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Suppose I have two real numbers $a$ and $b$ with decimal expansions

$a = \sum_{i=0}^{N_a} a_i 10^i + \sum_{i=1}^\infty a_{-i} 10^{-i}$ and $b = \sum_{i=0}^{N_b} b_i 10^i + \sum_{i=1}^\infty b_{-i} 10^{-i}$ ($a_i, b_i \in \{0,\dots,9\}\; \forall i$).

Is there a general closed formula for the decimal expansions of $a +b$, $a -b$, $a\cdot b$ and $a/b$ (if defined)?

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Nothing nice. That's why defining the reals via infinite decimals expansions, or infinite binary expansions, is not a good way to proceed. –  André Nicolas Oct 26 '12 at 14:52

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