CHAPTER 3 : THE USE OF LITERAL SYMBOLS
Felix Klein (Germain mathematician ) said that "Real mathematics begins with operations with letters". The introduction of letters into mathematics helps us to think in more general terms. In arithmetic we deal with numbers like 1, 3, 5, … ½ , ¾ , ..... . Each number has a fixed, definite value.
In algebra, in addition to these numbers, we use letters like a, b, c, .... x, y, z etc to represent any number. Following examples illustrate this: -
Example 1 A man’s present age is 40 years.
His age before 3 years was 40 - 3
5 years hence will be 40 + 5
n years hence will be 40 + n
The use of the letter ‘ n ’ generalizes the statement about his age.
Example 2 The area of a rectangle
can be written as :
Area = length ´ breadth
Thus, for a rectangle with the length
20 m and breadth 10 m :
Area = 20 ´
10 = 200 sq.m
Now if l
is the length and b is the breadth, then
Area = l´ b .....
This is called a formula . This has two advantages.
Firstly, it is a general statement true for any particular set of
values of l and b. Secondly the
relation is neatly expressed. Thus the use of letters enables us
to express results in arithmetic in a general form. From this point
of view, Algebra may be considered as "generalized Arithmetic".
As discussed, these letters represent numbers, so they
must obey all the rules of signs and of the operations, such as addition,
subtraction, multiplication and division and also the properties of
these fundamental operations.
These letters, used to stand for numbers which are called ‘ Variables ’. They are used to transform verbal expressions into algebraic expressions.
1) The letters a, b, c, .... x, y, z used to denote numbers, as well as the signs +, - etc used to denote their relations to each other, are called ‘Symbols’.
2) Any collection of such symbols is called
an expression, thus 3 a2 - bc + 5a ¸
3 is an expression.
3) The parts of an expression connected by + or - signs are called the ‘terms’ of the expression. Thus 3 a2, bc and 5 a ¸ 3 are terms of the above expression.
4) Further, in 3a2, the numerical factor 3 is called the coefficient of a2 ; similarly 1 and 5 are coefficients of terms bc and a respectively.
5) The process of finding the numerical value of an expression when the letters in it stand for a given number is called ‘ substitution ’
H.C.F and L.C.M