Issqrt 2arationalnumber? Choices: (A)yes (B)no

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Issqrt 2arationalnumber?    Choices: (A)yes (B)no

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Definition of Rational Numbers: A rational number is a number that can be expressed as the quotient of two integers, where the denominator is not zero. To determine if sqrt(2) is rational, we need to check if it can be written in the form of a fraction with integer numerator and denominator.Assumption of sqrt(2) as Rational: The number sqrt(2) is known to be the positive solution to the equation x^2 = 2. If sqrt(2) were rational, it could 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.Squaring the Equation: Assuming sqrt(2) is rational, we would have sqrt(2) = a/b. Squaring both sides of the equation gives us 2 = a^2/b^2. This implies that a^2 = 2b^2.Evenness of a: The equation a^2 = 2b^2 suggests that a^2 is an even number since it is two times another integer. Therefore, a must also be even because the square of an odd number is odd.Substitution and Simplification: If a is even, then we can write it as a = 2k, where k is an integer. Substituting this into the equation a^2 = 2b^2 gives us (2k)^2 = 2b^2, which simplifies to 4k^2 = 2b^2, and then to 2k^2 = b^2.Evenness of b: The equation 2k^2 = b^2 implies that b^2 is even, and hence b must also be even. This means that both a and b have at least the factor 2 in common, which contradicts our initial assumption that a and b have no common factors other than 1.Contradiction and Conclusion: Since our assumption that sqrt(2) is rational leads to a contradiction, we conclude that sqrt(2) cannot be expressed as a fraction of two integers. Therefore, sqrt(2) is not a rational number.

Is $\sqrt{2}$ a rational number? $\newline$ Choices: $\newline$ (A) yes $\newline$ (B) no

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