To the Student
The bashful cannot learn, nor the irritable teach.
- Hillel, Sayings of the Fathers, II, 6
With common sense and some algebra you can understand the world better
than you can with common sense alone. That is the theme of this book
on using algebra. Almost every problem is an application, a
description in words of a piece of the world. No problems appear just
in order to illustrate mathematical ideas. We will study the
mathematics of automobile fuel economy, of inflation, of radioactive
decay, and of population growth. You will learn to estimate numbers
that are hard to measure, and to manipulate very large and very small
numbers.
To study this material you must already know some algebra, and you
will master more while you learn to see the world in a mathematical
way. But you will probably find less emphasis on the rules of algebra
than you are used to. We concentrate more on how to read problems and
make sense of what they say than on the laws that tell how to move
mathematical symbols around on paper. The variables will rarely be
the traditional x and y. They will more often be P (for, say, electric
power consumption, measured in kWh per person per year) or T (for
time, measured in years or seconds).
The text is a sequence of worked examples designed to introduce new
ideas in concrete settings. Read them. Listen to lectures about
them. Then work problems. Mull them over, play with them, and talk
about them with your friends. Just watching is not enough.
Mathematics is not for spectators. If something in the text puzzles
you, ask about it. Perhaps my estimate or my models do not match your
experience. Perhaps something could be more clearly explained.
Question. Argue. Look for understanding, not for revealed truth.
Each problem, each technique, each answer is supposed to make sense.
You may find that idea strange. Many students have been convinced by
their experiences that the last thing mathematics can be is sensible.
If you're one of them, I hope I can change your mind.
While working through this book you will learn to use a calculator.
For many of you that will be a new tool, and novelty makes learning
exciting. We can use real data rather than simple numbers in the
probs. Moreover, using a calculator does more than make arithmetic
rable. It provides new routes toward understanding difficult ideas.
The material that directly addresses your increasingly sophisticated
use of the calculator is in indentations like this one.
You will need a scientific or slide rule calculator, one with a key
labeled "y to the power x," or something similar. Right now (1982)
such calculators cost between $15 and $40. Your calculator must
provide:
* both common and natural logarithms (LOG and LN),
* parentheses (or RPN, as on Hewlett-Packard calculators),
* scientific notation, to display large numbers,
* at least one memory location.
If it uses rechargeable (or 1000-hour) batteries you will not be
replacing them frequently. You will find more than one memory
register very useful, though not essential. Other features may meet
your long-term needs. Some calculators have built-in statistical
tools. Some do common business calculations efficiently. Some are
particularly suited for scientific work. The most exciting option you
could select would be programmability, although a programmable
calculator might cost more than $40 and is not necessary for this
material. Buy the best calculator you can afford. Plan to keep it.
This book was designed to cover useful and interesting material, the
studying for its own sake. It may not be suitable preparation for
advanced work in mathematics. In particular, this is not a standard
calculus text, although it covers many of the topics included in such
texts. It can, though, prepare you to take a short, applied calculus
course. I hope that you enjoy learning the mathematics in this
book. I've tried to find examples to interest you if you care about
biology, economics, management, or sociology, or if you want to know
how schematics can help you see the world in a new way.