NCERT Class X Chapter 1: Real Numbers Exercise 1.2 Question 3(i)
NCERT Class X Chapter 1: Real Numbers Exercise 1.2 Question 3(i)
Question:
Prove that \( \frac{1}{\sqrt{2}} \) is irrational.
Given:
The number \( \frac{1}{\sqrt{2}} \).
To Find:
Prove that \( \frac{1}{\sqrt{2}} \) is irrational.
Formula:
We use the definitions of rational and irrational numbers.
Definitions:
- Rational Number: A number that can be written as \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).
- Irrational Number: A number that cannot be written as the ratio of two integers.
Solution:
Step 1: Assume, for contradiction, that \( \frac{1}{\sqrt{2}} \) is rational.
Step 2: By definition, it can be written as \( \frac{p}{q} \), where \( p \) and \( q \) are integers, \( q \neq 0 \), and \( \gcd(p, q) = 1 \).
$$ \frac{1}{\sqrt{2}} = \frac{p}{q} $$Step 3: Cross-multiply to eliminate denominators.
$$ 1 \times q = p \times \sqrt{2} $$ $$ q = p \sqrt{2} $$Step 4: Divide both sides by \( p \) (assuming \( p \neq 0 \)) to solve for \( \sqrt{2} \).
$$ \frac{q}{p} = \sqrt{2} $$Step 5: Here, \( \frac{q}{p} \) is a ratio of two integers, so it is rational. This would mean \( \sqrt{2} \) is rational.
Step 6: But it is a well-known fact that \( \sqrt{2} \) is irrational. This is a contradiction.
Step 7: Therefore, our assumption is false. Thus, \( \frac{1}{\sqrt{2}} \) is irrational.
Result:
\( \frac{1}{\sqrt{2}} \) is irrational.
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