Addition in Other Domains


  1. \frac{1}{1/A+1/B}
  2. \sqrt{A^2+B^2}
  3. e^{\ln{A}+\ln{B}}

These expressions are all commonly used. Number 3 is so common that it has it’s own shorthand: A \times B .

Let’s define a family of operators


where f is a monotonic univariate function that takes us to the domain in which the summing happens, and f^{-1} brings us back. So by definition…

A +_f B=f^{-1}(f(A)+f(B))

Those three expressions listed in the beginning then become:

  1. A+_{-1}B         (  because (A^{-1}+B^{-1})^\frac{1}{-1}    )
  2. A+_{2}B           (  because (A^2+B^2)^\frac{1}{2}    )
  3. A+_{\ln}B          (  because \ln^{-1}(\ln{A} + \ln{B})    )

We can combine, for example, parallel resistors like this:

ResistorA +_{-1} ResistorB +_{-1} ResistorC

and independent Gaussian deviates like this:

ErrorA +_2 ErrorB + _2 ErrorC

Short-hand could be useful in itself, but what properties/identities might these operators permit? It isn’t hard to verify that the entire family inherits the associative and commutative properties of it’s inner champion + . This symmetry allows us to construct multivariate expressions in the exact same way as it works for + :

B +_f (C +_f A) = A +_f B +_f C

In calculus we have the product rule to help us with d[A \cdot B] = dA \cdot B + A \cdot dB . The generalization for +_f turns out to be

d[A +_f B] = \frac{f'(A)dA+f'(B)dB}{f'(A +_f B)}

and again, this generalizes straightforwardly to higher dimensions.

The proof isn’t difficult, but for now let’s see how the product rule emerges as a special case of this rule:

d[A \cdot B] = d[A +_{\ln} B] = \frac{\ln'(A)dA + \ln'(B)dB}{\ln'(A +_{\ln} B)} = \frac{dA/A+dB/B}{\frac{1}{AB}} = B \cdot dA + A \cdot dB

Might this family yield more identities that allow it to share in the abilities of its special cases (+ and \times ) to reduce complex expressions to simple insights?

Giving these expressions full multivariate-function status just seems like overkill. The area and the diagonal length of a (b&w) photograph are simpler things than the photograph itself, and deserve a simpler abstraction.

TODO: Check more identities.

TODO: Note on nomograms. Each of +_{-1} , +_2 , +_{ln} has a simple visualization using perpendicular axes.

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