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If the continued fractions of two irrational numbers have a common completequotient, then these two numbers are in the same orbit under the action of$\mathrm{PGL}(2,\mathbb{Z})$. The converse is Serret's well-known theorem, butwe give a simpler and purely algebraic proof of it.

Preuve simplifi\'ee du th\'eor\`eme de Serret sur les nombres \'equivalents