On May 15, 10:46=A0am, Pubkeybreaker <pubkeybrea...@[EMAIL PROTECTED]
> wrote:
> There is no =A0canonical mathematical definition of the word 'simplest'.
I believe Pubkeybreaker has a point here.
This thread inspired me to determine what exactly the
word "simplify" means in mathematics.
Let's begin with pure numerical expressions, with no
variables at all.
In the set Z, the word "simplify" obviously means to
rewrite the expression as a single number, with
possibly a unary negative sign, followed by the
decimal digits, with no explicit addition or other
operation symbols.
Even in the set Q, there is some controversy as to
whether the "improper fraction" or the "mixed number"
is the "simpler" form. Once a decision is made, then
"simplify" is easy to figure out. In the improper
fraction case, "simplify" means to write the fraction
a/b, such that aeZ, beN, and gcd(a,b) =3D 1. In grade
school mathematics, the word "simplify" emphasizes
the fact that the gcd (or GCF) of the numerator and
the denominator should be one.
Once we reach the set of algebraic reals it becomes
even more complicated. There's the concept of
"rationalizating the denominator," such that an
expression such as sqrt(1/2) must be rewritten to
become sqrt(2)/2. Of course, Abel's impossibility
theorem tells us that not every algebraic number
can be written in such a "simplified" form. Since
the OP is dealing with algebraic expressions, we
shall not go any further than this.
Now we return to the OP, where we are dealing with
algebraic expressions. The example we were given is:
Simplify (x^2-1) / (x+1).
A little common sense tells us that the test maker
wants us to divide the polynomial x^2-1 by x+1. Then
why didn't the test maker simply say "divide" and
avoid the ambiguity of the word "simplify"? Most
likely, it's because this test contains problems
where one should add, subtract, multiply, and
divide various polynomials, and the blanket word
"simplify" is intended to cover all four cases.
On a test where division is the only operation to
be performed, sometimes the instructions will state
"Divide. Assume that no denominator equals zero," or
"The domain of every variable is assumed to the the
set of all real numbers such that the expression in
which it appears is defined."
We must remember that the context of the OP's
question is a _multiple-choice_ exam. This means
that the test maker has already decided what
constitutes a "simplified expression." And most of
the time the test taker is not to choose between
two equivalent expressions (or two that agree for
all but finitely many reals), but among choices
that are vastly different.
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