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Prove 2-root 3 is irrational?
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Prove 2-root 3 is irrational?
Proof that 2-root 3 is irrational:
- Assumption: Assume that 2-root 3 is a rational number.
- Definition of Rational Number: Since 2-root 3 is a rational number, it can be expressed in the form of p/q, where p and q are integers and q is not equal to zero.
- Squaring: Squaring both sides of the equation 2-root 3 = p/q, we get:
- 4 - 4root3 + 3 = p^2/q^2
- 1 - 4root3 = p^2/q^2
- Multiplying by q^2: Multiplying both sides of the equation by q^2, we get:
- q^2 - 4root3q^2 = p^2
- Divisibility: Since q^2 is divisible by q, p^2 must also be divisible by q.
- Prime Factors: Let p^2 have prime factors of 2 and/or 3. Then, q^2 must also have factors of 2 and/or 3, as p^2 is divisible by q^2.
- Contradiction: This implies that p and q both have factors of 2 and/or 3, which contradicts the assumption that p and q are coprime.
- Conclusion: Therefore, the assumption that 2-root 3 is a rational number is false, and hence it is an irrational number.
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Prove 2-root 3 is irrational?
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Prove 2-root 3 is irrational? 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 2-root 3 is irrational? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prove 2-root 3 is irrational?.
Prove 2-root 3 is irrational? 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 2-root 3 is irrational? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prove 2-root 3 is irrational?.
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