What are Rational and Irrational Numbers with Examples ?

Here you will learn what are rational and irrational numbers with examples.

Let’s begin –

What are Rational and Irrational Numbers ?

Rational Number

A rational number is defined as number of the form a/b where a and b are integers and b \(\ne\) 0.

The set of rational numbers encloses the set of integers and fractions.

Rational numbers that are not integral will have decimal values. These values can be of two types :

(a) Terminating (or finite) decimal fractions : For example, 17/4 = 4.25, 21/5 = 4.2 and so forth.

(b) Non-terminating decimal fractions : Amongst non-terminating decimal fractions there are two types of decimal values.

(i) Non-terminating periodic fractions : These are non-terminating decimal fractions of the type \(x.a_1a_2….a_na_1a_2….a_n\). For example \(16\over 3\) = 5.3333, 15.23232323, 14.28762876….  and so on.

(ii) Non-terminating non-periodic fractions : These are of the form \(x.b_1b_2….b_nc_1c_2…c_n\). For example, 5.273168143725186….

Of the above categories,terminating decimal and non-terminating periodic decimal fractions belong to the set of rational numbers.

Irrational Number

Fractions, that are non-terminating, non-periodic fractions are called irrational numbers.

Some examples of irrational numbers are \(\sqrt{2}\), \(\sqrt{3}\) etc. In other words, all square and cube roots of the natural numbers that are not squares and cubes of natural numbers are irrational. Other irrational numbers include \(\pi\), e and so on.

Every positive irrational number has a negative irrational number corresponding to it.

All operations of addition, subtraction, multiplication and division applicable to rational numbers are also applicable to irrational numbers.

You should realise that once an irrational numbers appears in the solution of question, it can only disappear if it is multiplied or divided by the same irrational number.

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