This paper looks at how ancient mathematicians (and especially the Pythagorean school) were faced by problems/paradoxes associated with the infinite which led them to juggle two systems of numbers: the discrete whole/rationals which were handled arithmetically and the continuous magnitude quantities which were handled geometrically. We look at how approximations and mixed numbers (whole numbers with fractions) helped develop the arithmetization of geometry and the development of mathematical analysis and real numbers.

翻译：本文探讨古代数学家（尤其是毕达哥拉斯学派）如何面对与无穷相关的难题/悖论，这促使他们在两套数字系统间周旋：算术处理的离散整数/有理数与几何处理的连续量。我们考察近似值和带分数（整数与分数的组合）如何促进几何算术化，推动数学分析与实数概念的发展。