When an integer a is divided by another non-zero integer b. The following cases arise:

1. When a is a multiple of b

Suppose a = mb, where m is a natural number or integer, then

a/b = m

2. When a is not a multiple of b

In this case a/b is not an integer, and hence it is a new type of number. Such a number is called a rational number. Thus, a number which can be put in the form p/q, where p and q are integers and q ≠ 0, is called a rational number.

Positive and Negative Rational Numbers

A rational number p/q is said to be a positive rational number if p and q are both positive or both negative integers. If the integers p and q are of different signs, then p/q is said to be a negative rational number.

Important Results

  1. Every natural number is a rational number but the vice-versa is not always true.
  2. Every whole number and integer is a rational number but vice-versa is not always true.

Standard Form or Lowest Form

A rational number p/q, where p and q are integers and q ≠ 0, in which q is positive (or made positive) and p and q are co-prime (i.e. when they do not have a common factor other than 1 and –1) is said to be in standard form. A rational number in standard form is also referred to as a rational number in its lowest form.

For example, the standard form of the rational number 2/-3 is -2/3. The rational number 27/18 can be written as 3/2 in the standard form or the lowest form.

Equivalent Forms

A rational number can be written in an equivalent form by multiplying or dividing the numerator and denominator of the given rational number by the same number.

For example,

Comparison of Rational Numbers

In order to compare two rational numbers, follow any of the following methods:

  1. If two rational numbers, to be compared, have the same denominator, compare their numerators. The number having the greater numerator is the greater rational number. For example, for the two rational numbers 9/17 and 5/17, as 9 > 5, 9/17 is greater rational number.

  2. If two rational numbers are having different denominators, make their denominators equal by taking their equivalent form and then compare the numerators of the resulting rational numbers. The number having a greater numerator is greater rational number. For example, to compare two rational numbers 3/7 and 6/11, first make their denominators same. Numbers will be 33/77 and 42/77. As 42 > 33, 6/11 > 3/7.