Real numbers as infinite decimals -- theory and computation

In the 16th century, Simon Stevin initiated a modern approach to decimal representation of measuring numbers, marking a transition from the discrete arithmetic practised by the Greeks to the arithmetic of the continuum taken for granted today. However, how to perform arithmetic directly on infinite decimals remains a long-standing problem, which has seen the popular degeometrisation of real numbers since the first constructions were published in around 1872. Our article is devoted to solving this historical problem. An issue that Hardy called "a fatal defect" is also settled.