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.