(5.4.25) Flawed Mathematical Induction Proof That All Powers of Nonzero Real Numbers Equal 1

This presentation examines a flawed mathematical proof which claims that every nonnegative integer power of any nonzero real number equals 1. The proof starts by defining a property P(n) that posits r to the power of n equals 1 for any nonzero real number r and for all nonnegative integers n. The proof begins correctly by confirming that P(0) is valid, as r raised to the power of 0 is universally 1, following the zero exponent rule. The purported proof then attempts to use strong induction, stating that if P(i) is true for all integers i from 0 to k, then P(k+1) must also be true. It incorrectly manipulates exponents to demonstrate that r raised to the power of k+1, calculated as r to the power of k times r, equals 1 by misapplying the inductive hypothesis. The critical error occurs when this method is applied starting from k=0, leading to an unverified conclusion that r to the power of -1 equals 1, which does not logically follow from the given assumptions. This session outlines the necessity of careful verification at each step of inductive reasoning, especially in the handling of base cases linked to broader inductive claims.

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