A Class of Functions that I made
Let us define G and H as repeated exponents and repeated natural logarithms, such that: G(x, 0) = x G(x, n+1) = exp( G(x, n) ) and H(x, 0) = x H(x, n+1) = ln( H(x, n) ) Note that: G(x, -n) = H(x, n) Next, let us define F in terms of G and H F(x, n, y) = G( ( H(x, n) + H(y, n) ), n) Addition and Multiplication fall out F(x, 0, y) = x + y F(x, 1, y) = x * y Taking it to the next step, F(x, 2, y) = exp(exp( ln(ln(x)) + ln(ln(y)) )) Apply the laws of logs = exp(exp( ln( ln(x) * ln(y) ) )) = exp(exp( ln(ln( x ^ ln(y) )) )) = x ^ ln(y) However, notice we could have equally chosen ln(x) as the power, so x ^ ln(y) = y ^ ln(x) F is commutative and associative for all natural n One neat thing follows when y = x F(x, 0, x) = x + x = x * 2 = F(x, 1, 2) F(x, 1, x) = x * x = x ^ 2 = F(x, 2, exp(2)) and in general F(x, n, x) = F(x, n+1, G(2, n)) F also has identities for all natural n, and they’re all related to eachother in an interesting way F(x, 0, 0) = x F(x, 1, 1) = x F(x, 2, e) = x in general F(x, n, G(0, n)) = x Inverses are a little tricky to visualise in the general case, but: F(x, 0, x * -1) = x + -x = 0 F(x, 1, x ^ -1) = x * 1/x = 1 F(x, n, F(x, n+1, G(-1, n))) = G(0,n) Finally, there’s one special case that I built this whole thing for. n = -1 It should be clear that: F(x, -n, y) = H( ( G(x, n) + G(x, n) ), n) So: F(x, -1, y) = ln( exp(x) + exp(y) ) And here’s where it gets interesting: F(x, -1, x) = ln( exp(x) + exp(x) ) = ln( exp(x) * 2 ) = ln( exp(x) ) + ln(2) = x + ln(2) F(x, -1, y) is a function that addition distributes over. That’s wild to me. In general, F(n) and F(n+1) always form a ring for natural n
