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For each of the statements below state whether it is True or Fake. If True, then give a proof. If Fake then explain why, e.g., by giving a counterexample The sum of the squares of any two rational numbers is a rational number For all positive x R, if x is irrational then squareroot x is irrational (c) For all x, y K, if x and y are irrational then x^2 + y is irrational log_10(2) is irrational

Solution

(a) True

Proof : (5)² + (-0.1)² = 25 + 0.01 = 25.01 = 2501/1000

So the sum of the squares of any two rational numbers is a rational number.

(b) True

Proof : x = Pi = 3.1415926535897932384626433832795 (and more ...) is irrational number then root of x is also irrational number because it cannot be written as a simple fraction.

(c) True

Proof : x and y are irrational.

x = Pi = 3.1415926535897932384626433832795 (and more ...)

y = e (euler\'s numner) = 2.7182818284590452353602874713527 (and more ...)

then x² + y is also irrational because it cannot be written as a simple fraction.

(d) True

Proof : log₁₀(2) = 0.30102999566...(and more)

In decimal places there is no pattern so we can not express this as a fraction so it is a irrational number.