CHAPTER 9
FUNDAMENTALS OF ALGEBRA
The numbers and operating rules of arithmetic form a part of a very important branch
of mathematics called ALGEBRA. Algebra extends the concepts of arithmetic
so that it is possible to generalize the rules for operating with numbers and use these rules in
manipulating symbols other than numbers. It does not involve an abrupt change into a
distinctly new field, but rather provides a smooth transition into many branches of mathematics
with a continuation of knowledge already gained in basic arithmetic.
The idea of expressing quantities in a general way, rather than in the specific terms of
arithmetic, is fairly common. A typical example is the formula for the perimeter of a
rectangle, P = 2L + 2W, in which the letter P represents perimeter, L represents length, and W
represents width. It should be understood that
2L = 2(L) and 2W = 2(W). If the L and the W
were numbers, parentheses or some other multiplication sign would be necessary, but the
meaning of a term such as 2L is clear without
additional signs or symbols.
All formulas are algebraic expressions, although they are not always identified as such.
The letters used in algebraic expressions are
often referred to as LITERAL NUMBERS (literal implies "letteral").
Another typical use of literal numbers is in
the statement of mathematical laws of operation.
For example, the commutative, associative, and
distributive laws, introduced in chapter 3 with
respect to arithmetic, may be restated in general terms by the use of algebraic symbols.
COMMUTATIVE LAWS
The word "commutative" is defined in chapter 3. Remember that the commutative laws
refer to those situations in which the factors
and terms of an expression are rearranged in a
different order.
ADDITION
The algebraic form of the commutative law
for addition is as follows:
a+b=b+a
From this law, it follows that
a + (b + c) = a + (c + b) = (c + b) + a
In words, this law states that the sum of two or
more addends is the same regardless of the
order in which the addends are arranged.
The arithmetic example in chapter 3 shows
only one specific numerical combination in
which the law holds true. In the algebraic example, a, b, and c represent any numbers we
choose, thus giving a broad inclusive example
of the rule. (Note that once a value is selected
for a literal number, that value remains the
same wherever the letter appears in that particular example or problem. Thus, if we give a
the value of 12, in the example just given, as
value is 12 wherever it appears.)
MULTIPLICATION
The algebraic form of the Commutative law
for multiplication is as follows:
ab = ba
In words, this law states that the product of
two or more factors is the same regardless of
the order in which the factors are arranged.
ASSOCIATIVE LAWS
The associative laws of addition and multiplication refer to the grouping (association) of
terms and factors in a mathematical expression.
ADDITION
The algebraic form of the associative law
for addition is as follows:
a+b+c=(a+b)+c=a+(b+c)
In words, this law states that the sum of three
or more addends is the same regardless of the
manner in which the addends are grouped.
MULTIPLICATION
The algebraic form of the associative law
for multiplication is as follows:
a · b · c = (a ·
b) · c = a · (b ·
c)
In words, this law states that the product of
three or more factors is the same regardless
of the manner in which the factors are grouped.
DISTRIBUTIVE LAW
The distributive law refers to the distribution of factors among the terms of an additive
expression. The algebraic form of this law is
as follows:
a(b + c) = ab + ac
From this law, it follows that: If the sum of two
or more quantities is multiplied by a third
quantity, tine product is found by applying the
multiplier to each of the original quantities
separately and summing the resulting expressions.
ALGEBRAIC SUMS
The word "sum" has been used several times
in this discussion, and it is important to realize
the full implication where algebra is concerned.
Since a literal number may represent either a
positive or a negative quantity, a sum of several literal numbers is always understood to be
an ALGEBRAIC SUM. That is, it is the sum
that results when the algebraic signs of all the
addends are taken into consideration.
The following problems illustrate the procedure for finding an algebraic sum:
Let a = 3, b = 2, and c = 4.
Then a + b + c = (3) + (2) + (4)
= 5
Also, a  b  c = a + (b) + (c)
= 3 + (+2) + (4)
=1
The second problem shows that every expression containing two or more terms to be
combined by addition and subtraction may be rewritten as an algebraic sum, all negative signs
being considered as belonging to specific terms
and all operational signs being positive.
It should be noted, in relation to this subject,
that the laws of signs for algebra are the same
as those for arithmetic.
