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 Wrepresents 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 inthe 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.
The word "commutative" is defined in chapter 3. Remember that the commutative lawsrefer to those situations in which the factors and terms of an expression are rearranged in a different order.
The algebraic form of the commutative lawfor addition is as follows:
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 ormore 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.)
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.
The associative laws of addition and multiplication refer to the grouping (association) of terms and factors in a mathematical expression.
The algebraic form of the associative law for addition is as follows:
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.
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.
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.
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)
Also, a - b - c = a + (-b) + (-c)
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.