Algebra^{3.1} is a *reasoning process* that is one of
the fundamental cornerstones of physical science and engineering. As
far as we are concerned, it consists of two things:

- Representing
*numbers*(quantities) of one sort or another with*symbols*.In the sciences we don't only use symbolic representation for ``unknown'' parameters - we will often use algebraic symbols for numbers we know or for parameters whose value is actually given in problems. In fact, with only a relatively few exceptions, we will

*prefer*to use symbols as much as we can to permit our algebraic manipulations to eliminate as much eventual*arithmetic*(computation involving actual numbers) as possible. We hate doing arithmetic, right? - Performing a sequence of
*algebraic transformations*of a set of equations or inequalities to convert it from one form to another (desired) form.The permitted transformations are generally based on the set of arithmetic operations defined over the field(s) of the number types being manipulated that we talked about above and extended by means of suitable definitions that help us ``compress'' our representation a little bit.

There are two ways students are usually taught algebra. One is by being given all sorts of problems of the form:

- Given
, find the two roots and .
- Given a straight line that passes through the points and , find and in the equation .
- If , what is ?

Of course, students aren't given only three such problems; they're given three hundred! And each problem is usually given only in groups of other problems of the same kind, to drill the student in some aspect of algebra. This is bo-ring and appears to be pointless to many young students. Even if they dutifully do all of the exercises, they quickly forget.

The second is by being given problems in words, like:

- At time a truck is observed 20 meters down the road from a
stoplight. A policeman clocks it with radar travelling at 32 m/sec and
accelerating at a rate of 6 m/sec. Assuming that its acceleration
has been constant since it began moving, at rest, a short time earlier
at what time was the truck at the intersection?
- An airplane travelling north at a constant speed is observed two
miles due north of the Washington Monument at midnight and is five miles
north six minutes later. Find the speed of the airplane, and the time
it was right over the Washington Monument.
- The area of one side of a cube is 9 square centimeters. What is
its volume?

These are, of course, the *same three problems* as far as their *algebra* is concerned, but they all *mean* something. They all tell
a little story in words that you must *visualize* and *transform
into equations* so that you can use the rules of algebra to solve for a
desired quantity.

It is impossible to give any *completely* general set of rules for
transforming words into equations - often the right transformation is
``domain specific'' - it is a big part of what you learn when you learn
physics, or chemistry, or economics. One tends to use certain symbols
to stand for certain quantities, e.g. for time in the first
equation/problem above or and for area and volume respectively
in the third, and if we fail to use the right symbol - for example the
we used instead of in the *second* equation/problem, it is
harder to see the correspondance. We're much more likely to interpret
the equation for as a line in a two dimensional plane with
*no* time involved.

This leads to a bit of a paradox. If you learned your algebra mostly
from word problems, you may have a hard time making sense out or knowing
all of the operations you need to know, because it isn't always easy to
come up with a word problem that requires you to use algebra to resolve
relations like
into an expression for (until
the day you encounter one in some discipline or other), so certain kinds
of solution methodology are undertaught. Yet if you (more than likely)
learned it mostly by merely *drilling* the operations in through
your skull utterly devoid of meaning (quite possibly with indifferent
success) you may have no idea how to interpret word problems as algebra
and vice versa.

To do science you have to do *both*, and do them well!

In this review, however, we can't really do both at the same time - we
have to pick one or the other to do *first*. For better or worse,
we're going to start off by thinking about how to convert word problems
into algebraic form and the *idea* of how to manipulate (transform)
those forms using simple rules, since without these two concepts it is
difficult to even discuss formal algebra. It will turn out that when we
write down an equation, *no matter what* we've written down we can
very likely find a circumstance where that very equation occurs and has
*meaning* (as we demonstrated above). With this as motivation
perhaps we'll be able to look at lots of equations and rules with a
little more faith that there is some point to it all.

Here are a *few* rules that may help you in the process of *encoding a problem* in algebra: putting a word problem into the form of
a set of (hopefully simple) algebraic relations one can then manipulate
according to the rules of algebraic transformation to find a solution,
and then translating the result (if necessary) back into words.

- Identify all the
*quantities*given in the problem. Algebra doesn't work too well with qualities such as being ``green'' (unless greenness can somehow be converted into a numerical scale) but it can certainly deal with the ``number of green balls in an urn containing 1000 balls''. Common quantities that occur in problems include length, time, mass, temperature, count, an angle's size in degrees or radians, cost, age, area, volume (all of which carry*units*, sort of), and all sorts of pure (scalar, dimensionless) numbers like 4, , , . Ultimately, quantities are things that can be expressed as a*number*, which is why we started this book by talking about numbers. Well, duh! - Assign a
*symbol*for all of the quantities. Often it is a good idea to include quantities for which values are given as well as the unknowns. For example, we might let stand for ``number of green balls in the urn'' and stand for ``the total number of balls of all colors in the urn'' whose value in this particular problem*happens*to be 1000. Maybe will end up being important in solving the problem; maybe not, but it is better to make it a symbol early. We can always substitute in any particular value such as*later*when it is time to do*arithmetic*. - Sometimes you will have to
*make symbols up*for quantities that are*not*given in a problem but that you know intuitively are important. Even if the total number of balls isn't given, we know that if we're counting balls in an urn identified by color, we'll might well need symbols for each color the urn might contain as well as a symbol for the total. In fact, you should feel free to introduce new symbols at will as the problem develops, even temporary ones that you only keep for a few steps! - Read the problem
*carefully*to determine relevant*relationships*. This part is often very domain specific, as noted. For example, by talking about balls and urns, it seems likely that I'm about to talk about*probability*since drawing balls at random from an urn is a classic example used in many probability courses. In this case the relevant relationship might be , the probability of drawing a green ball is the total number of green balls divided by the total number of balls of all colors.In another context, it may be known that green balls have a mass of 20 grams, red balls have a mass of 10 grams, and blue balls have a mass of 30 grams, and you may be asked to find the maximum or minimum possible mass of all the balls in the urn. In a problem like this you might need all sorts of symbols and relations: ( and and and and so on to lead you to conclude that ,

- Once you have written down a symbolic representation of all the
relationships you know that might be relevant to the problem, it is time
to
*think*. Visualize what's going ont. Identify the quantity (or quantities) you are trying to solve for. Then identify a strategy - a path for performing algebraic steps that isolate that quantity and allow you to evaluate it from known information substituted into the algebra. *Do not do arithmetic*in general until this very last step. If you reason with symbols, especially with ``standard'' symbols or symbols with implicit units, you can perform a number of checks on the result by just looking at the answer to see if it has the right units, varies the way you intuitively expect with the parameters of the problem. If you reduce the number of green balls and the probability of drawing a green ball goes*up*, there may be a problem! This also helps people who check your work - the algebra is*your reasoning*. Arithmetic is just a tedious but necessary step required to turn the reasoning into an actual answer.

Let me emphasize this latter point. Most *good* instructors will
weight the reasoning far more strongly than the actual answer,
especially in physics. I personally have been known - not even all
that infrequently, actually - to give a student full credit for getting
the right *algebraic* answer for a particularly difficult problem
even if they then turn around and mispunch the numbers into their
calculator when computing the final (numerical) answer. By the same
token I've been known to give a student a *zero* on a problem where
all they do is write down the exactly correct answer - with no work or
exposition of their reasoning process whatsoever.

A correlary of this is that you should never try to ``do algebra'' with
your calculator, by punching numbers into it in an order that you think
untangles some expression hoping that you punch through to the right
number. You can't check your work and any tiny error in keying will
give you the wrong answer. Worse than wrong, as *nobody* - not
you, not your instructor, not your grader - will be able to tell what
you did wrong. Do algebra on *paper* by manipulating *symbols*
through small steps that lead to your answer. You can then easily
check, and so can your instructor!

If you practice the steps above in the context of whatever it is that
you are studying, you will rapidly learn the right symbols to use to
express things and become adept at transforming word problems into them.
Once you have correctly reduced a problem to an algebraic formulation,
however, it is usually still necessary to solve for an answer using the
*steps* of algebra, the ways of transforming equalities (or
inequalities) into *new* equalities that gradually isolate some
desired variable as ``an answer''.

Let's review these rules for transformation.

- Algebraic Transformations of Equations

- Addition Rules
- Multiplication Rules

- Order Convention for Mixed Forms
- Distributivity
- Power Rules

- Consistency of Units
- Placing the two sides of any equality into
*almost*any functional or algebraic form as if they are variables of that function - Inequalities

- Placing the two sides of any equality into
- Powers