BELIEVE ME NOT! - - A SKEPTIC's GUIDE
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A handy trick for introducing Algebra to young children
(who have not yet learned that it is supposed to be too hard for them)
is to phrase a typical Algebra problem in the following way:
"I'm thinking of a number, and its name is `x' . . .
so if
2x + 3 = 7, what is x?"
(You may have to spend a little time explaining the notational
conventions of equations and that 2x means 2 times x.)
Most 7-year-olds can then solve this problem by inspection
(my son and daughter both could!)
but they may not be able to tell you how
they solved it. This suggests either that early Arithmetic
has already sown the seeds of algebraic manipulation conventions
or that there is some understanding of such concepts "wired in"
to our brains. We will never know how much of each is true,
but certainly neither is entirely false!
What we learn in High School Algebra is to examine how
we solve problems like this and to refine these techniques
by adapting ourselves to a particular formalism and technology.
Unfortunately our intuitive understanding is often trampled upon
in the process - this happens when we are actively discouraged
from treating the technology as a convenient representation for
what we already understand, rather than a definition of correct
procedure.
In Algebra we learn to "solve" equations.
What does that mean? Usually it means that we are to take
a (relatively) complicated equation that has the "unknown"
(often but not always called "x") scattered all over the place
and turn it into a (relatively) simple equation with x on the
left-hand side by itself and a bunch of other symbols
(not including x) on the right-hand side of the "=" sign.
Obviously this particular format is "just" a convention.
But the idea is independent of the representation:
"solve" for the "unknown" quantity, in this case x.
There are a few basic rules we use to "solve" problems in Algebra;
these are called "laws" by Mathematicians who want to emphasize
that you are not to question their content or representation.
- Definition of Zero:
- Definition of Unity:
|
(4.3) |
- Commutative Laws:4.5
a + b |
= |
b + a |
(4.4) |
ab |
= |
ba |
(4.5) |
- Distributive Law:
- Sum or Difference of Two Equations:
Adding (or subtracting) the same thing from both sides of an equation
gives a new equation that is still OK.
|
(4.7) |
|
(4.8) |
- Product or Ratio of Two Equations:
Multiplying (or dividing) both sides of an equation by the same thing
also gives a new equation that is still OK.
|
(4.9) |
|
(4.10) |
These "laws" may seem pretty trivial (especially the first two)
but they define the rules of Algebra whereby we learn to manipulate
the form of equations and "solve" Algebra "problems."
We quickly learn equivalent shortcuts
like "moving a factor from the bottom of the left-hand-side
[often abbreviated LHS] to the top of the right-hand side [RHS]:"
|
(4.11) |
and so on; but each of these is just a well-justified concatenation
of several of the fundamental steps. (Emergence!)
You may ask, "Why go to so much trouble to express the obvious
in such formal terms?" Well, as usual the obvious is not necessarily
the truth. While the real, imaginary and complex numbers may all
obey these simple rules, there are perfectly legitimate and useful
fields of "things" (usually some sort of operators)
that do not obey all these rules, as we shall see
much later in the course (probably). It is generally a good idea
to know your own assumptions; we haven't the time to keep reexamining
them constantly, so we try to state them as plainly as we can and keep
them around for reference "just in case . . . . "
Next: Trigonometry
Up: The Language of Math
Previous: Solid Geometry
Jess H. Brewer -
Last modified: Fri Nov 13 16:17:50 PST 2015