# Rational Number – An Idea of Number System

Number System / Friday, November 22nd, 2019

# What is rational number?

A number of the form p/q, where p, q are integers, prime to each other and q ≠ 0 is called a rational number. Here q is generally assumed to be positive.

The set of rational numbers is denoted by Q. We observe that N ⊂ Z ⊂ Q.

But before diving into the concept let us understand how we came to know about rational numbers.

Let us consider the fourth fundamental operation – division. Given any two numbers x and y if a number z is uniquely determined by the relation x = yz (y ≠ 0), then z is called the quotient when x is divided by y and we write

$z=\frac{x}{y}~~\left( y\ne 0 \right)$

We know that 10 ÷ 2 = 5, (-35) ÷ 7 = -5. Here all the quotients are integers, positive or negative. But an integer divided by an integer (other than zero) does not always give an integer as quotient (3/7), (-52/7) are example of such situations.

To overcome this limitation of integers, new numbers – positive and negative fraction – were introduced. Thus the totality of all integers (positive, negative or zero) and fractions (positive or negative) is called the domain of rational numbers.

## Geometrical Representation of Rational Number

In this article the mode of representing rational numbers by points along a straight line or by segments of a straight line, which may called Number line, will be explained.

To start with, a straight line of indefinite length X’OX is drawn and an arbitrary point O on it is marked. The point O is called origin or the zero point. The number Zero (0) will be represented by the point O.

The point O divides the number line into two parts or sides. It is usual convention to take the portion of the number line on the right hand side of the origin O as positive, while the portion on the left hand side of O negative.

On the positive side of the number line, we take an arbitrary length OA, and call it the unit length. We say that the number 1 is represented by the point A. Thus a correspondence between the number 1 and the point A is established.

After having fixed an origin, positive sense and unit length on the number line, as indicated above, we are in a position to determine the unique point to represent a given rational number.

Let us consider a rational number p/q, where q is a positive integer. Let OA be divided into q equal parts, OQ being one of them. We take a point on the positive or negative side of O, according as p is positive or negative such that the distance of this point from O is p times the distance OQ. The point so obtained represents the number p/q. In the above figure, the point D represents the rational number 5/2 or 2.5.

 Example 01

Prove that √3 is not a rational number.

Solution:

Since 1 < 3 <4, 1< √3 < 2, which shows that √3 cannot be an integer.

Now, if possible, let 3 be a rational number and we assume √3 = p/q, where p and q are positive integers prime to each other and q > 1.

$\frac{{{p}^{2}}}{{{q}^{2}}}=3\Rightarrow \frac{{{p}^{2}}}{q}=3q…………(1)$

Since p and q are positive integers prime to each other, p2 and q are also positive integers prime to each other. Again since q > 1, p2/q represents a rational number which is not an integer, but 3q represents a positive integer. So from (1) we get a positive rational number which is not an integer, but 3q represent a positive integer. So, our initial assumption is not true, i.e., 3 cannot be a rational number.

 Example 02

If p is any prime number, show that p is not a rational number.

Solution:

If possible, let p be a rational number. Then we can express √p = m/n, where m and n are integers prime to each other.

Then m2 = pn2, so that m2 is a multiple of p which requires that m also a multiple of p, since p is a prime number.

Let, m = rp, for some r in Z.

Then

${{m}^{2}}=p{{n}^{2}}$

$\Rightarrow p{{n}^{2}}={{r}^{2}}{{p}^{2}}$

$\Rightarrow {{n}^{2}}=p{{r}^{2}}$

So n2 is a multiple of p and so n is also a multiple of p (since p is prime).

Thus we find that p is a common factor of both m and n. This contradicts the assumption that m and n are prime to each other, i.e., they have no common factor.

Hence our assumption that p is a rational number is not true.

 Example 03

Examine whether log10 5 is a rational number.

Solution:

If possible, let log10 5 represents a rational number and

${{\log }_{10}}5=\frac{p}{q}$

where p and q are positive integers prime to each other and q > 1.

Then, from definition of logarithm,

${{\left( 10 \right)}^{\frac{p}{q}}}=5\Rightarrow {{10}^{p}}={{5}^{q}}$

$\Rightarrow {{2}^{p}}{{.5}^{p}}={{5}^{q}}$

$\Rightarrow {{2}^{p}}={{5}^{q-p}}$

Since p and (q – p) are both positive integers and 2 and 5 are prime to each other, the above relation cannot hold.

Hence log10 5 cannot be rational, i.e., log10 5 is an irrational number.

;