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

triangles.

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

incomprehensible.

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.

1/2(grateDAY )!