Различные способы представления ...

[Solunac]

$a, b\in\mathbb{R}$

How many ways can be represented $abs(a-b)$, i.e. $|a-b|$. I would like to know if there is some elegant way to represent this. Let me show a little bit concrete...

$|a-b|=\left\{\begin{array}{l} a-b, a>b \\ b-a, a\leqslant b\end{array}$
$|a-b| = \max(a-b,b-a)$
$|a-b| = \frac{|a^2-b^2|}{a+b} = \frac{\max (a^2-b^2, b^2-a^2)}{a+b} = \frac{a-b}{sgn (a+b)}$ but I think that these ways are not elegant like the maximum idea, but...
$|a-b| = \sqrt{(a-b)^2}$
$|a-b| = min_{c\in\mathbb{R}}(|a-c|+|c-b|)$ but this representation already uses $abs(x-y) $form in itself (it's recursive in some sense), although it can be extended to $\mathbb{C}$, or $\mathbb{R}^2$.

So, does anybody know some other representation, or to come up with something?

Спасибо.

[gris]

$$|a-b| = \int\limits_0^{a-b} sgn (t) \,dt$$

$$|a-b| = e^{\int\limits_0^{a-b} \dfrac {dt}{t}}$$

[Профессор Снэйп]

$$ |a-b| = \sqrt[4]{(a-b)^4} = \mu \big([a,b] \cup [b,a]\big) = (a\mathop{\overset{\boldsymbol\cdot}{\smash-\vrule width 0pt height 1pt}}b) + (b\mathop{\overset{\boldsymbol\cdot}{\smash-\vrule width 0pt height 1pt}}a) = \exp \left( -C + \int \frac{dx}{x} \right) (a-b) = \ldots $$

Что за дурацкая тема?!