Find the mistake in the following proof that purports to show that
every nonnegative integer power of every nonzero real number is 1.
Let $r$ be any nonzero real number and let the property $P(n)$ be the equation
"$r^n=1$".
Basis
The property is true for $n=0$ by definition of the zeroth power.
Hypothesis
Let $k>0$ be an integer and suppose that $r^i=1$ for all integers $i$ with $0\leq i <k.$
Now
$r^k = r^{(k-1)+(k-1)-(k-2)}$
$ = \frac{r^{(k-1)} \cdot r^{(k-1)}}{r^{(k-2)}}$ by laws of exponent
$= \frac{1* 1}{1}=1$ by hypothesis
Thus, $r^k=1 $ as was to be shown.
Since we have proved the basis step and inductive step, we conclude that
$r^n =1$ for all integers $n\geq1$.
I know it's obviously wrong, but the logic looks so convincing. Nonetheless, my guess is that it is misusing the laws of exponents; however, I don't know how to explain the deception of the proof in thorough detail.So, how is it wrong, and where?
Best Answer
You need two base cases ($n=0$ and $n=1$), since you're using $r^{k-2}=1$. The $n=1$ case is clearly false, so the proof is wrong.