My intuitive way of thinking about it is that it is $2/2/2$ or $2/2^2$, So why then is it $1/2^2$? what is the flaw in my thinking?


Repeated multiplication can be seen as$$ overset extm termsa cdots a = a^m.$$

Dividing this term repeatedly can then be seen as subtracting from $a^m$, because$$ fraca^ma = fracoverset extm termsa cdots aa = overset extm - 1 termsa cdots a = a^m-1.$$

So $$frac12^2 = frac2^02^2 = 2^-2.$$

Well, $2^-n := 1 over 2^n (n geq 0)$ is a definition.

Why this definition?

Because the usual rules (e.g. $2^n+m=2^n2^m$) which were established for ($n,min magdalenarybarikova.combbN$) are now true for all $n,m in magdalenarybarikova.combbZ$. Which is nice, và shortens a lot of proofs.

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$2^2 = 2cdot 2 = 4$, $2^1 = 2$, $2^0 = 1$, $2^-1 = 1/2$, $2^-2 = 1/4$.

I guess the problem in your way of thinking is that when you"re thinking about multiplication, you should "start" from $1$, whereas you"re "starting" from $2$. The first $2$ in your expression $2/2/2$ is playing a different role than the other two $2$s - it should really be a $1$.


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