second half of 1st millennium BC — The
Lo Shu Square, the unique normal
magic square of order three, was discovered in
China.
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.
c. 250 BC — late
Olmecs had already begun to use a true
zero (a shell glyph) several centuries before
Ptolemy in the New World. See
0 (number).
150 BC —
Jain mathematicians in
India write the “Sthananga Sutra”, which contains work on the theory of numbers, arithmetical operations,
geometry, operations with
fractions, simple equations,
cubic equations, quartic equations, and
permutations and
combinations.
300 — the earliest known use of
zero as a decimal digit in the Old World is introduced by
Indian mathematicians.
c. 400 — the
Bakhshali manuscript is written by
Jaina mathematicians, which describes a theory of the infinite containing different levels of
infinity, shows an understanding of
indices, as well as
logarithms to
base 2, and computes
square roots of numbers as large as a million correct to at least 11 decimal places.
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.[2]
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.