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prove that root 2 plus root5 is an irrational number
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prove that root 2 plus root5 is an irrational number Related: Chapter...
Proof that √2 + √5 is an irrational number
Assume that √2 + √5 is a rational number.
Step 1: Definition of Rational Number
A rational number is a number that can be expressed as the ratio of two integers.
Step 2: Expressing √2 + √5 as a Ratio
Let √2 + √5 = p/q, where p and q are positive integers with no common factors.
Step 3: Squaring Both Sides
Squaring both sides of the equation (√2 + √5 = p/q), we get:
2 + 2√10 + 5 = p²/q²
7 + 2√10 = p²/q²
Step 4: Isolating √10
Isolating √10, we get:
√10 = (p² - 7q²)/2q²
Step 5: Checking for Even and Odd
Since p² - 7q² is even, it follows that √10 is also even.
Step 6: Division by 2
Dividing both sides of the equation (√10 = (p² - 7q²)/2q²) by 2, we get:
√10/2 = (p² - 7q²)/4q²
Step 7: Checking for Even and Odd
Since p² - 7q² is even, it follows that √10/2 is also even.
Step 8: Contradiction
This contradicts the fact that √10 is irrational, which means that our initial assumption (√2 + √5 is a rational number) is false.
Conclusion:
Therefore, we have proved that √2 + √5 is an irrational number.
Assume that √2 + √5 is a rational number.
Step 1: Definition of Rational Number
A rational number is a number that can be expressed as the ratio of two integers.
Step 2: Expressing √2 + √5 as a Ratio
Let √2 + √5 = p/q, where p and q are positive integers with no common factors.
Step 3: Squaring Both Sides
Squaring both sides of the equation (√2 + √5 = p/q), we get:
2 + 2√10 + 5 = p²/q²
7 + 2√10 = p²/q²
Step 4: Isolating √10
Isolating √10, we get:
√10 = (p² - 7q²)/2q²
Step 5: Checking for Even and Odd
Since p² - 7q² is even, it follows that √10 is also even.
Step 6: Division by 2
Dividing both sides of the equation (√10 = (p² - 7q²)/2q²) by 2, we get:
√10/2 = (p² - 7q²)/4q²
Step 7: Checking for Even and Odd
Since p² - 7q² is even, it follows that √10/2 is also even.
Step 8: Contradiction
This contradicts the fact that √10 is irrational, which means that our initial assumption (√2 + √5 is a rational number) is false.
Conclusion:
Therefore, we have proved that √2 + √5 is an irrational number.
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prove that root 2 plus root5 is an irrational number Related: Chapter 15 Probability Ex 15.2 Q5 Class 10 Maths NCERT?
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prove that root 2 plus root5 is an irrational number Related: Chapter 15 Probability Ex 15.2 Q5 Class 10 Maths NCERT? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about prove that root 2 plus root5 is an irrational number Related: Chapter 15 Probability Ex 15.2 Q5 Class 10 Maths NCERT? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for prove that root 2 plus root5 is an irrational number Related: Chapter 15 Probability Ex 15.2 Q5 Class 10 Maths NCERT?.
prove that root 2 plus root5 is an irrational number Related: Chapter 15 Probability Ex 15.2 Q5 Class 10 Maths NCERT? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about prove that root 2 plus root5 is an irrational number Related: Chapter 15 Probability Ex 15.2 Q5 Class 10 Maths NCERT? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for prove that root 2 plus root5 is an irrational number Related: Chapter 15 Probability Ex 15.2 Q5 Class 10 Maths NCERT?.
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