# Why does $(10^4 - 10^2) \cdot 0.0012121212\dots = 12$?

When you answer this question $(10^4 - 10^2) \cdot 0.0012121212\dots$ you get $12$. However, that seems to defy PEMDAS. Please explain. Doing PEMDAS wouldn't you get $(10^4 - 10^2)$ = $10^2$ and then multiply that by $0.0012121212\dots$?

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How is $10^4-10^2=10^4$? –  Isaac Jul 5 '12 at 23:24
It's not that either. $10000 - 100 = 9900$ –  Robert Israel Jul 5 '12 at 23:27
What is PEMDAS? –  Chris Eagle Jul 5 '12 at 23:27
@AsafKaragila: except that it really should be: Parentheses, Exponentiation, Multiplication and Division, Addition and Subtraction (and many students mislearn order of operations because of PEMDAS). –  Isaac Jul 5 '12 at 23:28
@AsafKaragila: It's very hard to keep track of bad/wrong mnemonic devices when one didn't learn them and instead learned the correct concept. :) –  Isaac Jul 5 '12 at 23:32

I figure it might be easier to see like this.

$10^4\times0.001212...=12.1212..$

$10^2\times0.001212...=0.1212...$

Now subtract.

$(10^4\times0.001212...)-(10^2\times0.001212...)=12$

Using the distributive property, we can rewrite this as

$(10^4-10^2)\times0.001212...=12$

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First, let $n=0.0012121212\dots$ so that $100n=0.12121212\dots$. Subtracting, $100n-n=99n=0.12=\frac{12}{100}$, so $n=\frac{12}{9900}$ (I'm intentionally not simplifying those fractions).
Now, as pointed out in the comments, $10^4-10^2=10000-100=9900$, so $$(10^4-10^2)(0.0012121212\dots)=9900\cdot\frac{12}{9900}=12.$$