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The square of a irrational number is always irrational true or false justify your answer?
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The square of a irrational number is always irrational true or false j...
Explanation:
Statement: The square of an irrational number is always irrational.
Justification:
Definition of Irrational Numbers:
An irrational number is a number that cannot be expressed as a simple fraction or ratio of two integers. It is a non-repeating, non-terminating decimal.
Proof by Contradiction:
Let's assume that the square of an irrational number is rational.
So, if we have an irrational number "a" and its square is rational, then a^2 is rational.
Assumption:
Let's say a^2 = b, where a is irrational and b is rational.
Expressing the Square:
If we take the square root of b, we get a = √b.
Since √b is a square root of a rational number, it can be expressed as a fraction, which contradicts the definition of an irrational number.
Conclusion:
Therefore, the assumption that the square of an irrational number is rational leads to a contradiction. Hence, the square of an irrational number is always irrational.
Statement: The square of an irrational number is always irrational.
- True: The statement is true.
Justification:
Definition of Irrational Numbers:
An irrational number is a number that cannot be expressed as a simple fraction or ratio of two integers. It is a non-repeating, non-terminating decimal.
Proof by Contradiction:
Let's assume that the square of an irrational number is rational.
So, if we have an irrational number "a" and its square is rational, then a^2 is rational.
Assumption:
Let's say a^2 = b, where a is irrational and b is rational.
Expressing the Square:
If we take the square root of b, we get a = √b.
Since √b is a square root of a rational number, it can be expressed as a fraction, which contradicts the definition of an irrational number.
Conclusion:
Therefore, the assumption that the square of an irrational number is rational leads to a contradiction. Hence, the square of an irrational number is always irrational.
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The square of a irrational number is always irrational true or false justify your answer?
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The square of a irrational number is always irrational true or false justify your answer? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The square of a irrational number is always irrational true or false justify your answer? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The square of a irrational number is always irrational true or false justify your answer?.
The square of a irrational number is always irrational true or false justify your answer? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The square of a irrational number is always irrational true or false justify your answer? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The square of a irrational number is always irrational true or false justify your answer?.
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