What are some complicated ways to write 1+1?

abs(e^iπ^) + abs(i^2^)

\tan(\pi/4) + 1/42 \int_{-7\ln 6}^\infty \exp(-x/7) \dd{x}

The standard way when using ordinal arithmetic is: Take the ordinal 1, which is {{}}. Replace each element with a ordered pair of the form {{a},{a,b}} with second element being 0 (that is {}). Repeat with second element 1. Take a union. Take find the ordinal with this order. Overall: otp({ {{{}},{{},{}}}, {{{}},{{},{{}}}} }) Or simplified

otp({ {{{}}}, {{{}},{{},{{}}}} })

How complex you looking? https://en.m.wikipedia.org/wiki/Principia_Mathematica

Most of them.

I saw somewhere that someone had decoded how an AI had learnt to do basic arithmetic. And it appeared to be using a massive expression containing lots of sin & cosines to do basic addition

(10^googol^)^0^ + (TREE(3))^0^

Although that's fairly easy to write. It's hard to calculate, if you calculate the brackets first.

(fix add a b := match a with O => b | S x => add x (S b) end) (S O) (S O)

Perhaps: (lim_{n->\infty} \sum_{m=1}^n 1/2^m ) + dim(Im(matrix([1,3,4],[2,6,8],[3,9,12])))

Pi/pi + pi/pi