Module hyperreal 

Hyperreal numbers generated by the program’s infinitesimal dt — exact, ordered, decidable.

A program that declares Let dt be an infinitesimal. gets a positive number smaller than every positive standard number. Every hyperreal value built from dt with the field operations is a rational function of dt with real algebraic coefficients, N(dt)/D(dt), and that is how a Hyperreal is held: reduced (gcd(N, D) = 1) with D comonic (its lowest-order nonzero coefficient is 1), so equal values are structurally equal.

The order is decided exactly by the transfer principle for real closed fields: a rational function of dt has, for every small enough positive standard t, the sign of its lowest-order coefficient — so x > 0 exactly when the lowest-order coefficient of N is positive (D being comonic). From the same expansion come the standard notions:

• the valuation v(x) = ord N − ord D — positive for an infinitesimal, negative for an infinite value, zero for an appreciable one;
• the standard part — the leading coefficient when v = 0, zero when v > 0, and none at all (the value is not limited) when v < 0;
• x ≈ y — x − y is infinitesimal.

A value prints as the exact rational function in the infinitesimal’s name (3 + 2·dt, 1/dt, (1 + dt)/(2 - dt)), with rational coefficients scaled to integers when that is possible.

Structs

Hyperreal

A hyperreal number generated by the infinitesimal. See the module documentation.

Enums

PowerError

Why a power is refused.

Constants

MAX_POWER_DEGREE

The highest degree one power may reach, so a single operation’s work stays bounded.