A long time ago I noticed that $2+2 = 2 \times 2 = 2^2$, which is pretty cool because it’s the 3 basic arithmetical operations. It then recently occurred to me to try to prove that $2$ is the only real number for which this is true. Here is what I came up with:
I start with this double equality:
$r+r = r \times r = r^r$ or rewritten as $r+r = r^2 = r^r$.
Just examining the first part of the double equality:
$$r+r=r^2,$$ I divide by $r$ and get:
$$1+1 = r \implies r=2.$$
Looking at the second part of the double equality:
$$r^2=r^r$$
I divide by $r^2$ and get:
$$1=r^{r-2}$$
Next, I take the logarithm of both sides:
$$\ln(1) = \ln\left(r^{r-2}\right) \implies 0 = (r-2)\ln(r).$$
The only numbers that make this true are $r=2$ and $r=1$, since substituting in any other real number would mean that two non-zero numbers multiplied together would make $0$, which is clearly false. Furthermore $r=1$ does not satisfy the first part of the double equality so it has to be $2$. QED.
I was pretty proud of myself for solving this (and yes I'm sure to most of you this is no big deal but I'm not a math person). However a few hours later a serious problem occurred to me. Going back to this step:
$$0=(r-2)\ln(r).$$
What if I divide both sides by $(r-2)\ln(r)$, then I get:
$$\dfrac{0}{(r-2)\ln(r)} = \dfrac{(r-2)\ln(r)}{(r-2)\ln(r)} \implies 0 = 1.$$
I can't explain this away as division by zero since it's in the numerator.
Can someone tell me what I'm doing wrong?
Thank you
Answer
If $0=(r-2)\ln(r)$, then you can't divide by $(r-2)\ln(r)$, since it is equal to $0$ so you would be dividing by $0$.
More generally, any time you divide by an expression, that step is only valid under the assumption that the expression is not equal to $0$. If the expression involves a variable, this may be true for some values of the variable.
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