# Thread: What is a term?

1. ## What is a term?

Work with them all the time, but I don't recall a formal definition of one (such a definition should take into account complex fractions I feel).

2. I would define a term along the lines of any numbers/variables which are not separated by a + or - sign it's by no means a real definition. Instead

any distinct quantity contained in a polynomial; "the general term of an algebraic equation of the n-th degree": http://wordnetweb.princeton.edu/perl/webwn?s=term

3. I once tried to find a "precise" definition of term to put in a book I was writing, but a universal definition doesn't seem to exist (if it does, someone please tell me). But the idea is exactly what e^{ipi} said. expressions separated by + and - signs are terms.

There are of course other definitions of terms used in different branches of mathematics, but I think this is the one you mean.

In a complex fraction, for example, there will be terms in both the numerator and denominator of the complex fraction, and some of the terms will be fractions themselves.

4. Hello, wonderboy1953!

Work with terms all the time, but I don't recall a formal definition of one.

I was taught that a term is an expression consisting of multiplication only.

. . So that: .$\displaystyle 2x^3y^4,\;5\sqrt{a},\;x^3e^{2x}\ln(y)$ are all terms.

I was also told that terms are separated by addition and/or subtraction.

. . So that: /$\displaystyle 2a + 3b$ has two terms (a binomial).
. . . . . . . . .$\displaystyle x^2 - 5x + 6$ has three terms (a trinomial).

5. it's quite common for students to confuse "terms" and "factors"

6. Originally Posted by wonderboy1953
Work with them all the time, but I don't recall a formal definition of one (such a definition should take into account complex fractions I feel).
a term is something that is separated by a + sign. For 2x-2y+3. The terms are 2x, -2y, 3.

7. Originally Posted by Barthayn
a term is something that is separated by a + sign. For 2x-2y+3. The terms are 2x, -2y, 3.
(not picking on you) would you consider the following a term?: (x + y) as in (x + y) + 2 e.g..

There are so many situations where it's gotten to the point that maybe the word term can't be defined in a consistent manner.

8. Originally Posted by wonderboy1953
(not picking on you) would you consider the following a term?: (x + y) as in (x + y) + 2 e.g..

There are so many situations where it's gotten to the point that maybe the word term can't be defined in a consistent manner.
I would consider that as two terms (x+y) with (x+y)+2 as three terms.

Often we start off assuming it to be a single term to make it easier: $\displaystyle (x+y-2)(x+y+2) = ((x+y)-2)((x+y)+2)$. In initially treating it as one term we see it's the difference of two squares: $\displaystyle (x+y)^2-4$. However, that (x+y)^2 will need to be expanded using the binomial theorem

9. Originally Posted by e^(i*pi)
I would consider that a trinomial
Going by what you're indicating, then there would be altogether seven terms in the expression.

10. I believe that there necessarily has to be some ambiguity in the definition of a term. Here is a specific example showing why:

I think that we would all agree that the expression 2(x + y) has only 1 term (however it has 2 factors, and one of those factors has 2 terms)
So what about the expression 1(x + y)? As written it looks like it has only 1 term. But it is the same as x + y which clearly has 2 terms.

Thus, it seems that there is no well-defined function f from expressions to positive integers which says how many terms are in the expression. That is, there is no function f such that if the expression E is equivalent to the expression E', then f(E)=f(E').

Compare this situation to the model theoretic definition of term which is well-defined.