May 25th
May 25th
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. In fact, it isn’t too uncommon to use
special symbols which are not part of any alphabet. 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
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:
| + |
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| - |
“minus” or “subtract”
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| × |
“times” or “multiply” |
| ÷, / |
“divide by” |
| = |
“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
and
3a often means 3 × a
(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
it is clear that the right-hand side means “multiply
times r 2 ”. 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).)
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