prove that 1/root 2 is irrational

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Assume Rational Number: Assume that 1/√2 is a rational number. According to the definition of rational numbers, if a number is rational, it can be expressed as a fraction a/b, where a and b are integers with no common factors other than 1, and b is not zero. Let's express 1/√2 as a/b, where a and b are integers with no common factors. 1/√2 = a/bExpress as Fraction: To remove the square root from the denominator, multiply both sides of the equation by √2. (1/√2) * √2 = (a/b) * √2 √2/2 = a√2/bEliminate Square Root: Now, square both sides of the equation to eliminate the square root. (√2/2)^2 = (a√2/b)^2 2/4 = 2a^2/b^2 1/2 = a^2/b^2Square Both Sides: Multiply both sides by b^2 to get rid of the fraction on the right side. b^2/2 = a^2Substitute b=2k: This implies that b^2 is even since it is equal to 2 times some integer (a^2). If b^2 is even, then b must also be even because the square of an odd number is odd. Let's say b = 2k, where k is an integer.Find a Contradiction: Substitute b = 2k back into the equation b^2/2 = a^2. (2k)^2/2 = a^2 4k^2/2 = a^2 2k^2 = a^2This implies that a^2 is even, and therefore a must also be even. Let's say a = 2m, where m is an integer.We now have a contradiction because we assumed that a and b have no common factors other than 1, but we have shown that both a and b are even, which means they both have at least the common factor 2. This contradiction means our initial assumption that 1/√2 is rational is incorrect. Therefore, 1/√2 must be irrational.

prove that $\frac{1}{\sqrt{2}}$ is irrational

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prove that $\frac{1}{\sqrt{2}}$ is irrational