[[!redirects Sandbox > history]] [[!redirects Sandbox]] < [[nlab:Sandbox]] Forget about real analysis. If we are already using Archimedean ordered Artinian local $R$-algebras, we could continue and use the complexification $C = R[i]/(i^2 + 1)$ of the Archimedean ordered field $R$ instead. Synthetic complex analysis to define the complex exponential function on $C$, and the complex sine and cosine function straight into trigonometry. I was wrong about the foundations mattering. In discrete mathematics one simply introduces set theoretic concepts by fiat rather than proving them from axioms. category: redirected to nlab