Parallel (operator)

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The parallel operator, written as a ∥ b, is a way to combine two values as if they were connected in parallel, like two resistors in parallel. It is defined by a ∥ b = 1/(1/a + 1/b), which is the same as ab/(a + b) when a and b are nonzero.

Key ideas and cases
- If one value is 0, then a ∥ 0 = 0. If one value is ∞, the result is the other value. If a = −b, then a ∥ b = ∞.
- It is commutative: a ∥ b = b ∥ a, and it is associative: (a ∥ b) ∥ c = a ∥ (b ∥ c).
- ∞ is the identity for parallel, while 1 is the identity for ordinary multiplication. Every non-∞ a has a parallel inverse, which is −a, since a ∥ (−a) = ∞; 0 has no parallel inverse.
- For more than two values: a1 ∥ a2 ∥ ... ∥ an = 1 / (1/a1 + 1/a2 + ... + 1/an).

Common uses
- In electronics, it gives the total impedance of components connected in parallel.
- In mechanics, the reduced mass μ = mM/(m + M) can be written as m ∥ M.
- It is the dual operation to ordinary addition: parallel sum behaves like adding reciprocals.

Example
- 3 ∥ 6 = (3 × 6) / (3 + 6) = 18 / 9 = 2.

In short, the parallel operator is a simple rule for combining quantities that share a parallel path, based on summing reciprocals.