you will need to have at your fingertips in this course.
Appendix Table 1 in your book contains the values of a number of useful
physical and astronomical constants you will often be using in your calculations.
Familiarize yourself with what is there; with usage you should eventually
be able to give a rough value for each from memory. Notice that the
value of each constant is given in terms of some unit. Notice that
the number used to express the speed of light or the length of a light
year changes depending on which unit is used. You need not memorize
things, but you still will have to know how to use units. Certain constants
(the speed of light, a couple of distance like AU and ly in eters)
will come up often enough that you should have them at your fingertips.
We will make a provision for constants to be available during tests.
Units are actually a very useful way to check if you
are doing a calculation properly. As you use each number, carry along
its units and treat them them just like algebraic quantities (multiplying,
squaring, cancelling if both numerator and denominator, etc.).
This will force you to make sure that all units are consistent (eg. don't
use some distances in cm and some in kiloparsecs without proper conversion).
Then at the end you can examine the units of the answer to see if they
make sense. For example, did the distance to the moon come out in
km/sec (there should not be a time unit in a distance quantity)?
A numerical answer is never complete without appropriate units, and full
credit will not be given without them. Naturally as a precursor to
this, you should understand what kind of units are appropriate for a given
quantity in the first place.
How big is the sun? How far away is the Andromeda
galaxy? How small is an atom? These questions arise continually
in astrophysics, so we need a quick, convenient way of expressing answers,
and comparing these answers to things we already know.
Comparing numbers: The magnitude of a number
is its size. Numbers which are about the same size have the same
order of magnitude. A common way of deciding whether numbers
are the same order of magnitude is to see if they have the same power of
ten when expressed in scientific notation. Thus 300, 253, and 958
are the same order of magnitude, as are 3 x 1010
and 4.66775 x 1010, or 2 x 10-4
and 6.3 x 10-4. Their orders of magnitude
are, respectively 102, 1010,
and 10-4.
Note how easy it is to compare numbers when they are
written this way. All you have to do is look at the exponents.
Here is an example. The distance to New York is 3200 miles, and to
the moon is 240,000 miles. How many times farther is the moon than
New York? 3200 may be written as 3.2 x 103, and
so is approximately 103. Similarly, the moon is approximately
105 miles away. Dividing 105 by 103
gives 102, so the moon is about 100 times as far away as New
York.
Order of magnitude calculations can be done with a little
more precision simply by expressing numbers with the power of ten and
the first digit, then making approximations ("~" means approximately)
along the way, like pi ~ 3, 36 ~ 40, etc. In our example
with the moon and New York, we approximated 3200 as 103
and 240,00 as 105. Let's keep one decimal place: then
3200 ~ 3000 and 240,000 ~ 200,000. The the moon is thus (2 x 105
) / (3 x 103) = (2/3) x (105/103)
~ 0.7 x 100 =70 times further away than New York. This is a
good approximation.
What does it mean to say that an object is big,
heavy, or far away? Nothing! Yet we use
these expressions all the time.
These phrases only begin to have meaning when we are
comparing two things. The Empire State Building is big when compared
to a house, a football player is heavy when compared with a jockey, and
France is far away when compared with Nevada. In addition these
qualitative terms tend to lose their meaning if the quantities being compared
are of vastly different orders of magnitude. A football player is
indeed large when compared to an atom, but the athlete and the atom are
on such vastly different scales that the comparisonis meaningless.
One has to compare things that are in the same class.
Here are two astrophysical examples:
The age of the Earth is 4 billion years. A human
may live 100 years (more if you're Dick Clark, but he may not be human).
Suppose it takes 1 million years for a mountain to grow 1 km high.
Is this a quick or slow process? It takes 104 human
lifetimes for that mountain to form, a period so long that all of our recorded
history would not be enough time for anyone to notice even the smallest
change. The mountain, however, took only 10-3
of the age of the Earth to form. If the Earth's age were made into
one day, the mountain would rise in about one minute! So this
process is both fast and slow, depending on how we made the comparisons.
Not only that, but the process was extremely fast or slow, because
of the specific objects we took.
Light travels 3 x 108 meters per second.
Is this fast? Light can go all the way around the Earth seven times
in one second. Fast!! But even the nearest star (other than
the sun) is so far away that light takes over four years to get here from
there. If we wanted to communicate by radio (which travels at the
speed of light) to astronauts visiting that star, it would take over
eight years merely to have the conversation ``Hello!'' ``Hello!''.
Slow!!
It is extremely important that you are able to make these
comparisons. A good scientist or student of science should never
look at a number without comparing it with something else. That comparison
invests the number with more meaning, and is also an aid in remembering
it. Making comparisons when doing problems is also an important skill
to develop. You should always compare the number you get from solving
a problem with the number you expect, based on a comparison with
previous knowledge, or on an order-of-magnitude calculation. For
example, if you were calculating the diameter of the moon, and you came
out with 5 meters, you would then know that something had gone wrong.
Moons do not fit inside your apartment, and an order-of-magnitude solution
of the problem might have told you to expect a few thousand kilometers
as the right answer. Many people solve a problem and then do not
look at the result to see if it makes sense. You wouldn't do this
with your rent payment; don't do it here either.
Of course, you can't get this ``feel'' for the numbers
in astronomy without studying the material. You can get it quickly
by noting the order of magnitudes of the answers to your problems:
stellar temperatures are 103 to 105
K, planetary distances are 107 to 109
km, etc. A little effort in this direction will vastly improve your
skill in science.
There are two concepts you must learn if you are to
express answers to physical problems properly: precision
and accuracy. These are slightly different, so be careful
to know what each means. Precision is the degree to which a number
is defined. Accuracy is the degree to which a number is exact.
Now for some elaboration. Look at this list of
numbers: 3, 3.1, 3.14, 3.141, 3.1415, 3.14159. Each successive number
is more precise than the previous; that is, it is better defined or ``less
approximate.'' Each number has one more significant digit
than the one before it. Thus the precision of a number is represented
by the number of significant digits it has. We'll return to the idea
of significant digits later.
You probably have a vague feeling already for the meaning
of accuracy. Suppose there are 853,596 beans in a jar, and you want
to guess the number. If you guessed 850,000 you are close: your accuracy
is high. If you guessed 20,000, you would not be close at all: your
accuracy is poor.
Note that precision and accuracy are two different things.
Take the jar of beans again. Below are several ways you can guess
their number, with a description of your accuracy and your precision.
Your guess is 100,000.
You are neither accurate nor precise.
Your guess is 800,000.
You are accurate, but not precise.
Your guess is 456,500.
You are precise, but not very accurate
Your guess is 853,400.
You are both precise and accurate.
Do you get the picture? Try another example.
Suppose there are 29,867 students at Berkeley. If you asked various authorities
about the number of students, they would probably give different answers.
Suppose you ask five people and you get the following five answers:
1.) 30,000
2.) 12,000
3.) 21,970
4.) 31,000
5.) 123,564
Which guessas are precise, and which are accurate?
Rank the numbers in terms of their precision, and in terms of their accuracy.
We mentioned above the concept of significant
figures, which express the precision of a number. To get the
number of significant figures, merely count the digits, leaving off the
leading and trailing zeros. Thus:
One significant figure: 2, 10,
0.4, 0.0000008, 5 x 10-3.
Three significant figures: 123,
0.0178, 1.56 x 1010.
Five significant figures: 12345,
1.2345, 123.45, 0.0000012345, 1.2345 x 1039.
Sometimes leading and trailing zeros can be significant.
To show this, one can write the number in scientific notation, and leave
the trailing zeros: 3.900 x 10-4.
Trailing zeros on the right hand side of a decimal point are also considered
significant, eg. 3.00 has three significant figures. It is very important
for you to realize that if a number is quoted to a certain number of significant
digits, it means that the number is known to that precision.
If you know a number is about 0.5 it makes no sense to say the number is
0.5000000 . When you say a number is 0.5, you mean that it is between
0.45 and 0.55. When you say a number is 120, you mean it is between
115 and 125. (If you really want to say it is between 119 and 121,
you can write 1.20 x 102.) If you
list a number as 103 you mean that it is
larger than 102 and smaller than 104.
Problems arise when you use electronic calculators.
These fancy machines list numbers very precisely. If you puch 1 divided
by 7 you get 0.142857142857... The problem arises because all
those digits may not really mean anything. If you know that two are
numbers are about 1 and approximately 7, then their quotient is not
exactly 0.142857..., but rather about 0.1.
The most important rule in expressing numbers is this:
The number of significant digits obtained from multiplying, dividing,
taking roots, or raising to powers cannot be more than the least number
of significant digits in any of the numbers involved. (A numerical
factor such as the ``4'' in ``4 pi R2''
can be considered as having an infinite number of significant figures.).
Here are some examples:
34x21 is exactly 714, but since
ther are two significant digits in each number, you should express
this number as 710.
79.5 / 32.7 = 2.43 (Three significant
figures, though your calculator might say 2.431192661.)
sqrt{24} should be written
4.9
sqrt{24.0} should be
written 4.90 (three figures)
sqrt{24.00} should be
written 4.899 (four figures)
(47.35)4
should be written 5.027 x 106.
The rule for addition and subtraction is a bit different:
the number of significant places in the sum or difference in two
numbers cannot be more than the least number of significant places in the
numbers. Some examples:
34.79 + 20.1 = 54.9
2,375,000 - 1,900,000 = 500,000
A common source of nonsense calculations is the subtraction
of two numbers which are approximately the same size to obtain a result
which has many fewer significant figures (maybe none).
Example: What is
(a x b) / (c x d - e x f)
a = 49.3
b = 11.2
c = 37.1
d = 15.6
e = 21.2
f = 27.2
Do this straight out on the calculator and you get 260.45283.
Because all the numbers involved have three significant figures, you might
be tempted to write this as 260. But that's misleading, because in
the middle of the calculation you did a subtraction in the denominator
578.76 - 576.64 = 2.12
which, if done with proper significant figures is
579 - 577 = 2
which is a number with only one significant digit
(at best). So the result is not any more precise than 300, and even
that may be misleading. Consider, for example, the result if the
number "e" is changed to 21.3. Now we get for the denominator
578.76-579.36 = -0.60.
and the quotient straight out would be -920. The
answer is even of different sign! To get any kind of meaningful answer
out of this problem, we would have to know the numbers a-f to more
than 3 significant figures. This example shows that the rules above
show you the best precision you can expect in a calculation.
If the numbers conspire against you, the precision could be much worse.