NCERT Class X Chapter 1: Real Numbers Exercise 1.2 Question 3(ii)
NCERT Class X Chapter 1: Real Numbers Exercise 1.2 Question 3(ii)
Question:
Prove that \( 7\sqrt{5} \) is irrational.
Given:
The number \( 7\sqrt{5} \).
To Find:
Prove that \( 7\sqrt{5} \) is irrational.
Formula:
We will use the definition of rational and irrational numbers.
Definitions:
- Rational Number: A number that can be written in the form \( \frac{p}{q} \), 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 that \( 7\sqrt{5} \) is rational. Then it can be written as \( \frac{p}{q} \), where \( p \) and \( q \) are integers, \( q \neq 0 \), and \( \gcd(p, q) = 1 \).
$$ 7\sqrt{5} = \frac{p}{q} $$Step 2: Divide both sides by 7 to isolate \( \sqrt{5} \).
$$ \sqrt{5} = \frac{p}{7q} $$Step 3: Since \( p \) and \( q \) are integers and \( q \neq 0 \), \( 7q \) is also a nonzero integer. Therefore, \( \frac{p}{7q} \) is a rational number.
Step 4: This means \( \sqrt{5} \) is rational. But it is a well-known fact that \( \sqrt{5} \) is irrational. This is a contradiction.
Step 5: Therefore, our assumption is wrong. So, \( 7\sqrt{5} \) is irrational.
Result:
Hence, \( 7\sqrt{5} \) is irrational.
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