c. 400 BC — Jaina mathematicians in India write the “Surya Prajinapti”, a mathematical text which classifies all numbers into three sets: enumerable, innumerable and infinite. It also recognises five different types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually.
953 — The arithmetic of the Hindu–Arabic numeral system at first required the use of a dust board (a sort of handheld blackboard) because “the methods required moving the numbers around in the calculation and rubbing some out as the calculation proceeded.” Al-Uqlidisi modified these methods for pen and paper use. Eventually the advances enabled by the decimal system led to its standard use throughout the region and the world.
1030 — Ali Ahmad Nasawi writes a treatise on the decimal and sexagesimal number systems. His arithmetic explains the division of fractions and the extraction of square and cubic roots (square root of 57,342; cubic root of 3, 652, 296) in an almost modern manner.
c. 1400 — Ghiyath al-Kashi “contributed to the development of decimal fractions not only for approximating algebraic numbers, but also for real numbers such as pi. His contribution to decimal fractions is so major that for many years he was considered as their inventor. Although not the first to do so, al-Kashi gave an algorithm for calculating nth roots which is a special case of the methods given many centuries later by Ruffini and Horner.” He is also the first to use the decimal point notation in arithmetic and Arabic numerals. His works include The Key of arithmetics, Discoveries in mathematics, The Decimal point, and The benefits of the zero. The contents of the Benefits of the Zero are an introduction followed by five essays: “On whole number arithmetic”, “On fractional arithmetic”, “On astrology”, “On areas”, and “On finding the unknowns [unknown variables]”. He also wrote the Thesis on the sine and the chord and Thesis on finding the first degree sine.
1427 — Al-Kashi completes The Key to Arithmetic containing work of great depth on decimal fractions. It applies arithmetical and algebraic methods to the solution of various problems, including several geometric ones.