NCERT Class X Chapter 1: Real Numbers Example 7

NCERT Class X Chapter 1: Real Numbers Example 7

Question:

Show that \(3\sqrt{2}\) is irrational.

Given:

The number \(3\sqrt{2}\).

To Find:

To prove that \(3\sqrt{2}\) is irrational.

Formula:

We use proof by contradiction and the property that the product of a non-zero rational number and an irrational number is irrational.

Solution:

Step 1: Assume, to the contrary, that \(3\sqrt{2}\) is rational.

$$ 3\sqrt{2} = \frac{a}{b} $$

where \(a\) and \(b\) are coprime integers, \(b \neq 0\).

Step 2: Divide both sides by 3 to isolate \(\sqrt{2}\).

$$ \sqrt{2} = \frac{a}{3b} $$

Step 3: Note that \(\frac{a}{3b}\) is a rational number, since \(a\) and \(b\) are integers and \(3b \neq 0\).

Step 4: But it is a well-known fact that \(\sqrt{2}\) is irrational.

Step 5: This leads to a contradiction, since \(\sqrt{2}\) cannot be rational.

Step 6: Therefore, our assumption is false. Thus, \(3\sqrt{2}\) is irrational.

Result:

\(3\sqrt{2}\) is irrational. It cannot be written as the ratio of two integers.

© Kaliyuga Ekalavya. All rights reserved.

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