For a field $F$, define the binary operation $\parallel :(F\mathbb{P}^1 \times F\mathbb{P}^1 \setminus\{(0,0)\}) \to F\mathbb{P}^1$ by

$$a \parallel b = \frac{1}{\frac{1}{a} + \frac{1}{b}}.$$

This operation is conjugate to ordinary addition by means of the Möbius transformation $x \to 1/x$. As such, it is associative, commutative and has an identity $1/0=\infty$. It is also distributive with respect to multiplication, so $F\mathbb{P}^1\setminus{0}$ becomes a field equipped with this operation and the usual product.

It is related to the harmonic mean, and is used to find the equivalent resistance in a circuit of parallel resistors (which is where I got the notation from).

My question is, has this operation been given a name somewhere?

  • 2
    $\begingroup$ I used to call it (maybe improperly but really informally) the "parallel" of two number :D $\endgroup$ – the_candyman May 4 '16 at 19:18
  • $\begingroup$ aka $\frac{ab}{a+b}$ $\endgroup$ – user645636 May 1 '19 at 1:56

For two numbers, the harmonic mean of a set of numbers $a$, $b$, $c$, etc. is defined as the number of terms in the set divided by the sum of all the reciprocals of the terms in the set. For example, the harmonic mean of $a$, $b$, and $c$ is $$\frac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}$$ So if $$a \parallel b=\frac{1}{\frac{1}{a}+\frac{1}{b}}$$

Then $a \parallel b$ would just be the harmonic mean of $a$ and $b$, divided by $2$.


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