So most p-adic notes inevitably beat us with a club with Ostrowski's theorem on absolute values.
However what if we forsake absolute values, and instead use a digit system where the value $v(x)$ of a number $x$ equals a tuple $(a,b)$. Here $a$ is the smallest negative exponent of some prime $p$ in the representation of $x$, while $b$ is the smallest positive exponent. For instance for $p=2$, $v(9/4)$ = $v(1/4 + 2)$ = $(2,1)$.
However we allow infinite expansions too:
We define a sequence as 'left-convergent' if for $\lim v_2(x_n - x_{n-1})$ the first tuple value approaches infinity, and likewise for 'right-convergent'.
In other words, high positive or negative powers of a prime both converge to different zeros. And therefore we can write unconditionally convergent expressions of the form $\sum_{-\infty}^{\infty} p^n a_n$ where $a_n$ is between $0$ and $p-1$.
So we basically obtain real (base $p$) plus p-adic 'combined' numbers.
The number zero can e.g. be represented as either:
$\cdots 0.0 \cdots$
or
$\cdots (p-1)(p-1)(p-1).(p-1)(p-1)(p-1) \cdots$
Since the latter is divisible by all factors of $p-1$, for $p>2$ we aren't dealing with a field (at least by the usual definition) as there are zero divisors. Though I don't know what happens with $p=2$?
Something like $\frac{1}{x}$ for $(x,p)=1$ will have multiple solutions. With the 'new' solutions corresponding to linear combinations of the real and p-adic solution.
E.g.
$\frac{1}{3}$ for $p=2$ has representations:
$0.010101 \cdots$ (real)
$\cdots 0101011;$ (2-adic)
$\cdots 010101.1010101 \cdots$ (sum of both divided by two)
This sort of feels like 'field extension' since we are adding new solutions to an equation.
In these numbers, series like $\displaystyle \sum_{n=0} p^{(-1)^n n}$ converge to finite values. Also power series converge absurdly: $\sum_{n=-\infty}^{\infty} x^n = 0$ for all $x$ with a power of $p$ in the numerator or denominator.
How about products? (responding to Julian Rosen). In general we can use Cauchy products if at least one of the numbers has a finite representation. However multiplying an infinite p-adic and infinite real number together causes convergence issues. If we denote $\frac{1}{x}_p$ and $\frac{1}{x}$ for the p-adic and real representatives of a fraction. Then $\frac{1}{x}_p \cdot \frac{1}{x} = \frac{1}{x}_p \cdot \frac{x}{x^2} = \frac{1}{x^2} = \frac{x}{x^2}_p \cdot \frac{1}{x} = \frac{1}{x^2}_p$. Hence by implication there can't be a 'single' representative for this product.
My questions are:
- Does this approach make sense/does it have a name?
- Does it have any benefits or unique advantages?
E.g. tying p-adic results to real ones? For instance 'hypothetically' if we can demonstrate that the 'combined' expression (real + p-adic) is irrational and the real solution is rational, then the remaining term must be irrational.