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Prove that 1/root 2 is irrational?
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Prove that 1/root 2 is irrational?
Proof that 1/√2 is irrational:
- Assumption:
- Squaring both sides:
- Consequences of squaring:
- Consequences of q^2 = k:
- Contradiction:
- Conclusion:
Let's assume that √2 is rational.
This means that it can be expressed as a fraction in the form of p/q, where p and q are integers with no common factors other than 1, and q is not equal to 0.
If we square both sides of the equation √2 = p/q, we get:
2 = (p^2)/(q^2)
2q^2 = p^2
From the equation 2q^2 = p^2, we can observe that p^2 is an even number since it is divisible by 2.
According to the fundamental theorem of arithmetic, any even number can be expressed as the product of 2 and another integer.
Thus, we can write p^2 as p^2 = 2k, where k is an integer.
Substituting this back into the equation, we get:
2q^2 = 2k
q^2 = k
From the equation q^2 = k, we can conclude that q^2 is also an even number.
Following the same reasoning as before, we can express q^2 as q^2 = 2m, where m is an integer.
Substituting this back into the equation, we get:
2m = k
We have now concluded that p^2 = 2k and 2m = k, where both p and q are even numbers.
This implies that p and q have a common factor of 2, contradicting our initial assumption that p and q have no common factors other than 1.
Therefore, our assumption that √2 is rational must be false.
Since we have proved that √2 is irrational, we can conclude that 1/√2 is also irrational.
Summary:
By assuming that √2 is rational and following the logical consequences of squaring both sides, we arrived at a contradiction. This proves that √2 is irrational and, consequently, 1/√2 is also irrational.
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Prove that 1/root 2 is irrational?
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Prove that 1/root 2 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 that 1/root 2 is irrational? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prove that 1/root 2 is irrational?.
Prove that 1/root 2 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 that 1/root 2 is irrational? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prove that 1/root 2 is irrational?.
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