Neighbourhoods

To continue our discussion, we need certain prerequisites namely, constants, variables, independent variables, dependent variables, intervals and neighbourhoods. . . Constants and Variables A quantity that remains unaltered throughout a mathematical process is called a constant .

A quantity that varies in a mathematical process is called a variable . A variable is an independent variable when it takes any arbitrary (independent) value not depending on any other variables, whereas if its value depends on other variables, then it is called a dependent variable . We know the area A of a triangle is given by A = bh . Here is a constant and A, b, h are variables.

Moreover b and h are independent variables and A is a dependent variable. We ought to note that the terms dependent and independent are relative terms. For example in the equation x + y = , x, y are variables and is a constant. Which of x and y is dependent and which one is independent?

If we consider x as an independent variable, then y becomes dependent whereas if we consider y as an independent variable, then x becomes dependent. Further consider the following examples: (i) area of a rectangle A = ℓb . (ii) area of a circle A = πr . (iii) volume of a cuboid V = ℓbh.

From the above examples we can directly infer that b, h, ℓ, r are independent variables; A and V are dependent variables and π is a constant. . . Intervals and Neighbourhoods The system R of real numbers can be represented by the points on a line and a point on the line can be related to a unique real number as in Figure .

. By this, we mean that any real number can be identified as a point on the line. With this identification we call the line as the real line . – – – e –