# What is rational number?

A number of the form ** p/q**, where

**are integers, prime to each other and**

*p, q*

*q***is called a rational number. Here**

*≠ 0***is generally assumed to be positive.**

*q*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

**if a number**

*y***z**is

**uniquely**determined by the relation

*x = yz (y***, then z is called the**

*≠ 0)***quotient**when

**is divided by**

*x***and we write**

*y*\[z=\frac{x}{y}~~\left( y\ne 0 \right)\]

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

**on it is marked. The point**

*O***is called**

*O***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

**as positive, while the portion on the left hand side of**

*O***negative.**

*O*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

**. Thus a correspondence between the number 1 and the point**

*A**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

**be divided into**

*OA***equal parts,**

*q***being one of them. We take a point on the positive or negative side of**

*OQ***, according as**

*O***p**is positive or negative such that the distance of this point from

**is**

*O***times the distance**

*p***. The point so obtained represents the number**

*OQ***. In the above figure, the point D represents the rational number**

*p/q***.**

*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

**, where**

*√3 = p/q***and**

*p***are positive integers prime to each other and**

*q***.**

*q > 1*\[\frac{{{p}^{2}}}{{{q}^{2}}}=3\Rightarrow \frac{{{p}^{2}}}{q}=3q…………(1)\]

Since ** p** and

**are positive integers prime to each other,**

*q***and**

*p*^{2}**are also positive integers prime to each other. Again since**

*q***,**

*q > 1***represents a rational number**

*p*^{2}/q*which is not an integer*, but

**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.,**

*3q***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***, where**

*= m/n***and**

*m***are integers prime to each other.**

*n*Then ** m^{2} = pn^{2}**, so that

**is a multiple of**

*m*^{2}**which requires that**

*p***also a multiple of**

*m***, since**

*p***is a prime number.**

*p*Let, ** m = rp**, for some

**in**

*r***.**

*Z*Then

\[{{m}^{2}}=p{{n}^{2}}\]

\[\Rightarrow p{{n}^{2}}={{r}^{2}}{{p}^{2}}\]

\[\Rightarrow {{n}^{2}}=p{{r}^{2}}\]

So ** n^{2}** is a multiple of

**and so**

*p***is also a multiple of**

*n***(since**

*p***is prime).**

*p*Thus we find that ** p** is a common factor of both

**and**

*m***. This contradicts the assumption that**

*n***and**

*m***are prime to each other, i.e., they have no common factor.**

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

Example 03 |

**Examine whether log_{10} 5 is a rational number.**

**Solution:**

If possible, let ** log_{10} 5** represents a rational number and

\[{{\log }_{10}}5=\frac{p}{q}\]

where ** p** and

**are positive integers prime to each other and**

*q***.**

*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

**are both positive integers and 2 and 5 are prime to each other, the above relation cannot hold.**

*(q – p)*Hence ** log_{10} 5** cannot be rational, i.e.,

**is an irrational number.**

*log*_{10}5