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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