\appendix
\chapter{Fundamental Quantities}

There are various ways to describe the world in which we live.  Some
are qualitative and others are quantitative.  Qualitative descriptions
describe aspects of objects or events such as texture, and use words
like `rough', `smooth', `flat' etc.  Qualitative descriptions cannot
be described numerically.  One would not say that you looked tired
with a value of 3.0 unless someone had first set up some kind of
numerical scale to measure just how tired you were; tiredness is not
something we measure quantitatively. On the other hand, length is a
dimension that can be described either qualitatively or
quantitatively; one can qualitatively describe an object as long, or
one can quantitatively describe it as 10 feet in length.

All quantitative measurements are made in some kind of unit.  Length,
for example, can be measured in units of meters, feet, miles,
etc. Other fundamental metric units are the kilogram (a measure of
mass) and the second (a unit of time).  Other units of measurement are
combinations of these fundamental units.  An example of a combination
is velocity, expressed in units of meters per second (m/s) which
measures how far something has moved in a given direction over a given
period of time.

In astronomical studies, one sometimes uses units which express rather
large values in the fundamental metric units.  An example of this
would be the Solar Mass unit (notated as M$\odot$). The mass of our
Sun is, by definition, one Solar Mass or about\break
1,900,000,000,000,000,000,000,000,000,000 kilograms.  A star with 10
times as much mass can be written as 10 M$\odot$; this is clearly more
convenient to write than a number with all those zeros! Other units
used in astronomy are the light year (ly), parsec (pc), and the
astronomical unit (A.U.), all of which are units of distance.  The
unit you choose to use depends on the situation, and personal
preferences. When describing distances in the solar system the
astronomical unit is typically used since it is the average distance
from the Sun to the Earth.  In describing distances to stars the
parsec or light year is usually used.
 
As described in the introductory lab, the metric system allows easy expression 
of large multiples of the
fundamental units via prefixes. For example, 1,000 meters is called a
kilometer and is usually written as 1km. 

%Below is a table listing some
%of the most common prefixes and what they stand for.  These prefixes
%may be placed in front of any base unit to express larger quantities.
%
%\begin{table}[htb]
%\begin{center}
%\caption{} 
%\bigskip
%\begin{tabular}{lllc}\hline\hline
%PREFIX & MULTIPLE & & SYMBOL \\
%\hline
%Giga  & 1000000000  & $10^9$    & G \\
%Mega  & 1000000     & $10^6$    & M \\
%kilo  & 1000        & $10^3$    & k \\
%centi & 0.01        & $10^{-2}$ & c \\
%milli & 0.001       & $10^{-3}$ & m \\
%micro & 0.000001    & $10^{-6}$ & $\mu$ \\
%nano  & 0.000000001 & $10^{-9}$ & n \\
%\hline
%\end{tabular}
%\end{center}
%\end{table}

As described in section 1.4 scientists also use a notation system called 
scientific notation for
representing very large or very small numbers without having to write
lots of digits. As an example of how large numbers can get in science
let's look at the mass of Mars.  Using Kepler's laws of motion to
study Mars' moons, astronomers have determined that Mars has a mass
approximately equal to 640,000,000,000,000,000,000,000 kilograms.  Now
you can see that it is rather inconvenient to write down all those
zeroes, and it is confusing to use the prefixes above.  Imagine how
much more mass there is in the Galaxy and you can see that we need an
easy way to write very big (or very small) numbers.  This leads us to
the concept of Scientific Notation.

%\section{Scientific Notation}
%
%Scientific notation utilizes powers of 10 in order to write numbers
%which are very large or very small and take up too much space to write
%(such as the mass of Mars above).  For example 1,000 can be written as
%$1\times 10\times 10\times 10$, or $1\times 10^3$.  That is, the 3
%zeros were used to move the 1 three spaces from the decimal
%point. Applying this rule to the number 1,244, the final product is
%$1.244\times 10^3$.  Remember, it is the exponent (or power) over the
%10 which gives you the number of places you should move the decimal to
%the right in order to get a number expressed in standard notation.  If
%the exponent were negative, like $1\times 10^{-3}$, then the decimal
%should be moved to the left three spaces to get 0.001.  A simple
%rule-of-thumb to remember is that numbers greater than 1 have positive
%(+) powers of 10, while numbers less than 1 have negative (-) powers
%of 10. The following list of useful examples which will help you.
%
%{\it Examples of Scientific Notation}

%\noindent
% 0.15 = 1.5 $\times$ 10$^{-1}$ \\
%   15 = 1.5 $\times$ 10$^1$ \\
%1,500,000  = 1.50 $\times$ 10$^6$ \\
%3,483 = 3.484 $\times$ 10$^3$ \\
%    5 = 5 $\times$ 10$^0$ \hskip 1in (In fact, any number x$^0$ = 1, 
%except 0$^0$!) \\
%1 million = 1,000,000 = 1 $\times$ 10$^6$ \\
%1 centimeter = 0.01 meters = 1 $\times$ 10$^{-2}$ meters \\
%1 nanosecond = 0.000000001 seconds = 1 $\times$ 10$^{-9}$ seconds \\
%\bigskip
%
\chapter{Accuracy and Significant Digits}

The number of significant digits in a number is the number of non-zero digits
in the number. For example, the number 12.735 has \textit{five} significant 
digits; the number 100 has \textit{1}.  When computing numbers, people
today often use calculators since they give us precise answers quickly.
Unfortunately, many times they give us answers that are unnecessarily and
sometimes unrealistically precise.  In other words, they give us as many
significant digits as can fit on the calculator screen.  In most cases,
you will not know the numbers you are plugging in to the calculator to
this precise of a value, and therefore will get an answer that has too
many significant digits to be correct.  This will be the case for your
astronomy labs this semester. In general, you should only report the
accuracy of a calculation with the number of significant digits of the
least certain (smallest number of significant digits) of any of the
numbers which were the input into the calculation. For example,
if you are dividing 13.2 by 6.8, although your calculator gives 1.94117647,
you should only report two significant figures (i.e. 1.9), since that is
the number of significant figures in the input number 6.8.

\chapter{Unit Conversions}

Very often, scientists convert numbers from one set of units to
another. In fact, not only do scientists do this, but you do it as
well!  For example, if someone asks you how tall you are, you could
tell them your height in feet or even in inches.  If someone said they
are 72 inches tall, and 12 inches equals 1 foot, then you know they
are 6 feet tall! This is nothing more than a simple conversion from
units of inches to units of feet. Another everyday unit conversion is
from minutes to hours, and vice versa. If it takes you 30 minutes to
drive from Las Cruces to Anthony, then it takes you 0.5 hours. We know
this because 60 minutes are equal to 1 hour. However, how can we write
these unit conversions, with all the steps, so that we are sure we are
converting units correctly (especially when the units are foreign to
us (i.e. parsecs, AU, etc.))?

Let us begin with our everyday conversion of inches to feet.  Say a
person informs you that they are 72 inches tall and you want to know
how many feet tall they are.  First, we need to know the unit
conversion from inches to feet (12 inches = 1 foot).  We then write
the following equation:

\beq
72 \; inches \times \frac{1 \; foot}{12 \; inches} = 6 \; feet
\eeq

Note how the inches units cancel (one in the numerator and one in the
denominator) and the units which remain are feet. As for the
mathematics, simply use normal rules of division (72/12 = 6) and you
wind up with the correct result.

The second example, minutes to hours, can be performed using the
method above, but what if someone asked you how many days there are in
30 minutes?  You will need to use 2 unit conversions to do this (60
minutes = 1 hours, 24 hours = 1 day).  Here is how you may perform the
unit conversion:

\beq
30 \; minutes \times \frac{1 \; hour}{60 \; minutes} \times 
\frac{1 \; day}{24 \; hours} \approx 0.0208 \; days = 2.08 \times 10^{-2} \; days
\eeq

%\bigskip

Again, note that the minute units have cancelled as well as the hour
units, leaving only days.
     
You have now seen how to perform single and multiple unit
conversions. The key to performing these correctly is to 1) make sure
you have all the conversion factors you need, 2) write out all of the
steps and make sure the units cancel, and 3) think about your final
result and ask whether the final result makes sense (is 30 minutes a
small fraction of a day? Does 72 inches equal 6 feet?).

\chapter{Uncertainties and Errors}

A very important concept in science is the idea of uncertainties and
errors.  Whenever measurements are made, they are never made
absolutely perfectly. For example, when you measure your height, you
probably measure it only to roughly the nearest tenth of an inch or
so.  No one says they are exactly 71.56789123 inches tall, for
example, because they don't make the measurement this accurately.
Similarly, if someone says they are 71 inches tall, we don't really
know that they are exactly 71 inches tall; they may, for example, be
71.002 inches tall, but their measurement wasn't accurate enough to
draw this distinction.

In astronomy, since the objects we study are so far away, measurements
can be very hard to make. As a result, the uncertainties of the
measurements can be quite large. For example, astronomers are still
trying to refine measurements of the distance to the nearest galaxy.
At the current time, we think the distance is about 160,000 light
years, but the uncertainty in this measurement is something like
20,000 light years, so the true distance may be as little as 140,000
light years or as much as 180,000 light years. When you do science,
you have to always assess the errors on your measurements.

