# Combined real + p-adic numbers?

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:

1. Does this approach make sense/does it have a name?
2. 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.

• What is the object / the number $x$ in: "...and instead use a digit system where the value of a number $v(x)$ equals a tuple..." ?! (This is just a first unclear point, appeared immediately after reading the line with Ostrowski's Theorem, which is not needed any longer...) Oct 18, 2021 at 17:16
• If I wanted to "combine" $p$-adics and reals, my next construction is the direct product $\mathbb Q_p \times \mathbb R$. I can embed $\mathbb Q$ into it diagonally, i.e. represent $1/3$ as $(1/3, 1/3)$. I can also put some generalization of the absolute value on it via $(x,y) \mapsto (\lvert x\rvert_p, \lvert y \rvert_{\text{real}})$ and already catch more information than your construction there. It's up to you to show me how your construction beats my $\mathbb Q_p \times \mathbb R$ when solving some question. Oct 18, 2021 at 17:31
• I think it's a quite interesting question based on the common motivations given for $p$-adic numbers as "base $p$ expansions that can go on forever in the wrong direction." Totally natural to ask: "why not both?" Oct 18, 2021 at 18:31
• You are probably fishing for the ring of Adeles. It does combine real, complex and p-adic numbers, and is quite useful in the theory of arithmetic groups. Oct 18, 2021 at 18:58
• When are two expressions $\sum_n a_n p^n$ (with $a_n\in\{0,1,\ldots,p-1\}$) equal? Also, you speak of zero divisors, but is it clear how to multiply two combined numbers? Oct 19, 2021 at 4:18