(Mathematics:)
A proof (or other argument or technique) is said to be elementary if it involves no "higher mathematics"; say nothing beyond what is studied in high school. Usually no limits derivatives or integrals, no algebra beyond the properties of arithmetic operations on N, Z, Q, R or C, intuitive probability theory, and none of the harder techniques of mathematical logic. "Intuitive" uses of the axiom of choice might be allowed, depending on whom you ask.

Almost all of plane geometry is "elementary" in this sense, as are various very useful theorems and lemmas.

``elementary'' ≠ ``easy'' !!

The term is unfortunate: it sounds as though mathematicians are sneering at anyone outside their little club. In fact, almost all mathematicians hold elementary techniques in very high regard. To be able to solve a hard problem is great. To do so using only the techniques taught in high school is much better!! Elementary techniques are particularly challenging, because they're known to everybody! Even better, you can show them to almost anyone (except if they punch you as soon as you get started), so they are much better for bragging than techniques which begin "consider a relaxed manifold that is embedded in an infinite dimensional Hausdorff K-compact torus"...

El`e*men"ta*ry (?), a. [L. elementarius: cf. F. 'el'ementaire.]

1.

Having only one principle or constituent part; consisting of a single element; simple; uncompounded; as, an elementary substance.

2.

Pertaining to, or treating of, the elements, rudiments, or first principles of anything; initial; rudimental; introductory; as, an elementary treatise.

3.

Pertaining to one of the four elements, air, water, earth, fire.

"Some luminous and fiery impressions in the elementary region."

J. Spencer.

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