Mathematical minimalism

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Andrzej Odrzywolek recently posted an article on arXiv showing that you can obtain all the elementary functions from just the function

operatorname{eml}(x,y) = exp(x) - log(y)

and the constant 1. The following equations, taken from the paper’s supplement, show how to bootstrap addition, subtraction, multiplication, and division from the elm function.

begin{align*} exp(z) &mapsto operatorname{eml}(z,1) \ log(z) &mapsto operatorname{eml}(1,exp(operatorname{eml}(1,z))) \ x - y &mapsto operatorname{eml}(log(x),exp(y)) \ -z &mapsto (log 1) - z \ x + y &mapsto x - (-y) \ 1/z &mapsto exp(-log z) \ x cdot y &mapsto exp(log x + log y) end{align*}

See the paper and supplement for how to obtain constants like π and functions like square and square root, as well as the standard circular and hyperbolic functions.

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