Euler's formula and why EML needs complex arithmetic

One of the prettier facts in all of mathematics:

e^(iθ) = cos(θ) + i·sin(θ)

This is Euler's formula. Set θ = π and it gives Euler's identity e^(iπ) + 1 = 0.

Why it shows up in EML

To reach trig functions sin and cos from exp and ln, you invert Euler's formula:

cos(x) = (exp(ix) + exp(−ix)) / 2
sin(x) = (exp(ix) − exp(−ix)) / (2i)

Both require exp of a complex number. Since EML nests exp(x) − ln(y), and both exp and ln extend naturally to ℂ, the paper allows intermediate computations to pass through complex values, even when the final answer is real.

Why this is surprising

You might expect "the single operator for real-valued elementary functions" to stay in ℝ. It can't. The paper is explicit: complex arithmetic internally is necessary.

This is a feature, not a bug. It's what lets a single operator reach all the way to sin, cos, and π.

What this means for implementation

A practical EML engine must support complex arithmetic at evaluation time. For the V1 of this teaching app, we stay in ℝ, every tree in the Function Gallery is chosen to avoid complex intermediates. For the trig functions, we point to the paper rather than showing the tree directly.