Representing Simple Arithmetic Symbolically
The transition from mathematical expressions involving just
numbers as in
5 + 3
to mathematical expressions involving symbols (which stand for
unknown or unspecified numbers) such as
x + 3
is the most difficult step in mastering the methods of basic
algebra. We can easily visualize what “5 + 3” means:
The group of 8 dots in the figure on the right clearly
contains 3 more dots than the group of 5 dots on the left. It is
obvious from these simple diagrams that when we compare
that the group of 8 dots can be described in relation to the
group of 5 dots using any of the following phrases:
 3 dots added to 5 dots
 3 dots plus 5 dots
 3 dots more than 5 dots
 the sum of 3 dots and 5 dots
 5 dots increased by 3 dots
 a total of 5 dots and 3 dots
 the 8 dots exceed the 5 dots by 3 dots
and so on. The clear point is: we started with a group of 5
dots and ended up with a group consisting of those original 5
dots plus another 3 dots (which we know from our knowledge of the
meaning of words such as “one,” “two,”
“three,” etc. – which are themselves just symbols
– that the final group of dots happens to be what we call
“eight” dots or 8 dots).
The expression
x + 3
has much the same interpretation:
In this case, instead of starting out with a specific number
of dots, such as the 5 dots in the earlier example, we’re
just saying: “start with a certain number of dots,” and
for convenience, we’ll represent the number of dots with
which we actually start by the symbol x. Whatever the actual
number of dots present in this group of x dots, the notion of
adding another 3 dots to the pile of dots is easy to picture. We
would have the original pile of x dots and now there would be 3
more dots in the pile of dots. This is all that is meant by
 3 dots added to x dots
 3 dots plus x dots
 3 dots more than x dots
 the sum of 3 dots and x dots
 x dots increased by 3 dots
 a total of x dots and 3 dots
 the x + 3 dots exceed the x dots by 3 dots
In this way, the algebraic expression
x + 3
is a way of writing down the result of starting with a certain
number, x, of things, and adding to that collection 3 more of the
same things. It is very important to notice that the symbol x
stands for the number of dots. x does not represent “a
dot” or “the dots.” x represents the number
of dots.
By now, you’re probably wondering what possible point
there could be to using alphabetic symbols for numbers whose
value we either don’t know at present, or for numbers we do
not want to restrict to a specific actual value at present.
In many problems, we will be able to write down a very
specific and concise representation of the answer despite not
having enough information to reduce this final answer to a
specific numerical value. Notice that
 it doesn’t matter how many thing we start out with.
That is, it doesn’t matter what value x actually may
represent. If we include another three things, then the
total number of things we end up with can be represented
by the expression “x + 3” things.
 it doesn’t matter what kind of “things” we
are talking about as long as they are things we can count
or measure numerically. In the four examples above, the
“things” were numbers of dollars, numbers of
kilometres, numbers of music CDs, and numbers of meters
of rope. In each case, adding an additional three of
these things to some original number of the same things
can be represented by the same symbolic expression. This
is because the symbol x represents a number of things
rather than the things themselves.
These two observations give algebraic methods enormous
usefulness in solving all sorts of problems involving numerical
quantities. Not only do we have a way of representing unknown
numerical values very clearly and concisely, but since these
symbols represent actual numbers, we can to some extent use our
knowledge of basic arithmetic with numbers to do arithmetic with
these “unknowns.”
Of course, with numbers we can simplify expressions like
“2 + 6” to 8 before multiplication by 4, so ultimately
the use of brackets is avoided. However, with symbols, as in
“x + 3”, it is not possible to do this – hence the
requirement to use brackets is unavoidable.
Always, always, always, in algebra, the literal symbols
represent numerical values. Often, being able to describe exactly
what numerical value a symbol represents is the most important
step in solving a problem.
