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Judge Software Activation Code Card Number For All Microsoft Documents, Microsoft Student Plus Product Key.Q:

What is the equivalence statement for $\mathbb{Q}\leftrightarrow \mathbb{Q}_p$?

Is there an intuitive way to think about the following analogy?

When working in $\mathbb{C}$, we say that $z$ is equal to $\alpha$ (in $\mathbb{C}$) if and only if $z$ is equal to $\alpha$ (in $\mathbb{Q}$) and $z$ is equal to $\alpha$ (in $\mathbb{R}$).

A:

Yes. One way is if $z$ is rational, then $z = x/y$ for some integers $x,y$. Then $z$ is the rational number represented by the unique normal form string consisting of a single $0$ followed by the binary representations of $x$ and $y$.
In the complex numbers, we have $z = 0 + i = (0+0i)/(0+0i)$.
In the reals, we have $z = 0 + 0i = (0+0i)/1+0i$.
In the p-adic numbers, we have $z = 0 + 0 + \cdots = (0+\cdots)/(1+\cdots)$.
If $z$ is irrational, then it does not have such a normal form representation.

A:

The real story is:
$$z=\alpha \iff z=\alpha\in\mathbb{R} \iff z=\alpha\in\mathbb{Q} \iff z=\alpha\in\mathbb{Q}_p$$
for all $z,\alpha\in\mathbb{C}$.
We use the existence of the $\mathbb{R}$ and $\mathbb{Q}$ real numbers to prove that there are $\mathbb{Q}_p$ numbers as well.
But $\mathbb{Q} ot\equiv\mathbb{Q}_p$.

A:

Consider the simplest example of a statement in which a formula is valid in different circumstances

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ProfessorTeachesExcel2016ActivationCode
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