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Prove that √3 + √5 is irrational for class 10 CBSE NCERT maths 5 marks?
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Prove that √3 + √5 is irrational for class 10 CBSE NCERT maths 5 marks...
To prove that √3 + √5 is irrational, we will use the method of contradiction.
Assumption
- Assume that √3 + √5 is rational.
- This means it can be expressed as a fraction:
√3 + √5 = p/q
where p and q are integers, and q ≠ 0.
Rearranging the Equation
- Rearranging gives us:
√3 = p/q - √5
- Squaring both sides leads to:
3 = (p/q - √5)²
- This expands to:
3 = (p²/q²) - 2(p/q)√5 + 5
- Rearranging gives us:
2(p/q)√5 = (p²/q² + 5 - 3)
- Simplifying further results in:
2(p/q)√5 = (p² + 2)/q²
Isolating √5
- Isolate √5:
√5 = (q²/(2p))(p/q + 2)
- Here, (q²/(2p))(p/q + 2) is rational since it is expressed as the ratio of integers.
Contradiction Arises
- Since √5 is known to be irrational, this creates a contradiction.
- Our assumption that √3 + √5 is rational must therefore be false.
Conclusion
- Hence, we conclude that √3 + √5 is irrational.
This proof highlights the fundamental property of rational and irrational numbers, reinforcing the understanding of number theory.
Assumption
- Assume that √3 + √5 is rational.
- This means it can be expressed as a fraction:
√3 + √5 = p/q
where p and q are integers, and q ≠ 0.
Rearranging the Equation
- Rearranging gives us:
√3 = p/q - √5
- Squaring both sides leads to:
3 = (p/q - √5)²
- This expands to:
3 = (p²/q²) - 2(p/q)√5 + 5
- Rearranging gives us:
2(p/q)√5 = (p²/q² + 5 - 3)
- Simplifying further results in:
2(p/q)√5 = (p² + 2)/q²
Isolating √5
- Isolate √5:
√5 = (q²/(2p))(p/q + 2)
- Here, (q²/(2p))(p/q + 2) is rational since it is expressed as the ratio of integers.
Contradiction Arises
- Since √5 is known to be irrational, this creates a contradiction.
- Our assumption that √3 + √5 is rational must therefore be false.
Conclusion
- Hence, we conclude that √3 + √5 is irrational.
This proof highlights the fundamental property of rational and irrational numbers, reinforcing the understanding of number theory.
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Prove that √3 + √5 is irrational for class 10 CBSE NCERT maths 5 marks?
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Prove that √3 + √5 is irrational for class 10 CBSE NCERT maths 5 marks? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Prove that √3 + √5 is irrational for class 10 CBSE NCERT maths 5 marks? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prove that √3 + √5 is irrational for class 10 CBSE NCERT maths 5 marks?.
Prove that √3 + √5 is irrational for class 10 CBSE NCERT maths 5 marks? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Prove that √3 + √5 is irrational for class 10 CBSE NCERT maths 5 marks? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prove that √3 + √5 is irrational for class 10 CBSE NCERT maths 5 marks?.
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