The basic idea of data abstraction is to structure programs so that they operate on abstract data.

**Representation of Abstract datatype using Rational
numbers**

The basic idea of data abstraction is to
structure programs so that they operate on abstract data. That is, our programs
should use data in such a way, as to make as few assumptions about the data as
possible. At the same time, a concrete data representation is defined as an
independent part of the program.

**Note**

In
concrete data representation, a definition for each function is known

Any program consist of two parts. **The two parts of a program are, the part
that operates on abstract data and the part that defines a concrete
representation**, is** **connected by
a small set of functions that implement abstract data in terms of the concrete
representation. To illustrate this technique, let us consider an example to
design a set of functions for manipulating rational numbers.

**Example**

A rational number is a ratio of integers, and
rational numbers constitute an important sub-class of real numbers. A rational
number such as 8/3 or 19/23 is typically written as:

<numerator>/<denominator>

where both the <numerator> and
<denominator> are placeholders for integer values. Both parts are needed
to exactly characterize the value of the rational number. Actually dividing
integers produces a float approximation, losing the exact precision of integers.

8/3 =2.6666666666666665

However, you can create an exact representation
for rational numbers by combining together the numerator and denominator.

As we know from using functional abstractions,
we can start programming productively before you have an implementation of some
parts of our program. Let us begin by assuming that you already have a way of
constructing a rational number from a numerator and a denominator. You also
assume that, given a rational number, you have a way of selecting its numerator
and its denominator component. Let us further assume that the constructor and
selectors are also available.

We are using here a powerful strategy for
designing programs: **'wishful thinking'**.
We haven't yet said how a rational number is represented, or how the constructor
and selectors should be implemented.

**Note**

Wishful Thinking is the formation of beliefs
and making decisions according to what might be pleasing to imagine instead of
by appealing to reality.

**Example: An ADT for rational numbers**

- constructor

- constructs a rational number
with numerator n, denominator d

**rational(n, d)**

- selector

numer(x) → returns the numerator
of rational number x

denom(y) → returns the
denominator of rational number y

Now you have the operations on rational numbers
defined in terms of the selector functions numer and denom, and the constructor
function rational, but you haven't yet defined these functions. What you need
is some way to glue together a numerator and a denominator into a compound
value.

The pseudo code for the representation of the
rational number using the above constructor and selector is

x,y:=8,3

rational(n,d)

numer(x)/numer(y)

- - output : 2.6666666666666665

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12th Computer Science : Chapter 2 : Data Abstraction : Representation of Abstract datatype using Rational numbers |

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