WHAT ARE RATIONALS & PROPERTIES ?

rational numbers

Rational numbers: A number that can be written in the form p/q, where p and q are integers and q is non zero. Thus, 2/3, 5/9, 31/57, -73/1134 are such examples of rational numbers.

In the view of decimal expansion

As we know that any fraction can be converted into decimal number upon dividing the numerator by its denominator. For example: 1/2 = 0.5, 1/3 = 0.33333....., 1/4 = 0.25, 14/11= 1.272727272....,  etc.

In above examples we have seen either the decimal expansion is terminating or repeating but what about the decimal expansion which are neither terminating nor repeating. About the later case we'll discuss later on. Now, let us summarise our results in the following form:

The decimal expansion of a rational number is either terminating or non terminating recurring (repeating). Moreover, a number whose decimal expansion is terminating or non-terminating recurring is rational.

               The collection of rational numbers is denoted by Q.

Properties of rational numbers:

• All fractions are rational numbers.
• All natural numbers are rational numbers but all rational numbers are not natural numbers.
• The integer 0 is a rational number.
• All integers are rational numbers.
• Here we can't say what is next rational number, i.e. we can't say successor or predecessor of a rational number as there are infinitely many rational numbers, even countably infinite.
• Rationals have a denseness property, i.e. the closure of a collection of rational numbers is closed.
• Between any two given rational numbers, we can always find another rational number and hence infinitely many rationals between any two given rationals.