Meaning of Term in Algebra

Date: 09/16/2007 at 12:22:14
From: Erin
Subject: Terrible Terms

When I was in middle school, I was taught that in algebra we do not 
write 2 * x because multiplication is used so much.  Instead, we use 
2x.

However, as I went through the math course, they explained to me that 
something like 2x is considered one term, and that something like 
2 + x is considered two terms.  Because of this, each are treated 
differently.  For example, in the problem 3(2x) all you need to do is 
multiply to the one term to get 6x.  But in a problem like 3(2 + x), 
because there are two terms in parentheses, you need to distribute to 
get 6 + 3x.  This is just one example of how a problem with one term 
and a problem with two terms are treated differently.

My question is this:  why is it that when you multiply or divide two 
numbers or variables they become one term (such as 2*x becoming 2x), 
but when you add or subtract two numbers or variables they are still 
two terms (such as 2 + x)?  This was never really explained to me 
because all they told me was that it's easier to not put the 
multiplication dot (*) because it's used so much.  But I know that's 
not the only reason because there are so many different properties 
and different ways to treat the terms, that there must be another 
reason then just for convenience.  Can you help me with my curiosity 
on this?  Thank you. 

Take the problem 2(xy).  In 2(xy), since the value in parentheses is 
one term, the appropriate thing to simplify it to would be 2xy.  This 
makes sense.  Lets say xy is the area of a box that is x units wide 
and y units long.  If we combine two of those kind of boxes together, 
we would have one rectangle with an an area of 2xy.  But if we were 
to pretend that xy was two terms, and, using the two-term 
distribution rule, multiplied both x and y by 2, this would not make 
sense.  That would be saying that 2 xy boxes placed together have a 
width of 2x and a length of 2y.  This is not true because if 4 boxes 
were placed together, then the resulting box would be 2x wide and 2y 
long.  Hence, if we distributed 2(xy) wrongly and got 2x2y, it would 
have to be simplified to 4xy.  4 boxes have an area of 4xy, and 2 
boxes have an area of 2xy.

x[]
 y

x[][]
  2y

2x [][]
   [][]
    2y

Now take the problem 2(x + y).  Since x + y is two terms, if we were
to distribute the 2 to both the x and the y, we would get 2x + 2y. 
This makes sense.  Lets say x + y is the number of units x and the
number of units y in one group.  If we get two of these kind of
groups, we would have double the number of units x and well as the
number of units y.  But if we were to pretend that x + y was one term,
then we would simply the problem as 2x + y.  This does not make sense.
If we were to try to simplify it this way, we would be saying that if
we had two groups, only the number of units x would double.  This 
doesn't make sense since one group has both units x and units y, and 
if we were to double that group we would have double the units x as 
well as double the units y.

One group: xxyy
Two groups: xxyy xxyy
Two groups would not be: xx xxyy

My teacher always tells me "terms are terrible".  You must treat them
in different ways.  In my opinion "terrible terms" are where most
algebraic mistakes come from.  I feel knowing more about how terms
work will help me to understand and avoid these mistakes.  An example 
of a "terrible term" algebraic mistake is that you cannot distribute 
an exponent among two terms. (x + y)^2 does not equal x^2 + y^2.  But 
you can distribute an exponent over one term.  (xy)^2 = x^2*y^2.  
Another example is that if the same term is added or subtracted in the 
numerator and denominator, they cannot be canceled out. (5 + 2)/(10 + 
2) does not equal 5/10.  But if the same term is multiplied in the 
numerator and denominator, they can be canceled out. 
(5*2)/(10*2)=5/10.

Date: 09/16/2007 at 23:34:58
From: Doctor Peterson
Subject: Re: Terrible Terms

Hi, Erin.

You've said a lot of good things; it sounds like you have a pretty
good understanding of the essentials.  I'll try saying a little more
that might put things in perspective (which is what I think you want),
and then we can discuss more if there are still gaps.

The reason adding makes two terms is simply that we DEFINE a term as
something that is added to other terms.

In part due to the order of operations, we think of any expression as,
ultimately, a sum: that is the last operation we do, so if there are
any additions (that are not inside parentheses), they break up the
expression into a sum, and the pieces are called terms.

Many of the things you talk about are distributive properties.  We can
show that multiplication distributes over addition,

  (a+b)*c = ac + bc

and that exponents distribute over multiplication,

  (ab)^c = a^c b^c

But exponents do not distribute over addition, and multiplication does
not distribute over another multiplication.  So you simply have to
keep in mind the rules that are true, and not do things that LOOK
similar but are not valid.

One help is to relate these properties to the order of operations
(which I believe is a major reason that the order is what it is):

  exponentiation
    distributes over...
  multiplication
    distributes over ...
  addition

Also, I believe that we leave out the symbol for multiplication not so
much because it is common, but because it is powerful (higher in the
order of operations).  By writing ax + by with the a and the x close
together, we make the order of operations look natural; we SEE ax as a
single term, by as another, and the main operation in the expression
as addition, a sum of terms.

Does that help at all?

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 10/02/2007 at 18:28:23
From: Erin
Subject: Thank you (Terrible Terms)

Hello.  Thanks for answering my question.  What you said about the
order of operations really helps - that exponents distribute over
multiplication, which distributes over addition.  It helps make sense
of it to me and puts things into better perspective.  It seems like it
has a lot to do with how we define math operations.

It was really neat hearing back from a mathematician - I plan 
to major in math in college.  Thanks again! :)