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Prove that √𝑛 is not a rational number, if n is not a perfect square?
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Prove that √𝑛 is not a rational number, if n is not a perfect square?
Proof that √𝑛 is not a rational number, if n is not a perfect square
Assumption:
- Let's assume that √𝑛 is a rational number for a non-perfect square number n.
Definition of Rational Number:
- A rational number is a number that can be expressed as a fraction a/b, where a and b are integers and b is not equal to 0.
Expressing √𝑛 as a rational number:
- If √𝑛 is rational, then we can write it as √𝑛 = a/b where a and b have no common factors other than 1.
Squaring both sides:
- Squaring both sides of the equation, we get n = a^2/b^2.
Implication of n not being a perfect square:
- If n is not a perfect square, then n cannot be expressed as (a^2)/(b^2) for any integers a and b.
Contradiction:
- This leads to a contradiction as we assumed that √𝑛 is rational, but it cannot be expressed as a ratio of two integers when n is not a perfect square.
Conclusion:
- Therefore, we can conclude that if n is not a perfect square, then √𝑛 is not a rational number.
Assumption:
- Let's assume that √𝑛 is a rational number for a non-perfect square number n.
Definition of Rational Number:
- A rational number is a number that can be expressed as a fraction a/b, where a and b are integers and b is not equal to 0.
Expressing √𝑛 as a rational number:
- If √𝑛 is rational, then we can write it as √𝑛 = a/b where a and b have no common factors other than 1.
Squaring both sides:
- Squaring both sides of the equation, we get n = a^2/b^2.
Implication of n not being a perfect square:
- If n is not a perfect square, then n cannot be expressed as (a^2)/(b^2) for any integers a and b.
Contradiction:
- This leads to a contradiction as we assumed that √𝑛 is rational, but it cannot be expressed as a ratio of two integers when n is not a perfect square.
Conclusion:
- Therefore, we can conclude that if n is not a perfect square, then √𝑛 is not a rational number.
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Prove that √𝑛 is not a rational number, if n is not a perfect square?
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Prove that √𝑛 is not a rational number, if n is not a perfect square? 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 √𝑛 is not a rational number, if n is not a perfect square? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prove that √𝑛 is not a rational number, if n is not a perfect square?.
Prove that √𝑛 is not a rational number, if n is not a perfect square? 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 √𝑛 is not a rational number, if n is not a perfect square? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prove that √𝑛 is not a rational number, if n is not a perfect square?.
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