To: Logy,
Mathematics does not have a "king", but it does have a "queen",
and that queen is "number theory".
Number theory is different from other branches of mathematics in that
it deals primarily with the properties of non-negative integers.
Most importantly, in number theory, concepts such as "the fundamental theorem of arithmetic",
"common factors", "prime numbers", "perfect numbers", "abundant numbers", etc.
all become utterly meaningless if we allow unity to be both a "multiplier" and a "multiplicand".
For example, in number theory, a "perfect number" can be defined as:
"A positive integer such that the sum of its positive proper factors is equal to itself".
Thus, if we allow both "multiplication by unity" and "multiplication of unity",
then
can be factored as:
, where
,
and the entire concept of a "perfect number" simply collapses!
However, if we allow only "multiplication of unity" and strictly prohibit "multiplication by unity",
then
must be factored as:
, where
,
and the concept of a "perfect number" remains intact.
You see, the "queen of mathematics" herself is telling us that "multiplication by unity"
is a really dumb idea because it not only destroys important concepts in number theory,
but it also accomplishes absolutely nothing and shouldn't even be considered an "operation"!
Now, "coheseve terms" (also known as "Blazys terms") were developed for use in number theory.
They are superior to "conventional" algebraic terms in that they don't allow multiplication by unity,
They are also superior in other ways.
For instance, if we consider the actual meaning of the equation:
where all the variables represent non-negative integers,
then it is clear that the greatest possible factor
was "extracted", then "eliminated" (meaning "cancelled"),
so that both sides must now represent the same "prime number".
However, prime numbers do not have exponents, so we are now stuck with
the seemingly impossible task of eliminating the exponent.
Again, the "Blazys term" on the right comes to the rescue by telling us that
before we can allow
(which common sense tells us must be allowable),
we must first let
, then immediately "cross out" or "cancel out"
the expressions involving logarithms, which leaves us with:
,
which simplifies to:
.
So not only do "Blazys terms" provide us with a more meaningfull and accurate
representation of "common factors" (as we saw in post #92),
but they also give us a more meaningfull and accurate representation of factors....period!
To put it simply, without "Blazys terms", even our most simple equations will wrongly imply that
eliminating the greatest possible factor from a number does not necessarily result in a prime!
You know Logy, you are absolutely right in your assertion that a lot of "high level"
mathematicians don't really understand the underlying principles of the mathematics
that they use, depend on and teach every day.
They are like parrots, mynah birds and
who do everything by rote,
mindlessly regurgitate only what they have been taught,
and never ever bother to actually reason, or think things through for themselves.
They are absolutely blind to the fact that "Blazys terms" have the potential to unify
number theory (where multiplication by unity is prohibited), and set theory, algebra,
calculus, etc. (where multiplication by unity is allowed).
You are also right in that the "mathematical emergency" does indeed extend far beyond
the idiotic way in which factors and common factors are presented to students.
Now here is a question for you.
Since it has now been demonstrated that the way in which factors and common factors
are presented to students is every bit as unscientific as "creationism",
shouldn't the courts intervene and put an end to it, just as they did with
the "theory" of "intelligent design"?
Don.
