3 Arithmetic vs. Analysis

Let $\underset{¯}{x}\in {\mathbb{Z}}_{(10)}$ and $a_n{\to }_d\underset{¯}{x}$ be a convergent sequence of natural numbers. Also, let $\underset{¯}{y}\in {\mathbb{Z}}_{(10)}$ and $b_n{\to }_d\underset{¯}{y}$ be a convergent sequence of natural numbers. As we have seen in Problem 4. (Section 1.): ${\{a_n+b_n\}}_{n=1}^{\mathrm{\infty }}$ and ${\{a_n{b}_n\}}_{n=1}^{\mathrm{\infty }}$ are Cauchy-sequences, so they have limits $\underset{¯}{w}$ resp. $\underset{¯}{z}$. We define $\underset{¯}{x}+\underset{¯}{y}=\underset{¯}{w}$ and $\underset{¯}{x}\underset{¯}{y}=\underset{¯}{z}$. Problem 1. Show that the sum and the product are well-defined, that is, it does not depend on the particular choice of the convergent sequences. [This is much easier than the classical case we had in class!!] Problem 2. Subtraction is not defined on the natural numbers, $3-7$ is not a natural number. Surprisingly, subtraction can be defined easily for $10$-adic numbers. Show that for any pair $\underset{¯}{x},\underset{¯}{y}\in {\mathbb{Z}}_{(10)}$ there exists a unique $\underset{¯}{z}$ such that $\underset{¯}{y}+\underset{¯}{z}=\underset{¯}{x}$. [Hint: $1+\underset{¯}{x}=0$, what is $\underset{¯}{x}$?] Problem 3.

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  Show that there is no $\underset{¯}{x}\in {\mathbb{Z}}_{(10)}$ such that $10\underset{¯}{x}=1$.

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  On the other hand, there exists $\underset{¯}{x}\in {\mathbb{Z}}_{(10)}$ so that $17\underset{¯}{x}=1$. That is, one can define division by $17$ (or by any number that is relative prime to $10$) in the world of $10$-adic numbers. [Hint: show that for any $k\ge 1$ there exists a natural number $a_k$ such that $17a_k\equiv 1\text{mod}({10}^k)$.]