The Mathematics 11
Competency Test
Literal Numbers
The basic difference between algebra and simple numerical arithmetic is that in algebra we can
use written or literal symbols to represent numerical values. These symbols are often ordinary
alphabetic characters: a, b, c, d, … or A, B, C, D, …. . (Usually in algebra, uppercase characters
are not considered to be the same as lowercase characters – ‘A’ and ‘a’ are not equivalent
algebraic symbols.)
Sometimes symbols which are characters from other alphabets are used. Mathematicians are
particularly fond of characters from the Greek alphabet. (See the short description of the Greek
alphabet in a subsequent note provided for general information only. The BCIT Mathematics 11
Competency Test does not include questions about the Greek alphabet directly.) In fact, it isn’t
too uncommon to use special symbols which are not part of any alphabet. As far as the BCIT
Mathematics 11 Competency Test is concerned, we need not concern ourselves here with the
many more exotic symbols that have important meanings in more advanced mathematics.
However, regardless of how exotic the symbols used may be, in algebra each symbol still simply
represents a numerical value.
Actually, we can distinguish at least three types of literal symbols in algebra:
1) constants – these are symbols which by convention always represent the same
numerical value and are used more as a matter of convenience than anything else. For
example, the Greek character  (pronounced “pie”) is often used to represent the ratio
of the circumference of a circle to its diameter. This is an irrational number equal to
3.141592653589793 to fifteen decimal places. It is exactly the same value for every
circle. Obviously, it is much easier to write the simple symbol  than it is to write a
sixteen or more digit approximation to the actual value of .
2) variables – these are symbols that can either represent any numerical value, or which
represent only one or a few possible numerical values, but for the moment we don’t
know precisely what those values are. So, for example, in the formula for the area, A,
of a circle of radius, r
A = r2
the symbol  represents a constant, as explained earlier. However, the symbol r
represents the actual radius of the circle whose area you wish to calculate. The actual
value of r will be different when circles of different sizes are being considered, and in
principle, r could have any positive value. Similarly, the symbol A represents the area
of the circle under consideration. In principle, the value represented by A could also be
any non-negative number. Of course, once you focus on a specific circle with a
specific value of r, the specific value that the symbol A represents is fixed by this
formula.
3) arithmetic operations and relations – we don’t usually think of these symbols as
algebraic symbols, but they are present in algebra and contribute essential information.
At a basic algebra level, there are just a few of these:
+

, /
David W. Sabo (2003)
“add” or “sum”
“minus” or “subtract”
“times” or “multiply”
“divide by”
Literal Numbers
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=
“equals,” indicates that the quantities or expressions to the left and right
represent the same numerical value.
Numbers or symbols as superscripts indicate powers or exponents as with ordinary numerical
arithmetic expressions.
Generally these arithmetic operation symbols must always be explicitly present. The one
exception is that the multiplication of quantities represented by two symbols may be represented
by just writing the symbols adjacent to each other with no space in between. Thus
ab
often means
ab
3a
often means
3a
and
(This can be ambiguous, since it is often convenient to use multi-character symbols. For
example, if we wrote the formula for the area of a circle as
Area   r 2
it is clear that the right-hand side means “multiply  times r2.” However, the symbol “Area” on the
left-hand side is a single symbol for the value of the area of the circle, and does not mean “A
times r time e times a.” In particular, this rule that adjacent symbols means multiplication is never
used for two or more numerical digits adjacent to each other. So “37” always means “thirtyseven” and never “3 times 7.” To indicate “3 times 7,” you would need to write “3  7” or (3)(7).)
The “Point” of Algebra
There are a number of important advantages to developing skill in working with literal symbols to
represent numbers. Among them are
(i) We can write numerical relationships very concisely. Thus, the formula
A  r 2
for the area of a circle is shorthand for the recipe: “to calculate the area of a circle,
raise its radius value to the second power and then multiply the result by the constant
3.141592653589793 or so.” The algebraic formula makes it much easier to see the
precise nature of the relationship between A and r here.
(ii) We are able to simulate arithmetic operations involving numbers whose values we
don’t know at present. This makes it possible to analyze and solve all sorts of
problems involving numerical values that would be impossible to solve by any other
means. The rules for carrying out this arithmetic with symbols are based directly on
the rules for doing arithmetic with ordinary numbers, because, after all, the symbols
just represent ordinary numbers. Most of “algebra” is just learning or practicing
strategies for doing this sort of symbolic arithmetic.
The development of science and technology, and most ongoing research and development work
today would not be possible without an ability to represent numerical values symbolically. Skill in
working with such symbolic representations is absolutely indispensable for working in nearly
every technological field today.
David W. Sabo (2003)
Literal Numbers
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