Amusingly, the answers to both of the prompts "proof of the existence of irrational numbers" and "proof of the nonexistence of irrational numbers" are completel

Amusingly, the answers to both of the prompts "proof of the existence of irrational numbers" and "proof of the nonexistence of irrational numbers" are completely wrong, but the latter is considerably worse: > The proof of the nonexistence of irrational numbers is a mathematical proof that the real numbers cannot be partitioned into two sets such that all elements of one set are irrational numbers and all elements of the other set are rational numbers. In other words, every real number is either rational or irrational. > Assume there exists an irrational number a and a rational number b such that a = b. Since a is irrational, it can be written as a = c/d where c and d are integers and d is nonzero. Thus a = b = c/d, so c/d is a rational number. But this is a contradiction because the rational numbers are closed under division, so c/d is rational if and only if c and d are both rational. Since d is nonzero, it cannot be rational