Tutorial on the term Term
I have a question about terms. I am a sophomore in high
school and I usually go about my business solving problems, but
today I had some extra time and sat down and thought. What exactly
is a term? This brought up some interesting situations that I would
like to ask why they work, shown below. Ok, so my first question is
what is a term. For example in 5x, are 5 and x separate terms or are
they one term all together since they are being multiplied. The same
question arises in 5x+6. Is this two terms, three terms, or one term
of (5x+6)? This is where I really started to get confused. I know
that I have always done this stuff and my questions may seem silly,
but here I go. In 5x + 2x, I would usually say that the answer would
be 7x by combining like terms, but if you considered them to be two
different terms, like (5x) + (7x), then how would you add these terms
if they were not the same. Here is another example of my confusion.
If you have x/6= 5+x, I would first multiply 6 to both sides to get
rid of the six on the left hand side of the equation. When I multiply
the 6 to the right hand side of the equation, I would multiply it
like this: 6*(5+x). Each side of the equation is a term, right? So
when multiplying you have to distribute across the term. Well, what
if you had x+6=12. In my mind, since each side of the equation is a
term, I would think of it like this, (x+6)=(12). Here, each side is
a term, and usually I would just subtract the six, but how can you
do that since (x+6) and 12 are not like terms. I have always been
taught that you cannot break up a term, such as in (x+10)/5, so why
can you do it here? I know these questions may seem tedious and
unnecessary, but I do not understand how we do what we do in math
if we do not understand these basic principles of adding and
Very good question.
I began realize that maybe the easiest way to separate terms is
by the separation of the +'s and -'s; however, when equations
include variables like x, y, z, etc, you have to group the x's
and y's and z's, etc, TOGETHER. In the case of multiplication
and division, 5x and 5/x are both single terms. In the case of
complex equations like 5x/z, you must separate the x and z;
therefore (5/x) * (1/z) are TWO terms of a product.
To simplify my explanation, I'll use the examples you have provided.
5x is a single term because you cannot "separate" the 5 and the
x. It's simply the (x) term and in this case the 5x term. For
5x+6, you have an (x) term and a constant; therefore, 5x+6 has two
terms. Using your example, 5x + 2x: here you have two values of
"like" terms. Meaning, you have two (x) terms, and by the
distributive property 5x + 2x = (5 + 2) x = 7x and, therefore you
group the two (x) terms, resulting in one (x) term.
Hence x/6 = 5 + x, is x = 30 + 6x, and finally, 0 = 30 + (6-1)x,
and therefore this equation has two terms: a constant term and one
In short, terms are grouped values of similar notation. Group the
x terms with the x's and the y terms with the y's...and so on...
and do not forget...group the constants with other constants.
Be careful though. As mathematics become more complex, there are
times when some variables are a "don't care" or "imaginary".
Meaning, you could have in special circumstances when (5x+2)i,
where i is an imaginary number, and hence you have ONE term that
is "imaginary" and is simply (5x+2).
Hope that helps.
Yours is not a silly question, nor is it simple, because you are dealing
with some of the very fundamental questions of mathematics. Specifically:
What do the "symbols" mean? How are the operations defined? Those two
questions have a long and non-trivial history. Usually, I avoid recommending
specific books, after all, we are not book sellers, but occasionally, a book
that really addresses a question and is readable at the same time comes
along, that it is worth breaking that general rule.
"Math through the Ages" by Berlinghoff and Gouvea is such an instance. What
you will find is that mathematical symbols and operations as we know them,
generically call these "terms", is a rather recent invention, developing
between the 1600's and 1800's. Before then, algebra was expressed verbally.
For example, the simple arithmetical statement: (5+6)-7 = 4, in modern
notation, would have been expressed verbally, "When 7 is subtracted from the
sum of 5 and 6, the result is 4." Not very efficient for such a simple
arithmetical statement, and you can see how things could get very
complicated for even a "simple" algebraic expression with powers and roots.
You also must appreciate the understanding of unknowns, zero, negative
numbers, irrational numbers, and imaginary (complex) numbers are all recent
additions to the history of mathematics, as is the subject of your inquiry
Fun question, but don't let it confuse you. A term is a grouping of numbers
and symbols (variables), separated by an arithmetic sign (+ or -). And the
variable is central to the whole concept!
When we see 5x + 3x both are terms (note the separating +) and both are
terms of the variable x so they are “like terms”. The five and three are
not considered terms in their own right because they are linked to (or
modify) the x. Like terms may be added and subtracted.
So how come if we write x + 3 the 3 is a term? Well, not only the + sign a
hint, but here we can say we wrote the expression in descending order and 3
is actually 3x0 !
Now as far as mathematical rigor is concerned, this may not be exactly proper,
but I think it may make sense considering your question.
Hope this helps and thanks for a neat question!
B.S. Earth Sciences; M.S. Geophysics
Oklahoma State Univ. Inst. of Technology
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Update: June 2012