When do we have 1 + 1 = 11 and 2 + 2 =5?

The phrase "$2+2=5$" is a clich\'e or a slogan used in political speeches, propaganda, or literature, most notably in the novel "$1984$" by George Orwell. More recently, we came across a You-Tube short film comedy, Alternative Math, produced by IdeaMan Studios (see \cite{Danny}). It is a hilarious exaggeration of a teacher who is dragged through the mud for teaching that $2+2=4$ and not $22$, as Danny, a young student, kept on insisting. In the movie, Danny and the whole community sincerely believe $1+1=11$ and $2+2=22$. Jokes aside, we ask the question whether a polynomially defined group law "$\oplus$" defined over the field of rationals such that $1\oplus1=u$ and $2 \oplus2=v$ can simultaneously be satisfied for arbitrary integers $u$ and $v$. Answer to this question takes us through a fascinating journey from Brahmagupta all the way to the modern works of Louis Joel Mordell and Ramanujan!