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Prove that 2 + 3 √ 5 is an irrational number, it is being given that √ 5
is an irrational number.?
is an irrational number.?
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Prove that 2 + 3 √ 5 is an irrational number, it is being given that √...
Proof that 2 + 3√5 is an irrational number
Assumption:
Given that √5 is an irrational number.
Assume that 2 + 3√5 is a rational number
Definition of Rational Number:
A rational number is a number that can be expressed as a ratio of two integers.
Let's assume 2 + 3√5 is rational
Express 2 + 3√5 as a ratio of two integers:
Let 2 + 3√5 = a/b, where a and b are integers with no common factors.
Simplify the equation:
2 + 3√5 = a/b
3√5 = a/b - 2
3√5 = (a - 2b)/b
Square both sides:
(3√5)^2 = ((a - 2b)/b)^2
45 = (a^2 - 4ab + 4b^2)/b^2
45b^2 = a^2 - 4ab + 4b^2
Conclusion:
After squaring both sides, we have obtained a contradiction as the left-hand side is an integer (45b^2) while the right-hand side is not necessarily an integer. This implies that our assumption that 2 + 3√5 is a rational number is incorrect.
Therefore, 2 + 3√5 is an irrational number.
Assumption:
Given that √5 is an irrational number.
Assume that 2 + 3√5 is a rational number
Definition of Rational Number:
A rational number is a number that can be expressed as a ratio of two integers.
Let's assume 2 + 3√5 is rational
Express 2 + 3√5 as a ratio of two integers:
Let 2 + 3√5 = a/b, where a and b are integers with no common factors.
Simplify the equation:
2 + 3√5 = a/b
3√5 = a/b - 2
3√5 = (a - 2b)/b
Square both sides:
(3√5)^2 = ((a - 2b)/b)^2
45 = (a^2 - 4ab + 4b^2)/b^2
45b^2 = a^2 - 4ab + 4b^2
Conclusion:
After squaring both sides, we have obtained a contradiction as the left-hand side is an integer (45b^2) while the right-hand side is not necessarily an integer. This implies that our assumption that 2 + 3√5 is a rational number is incorrect.
Therefore, 2 + 3√5 is an irrational number.
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Prove that 2 + 3 √ 5 is an irrational number, it is being given that √ 5is an irrational number.?
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Prove that 2 + 3 √ 5 is an irrational number, it is being given that √ 5is an irrational number.? 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 2 + 3 √ 5 is an irrational number, it is being given that √ 5is an irrational number.? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prove that 2 + 3 √ 5 is an irrational number, it is being given that √ 5is an irrational number.?.
Prove that 2 + 3 √ 5 is an irrational number, it is being given that √ 5is an irrational number.? 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 2 + 3 √ 5 is an irrational number, it is being given that √ 5is an irrational number.? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prove that 2 + 3 √ 5 is an irrational number, it is being given that √ 5is an irrational number.?.
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