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Generalized Indian methods for extracting square and cube roots to include fourth, fifth and higher roots, noted existence of different sorts of cubic equations.
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Established that dividing by zero yields infinity, found solutions to quadratic, cubic and quartic equations (including negative and irrational solutions) and to second order Diophantine equations, introduced some preliminary concepts of calculus.
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Fibonacci Sequence of numbers, advocacy of the use of the Hindu-Arabic numeral system in Europe, Fibonacci's identity (product of two sums of two squares is itself a sum of two squares)
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Developed field of spherical trigonometry, formulated law of sines for plane triangle.
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Solutions to quadratic, cubic and higher power equations using a method of repeated approximations.
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Culmination of Chinese “magic” squares, circles and triangles, Yang Hui’s Triangle (earlier version of Pascal’s Triangle of binomial co-efficients)
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Applied theory of conic sections to solve optical problems, explored amicable numbers, factorization and combinatorial methods.
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System of rectangular coordinates, such as for a time-speed-distance graph, first to use fractional exponents, also worked on infinite series.
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Use of infinite series of fractions to give an exact formula for π, sine formula and other trigonometric functions, important step towards development of calculus.
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Influential book on arithmetic, geometry and book-keeping, also introduced standard symbols for plus and minus.