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I don't get where the + 99...9dk + part comes from especially.

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    99...9 = 100...0 - 12013-01-23

3 Answers 3

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$10^k=10000...0=9999999...9+1$

thus

$10^kd_k=(9999999...9+1)d_k=999...9d_k+d_k \,.$

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Look at an example and maybe it becomes clear:

$2000 = 2\cdot 1000 = 2\cdot(1+999) = 2+2\cdot 999$

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    In my example, $d_k = 2$.2012-12-17
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He is just saying that $10^k=(10^k-1)+1=(999\ldots 9)+1$ For example, $10^3=999+1$. He is also factoring out $d_k$ in this step.