♦♦ Ce que sait tout enfant

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Equality between two numbers poses no difficulty when both are defined by "finitary"* methods. But as soon as infinity enters the picture—represented by ellipses in formulas—the door opens to all manner of paradoxes. (* Finitism is an approach to mathematics that considers only finite objects.)

There are other ways to represent real numbers besides our decimal notation! One of the best known is to write a number as a continued fraction. The resulting representations, even when truncated, very quickly provide "good" approximations of the original number.

Among irrational numbers, transcendental numbers are not roots of any polynomial equation with integer coefficients. They seem to transcend common sense, hence their name. Cantor showed that they make up the overwhelming majority of the real numbers, without identifying a single one!

Beyond the "classical" convergence of numerical series, many methods—some particularly powerful—can accelerate the convergence of a series or even assign a value to the "sum of a divergent series." Here are a few notable examples.