Concept

Mean value theorem

The statement that a differentiable function's change across an interval equals its slope somewhere inside, times the interval's length. It converts a bound on the derivative into a bound on the function, which is how nearly every error estimate in analysis is obtained.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

Area is the undoing of slope

Two operations invented for unrelated reasons — measuring a region and measuring a rate — turn out to be inverse. The picture is two panels sharing one axis, and the claim is that the lower curve's steepness is the upper curve's height.

analysis · the integral $How closely a fraction can come, and the barrier that says no closer. Two panels at very different scales: the approximations to √2, which stay above the barrier a degree-two number obeys, and the truncations of a constructed number, which fall below every barrier drawn.$

Approached too fast to be algebraic

An algebraic number of degree d cannot be approached by fractions faster than the denominator's dth power. So a number that is approached faster than that is the root of no polynomial at all — and one can be built by choosing where its decimal digits go.

number · irrationality

Named alongside it

The objects these essays reach for when they reach for this one.

Accumulation Algebraic number Antiderivative Area Construction Continued fractions Continuity Degree Derivative Diophantine approximation Existence proof Fundamental theorem

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