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Next: Algebraic Transformations of Equations Up: intro_math_review Previous: Complex Numbers   Contents


Algebra3.1 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:

That's it. It is really - simple - simple enough to be the best way of reasoning about nearly anything. Even English is, in some sense, a system of algebraic reasoning, with symbols standing for things. We're just going to streamline the process and make it more consistent than English usually is.

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

  1. Given $3t^2 + 32t + 20 = 0$, find the two roots $t_1$ and $t_2$.

  2. Given a straight line $y(x)$ that passes through the points $(x,y)
= (2,0)$ and $(x,y) = (5,6)$, find $a$ and $b$ in the equation $y(x) =
ax + b$.

  3. If $A = 9$, what is $V = A^{3/2}$?

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:

  1. At time $t = 0$ 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$^2$. 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?

  2. 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.

  3. 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. $t$ for time in the first equation/problem above or $A$ and $V$ for area and volume respectively in the third, and if we fail to use the right symbol - for example the $x$ we used instead of $t$ in the second equation/problem, it is harder to see the correspondance. We're much more likely to interpret the equation for $y(x)$ as a line in a two dimensional $x-y$ 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 $x = z^\pi(z^{-1} + a)$ into an expression for $a$ (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.

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.

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Robert G. Brown 2009-07-27