= Mathematics
{wiki}
Mathematics studies patterns in quantity, structure, space, and change by giving objects precise <mathematical definitions> and examining what follows from them. <Arithmetic> starts with operations on <numbers>; <algebra> organizes relations and transformations; <geometry and topology> studies shapes and the structures preserved by maps; <analysis> studies <limits> and continuous change. These fields connect because the same structure can appear in different forms.
A small example shows the value of changing viewpoint. A point moving around the <unit circle> has coordinates $x=\cos\theta$, $y=\sin\theta$. Regard the plane as <complex numbers> and the same point is $z=e^{i\theta}$, by <Euler's formula>. Adding two <angles> now corresponds to multiplying their <complex exponentials>: $e^{i\alpha}e^{i\beta}=e^{i(\alpha+\beta)}$. A geometric rotation has become an algebraic operation.
The connection reaches <calculus> too. Differentiating twice gives $x''+x=0$ and $y''+y=0$, so the two coordinates solve the <harmonic oscillator equation>. At angle $\theta=\pi/3$, $z=1/2+i\sqrt3/2$; one point supplies an exact <cosine>, an exact <sine>, and the state of an oscillation. The <Pythagorean trigonometric identity> records that the <modulus> stays one. Seeing these as descriptions of one object makes each formula easier to remember and reuse.
\Image[/circle-exponential-oscillator.png]
{title=Uniform motion around the unit circle connects complex exponentials and real oscillations}
This is why a useful mathematical explanation provides both a reliable <mathematical proof> and a way to see what the result is doing. In his 1994 essay, <William Thurston> discusses mathematical progress in terms of advancing understanding, and uses the many interpretations of a <derivative> to illustrate this aim. https://arxiv.org/abs/math/9404236[Thurston, On proof and progress in mathematics, sections 1–2]{ref} A diagram, a computed example, and a concise argument can make different parts of the same idea accessible.
Applications ask us to select a <mathematical model> of a physical or other system. Its assumptions determine which conclusions transfer to that system. Within mathematics, the same practice of making assumptions explicit lets a result about <groups>, <vector spaces>, or <probability distributions> apply to many concrete examples. The beauty is the combination of precision and reuse: an argument can reveal more than the particular case that first suggested it.
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