Revisiting Irrational Numbers

Revisiting Irrational Numbers

In Class IX, you were introduced to irrational numbers and many of their properties. You studied about their existence and how the rationals and the irrationals together made up the real numbers. You even studied how to locate irrationals on the number line. However, we did not prove that they were irrationals. In this section, we will prove that 2, 3, 5  and, in general, p  is irrational, where p is a prime. One of the theorems, we use in our proof, is the Fundamental Theorem of Arithmetic.
Recall, a number‘s’ is called irrational if it cannot be written in the form  where p and q are integers and q ≠ 0. Some examples of irrational numbers, with which you are already familiar, are:

Before we prove that 2 is irrational, we need the following theorem, whose proof is based on the Fundamental Theorem of Arithmetic.

Revisiting Rational Numbers and Their Decimal Expansions

In Class IX, you studied that rational numbers have either a terminating decimal expansion or a non-terminating repeating decimal expansion. In this section, we are going to consider a rational number, say

,  and explore exactly when the decimal expansion of   is terminating and when it is non-terminating and when it is non-terminating repeating (or recurring). We do so by considering several examples.

Let us consider the following rational numbers:

(i) 0.375       (ii) 0.104        (iii) 0.0875          (iv) 23.3408.

As one would expect, they can all be expressed as rational numbers whose denominators are powers of 10. Let us try and cancel the common factors between the numerator an denominator and see what we get:

Theorem: Let x be a rational number whose decimal expansion terminates.
Then x can be expressed in the form

,  where p and q are coprime, and the prime factorization of q is of the form 2ⁿ5^m, where n, m are non-negative integers.
Theorem:   Let   be a rational number, such that the prime factorization of q is of the form 2ⁿ5^m, where n, m are non-negative integers. Then x has a decimal expansion which terminates.