发明新数学是怎样一种体验

问题:

如何使 \(1+2+4+\cdots+2^n+\cdots=-1\)


全新定义距离函数,同时保持这一函数的平移不变性: \[ \displaylines{dist(A, B)=dist(A+x, B+x) \qquad \forall x } \] 首先假设 \(0\)\(2^n\) 距离为 \(2^{-n}\),之后通过平移不变性确定其它数的距离。

由此可知 \[ \displaylines{\lim_{n\rightarrow \infty} 2^n = 0 } \] 所以原问题 \[ \displaylines{1+2+4+\cdots+2^n+\cdots = \lim_{n\rightarrow \infty} 2^n -1 = -1 } \] 实际上这是 \(p\)-adic number 在 p=2 下的结论。

给出正式定义:

let \(p\) be a prime in \(\mathbb{Z}\). The \(p\)-adic order for \(\mathbb{Z}\) is defined as \(v_p: \mathbb{Z}\rightarrow\mathbb{N}\) \[ \displaylines{v_p(n) = \begin{cases} \max\{v\in \mathbb{N}: p^v | n\} & \text{if } n\neq 0 \\ \infty & \text{if } n=0 \end{cases} } \] The \(p\)-adic absolute value is defined as \(|\cdot|_p\) \[ \displaylines{|x|_p=\begin{cases} p^{-v_p(x)} &\text{if } x\neq 0 \\ 0 &\text{if } x=0 \end{cases} } \] A metric space then can be defined by \[ \displaylines{d(x,y) = |x-y|_p } \]


参考:

  1. Video
  2. p-adic Order
  3. p-adic Number