Sometimes mathematicians play "fast and loose" with definitions. We probably will not find total
agreement about what a "polynomial" is. Some books will define "a polynomial in x" as opposed
to just a polynomial. And what about a "polynomial in two variables?" If y=2+x is x+y a binomial?
And what does "quotient" refer to? If we consider 7÷2 as a fraction, is 7/2 a quotient? Or is it
3.5? Or is the quotient 3 (with remainder 1)?
And what is a fraction? Which among the following are fractions?
x, 2/x, x/y, 2/3, 2÷3, x÷y, 2*(1/x), x/1, x/3, xy where y=1/z, z where z=1/y, etc.
Let x=1/y, y=1/3 and z=1/x. Which are fractions? x, y, 1/x, 1/y, y/3, xy, xz, yz, 1/yz, etc.
What is an arithmetic fraction, an algebraic fraction, etc.?
And in geometry is an equilateral triangle also isoseles? There has been disagreement on whether
to make the set of equilateral triangles a subclass of isoseles triangles or a separate category of
The best we find at times is a "local" definition where an author defines a term precisely for his
following discussion. And often other mathematicians may find fault with this.
Mathematics is a LANGUAGE and has not been (nor ever is likely to be) "nailed down" so as to be
without ambiguity or disagreement even for the most common and "simple" concepts.
Sometimes we just have to "roll with the punches" and at times ask for clarification of what the
author intends (as is often the case in this forum).
Often pushing for the exact meaning of all the terms we encounter may result in returning to the
undefined terms of a system. But then the expression we obtain for a "higher level" concept's
definition may be so long and involved with the elementary undefined terms as to be basically
As an example consider the "Sheffer stroke" or "Dagger" in T/F two valued logic. Each of these
can be used to define the usual AND, OR, if..then, if and only if, NOT, exclusive OR. But the
expressions for some of these are quite long and complicated using just the one stroke or dagger.
It is interesting that one "operation" can be used to define all the usual stuff, but it is way too
unwieldly to want to use it. As humans we work better with the AND, OR, etc.
As another example consider binary vs hexadecimal. Working with hexadecimal is much easier
for us humans than working with binary, especially when numbers are fairly large. The strings
of 1's and 0's just get too long and difficult to deal with.