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=//__Phase 1__//= **Brahmagupta** Brahmasphuta Wrote a book. Brahmasphuta has 4 1/2 Chapters of just math while the 12th chapter it about Arithmetic rogessions and a little of Geometry the 18th chapter deals with ax-by=c. Kuttaka means Alegbra. Brahmasphuta was the founder of the indeterminate equations of the second degree Ex: Nx^2+1=Y^2. He was the first to use algebra to sovle Astronomical problems Brahmagupta gave the solution of the general Linear Equation in chapter eighteen of Brahmasphutasiddhanta. 18.43 The difference between rupas, when inverted and divided by the difference of the unknowns, is the unknown in the equation. The rupas are [subtracted on the side] below that from which the square and the unknown are to be subtracted. Which is a solution equivalent to X = e-c/b-d, where rupas represents constants. He further gave two equivalent solutions to the general quadratic equation, 18.44. Diminish by the middle [number] the square-root of the rupas multiplied by four times the square and increased by the square of the middle [number]; divide the remainder by twice the square. [The result is] the middle [number]. 18.45. Whatever is the square-root of the rupas multiplied by the square [and] increased by the square of half the unknown, diminish that by half the unknown [and] divide [the remainder] by its square. [The result is] the unknown.

**Yang Hui** Yang acknowledged that his method of finding square roots and cubic roots using "Yang Hui's Triangle" The writing of the Chinese mathematician Yang Hui represents the first in which quadratic equations with negative coefficients of 'x' appear, although he attributes this to the earlier Liu Yi. The first appearance of the general solution in the modern mathematical literature is evidently in an 1896 paper by Henry Heaton. His name was Qianguang. Yangs teacher was Qin Jiushao and He Finished his first math book in 1275 AD. In his written work, Yang provided theoretical proof for the proposition that the complements of the parallelograms which are about the diameter of any given parallelogram are equal to one another. This was the same idea expressed in Euclid's forty-third proposition of his first book, only Yang used the case of a rectangle and gnomon There were also a number of other geometrical problems and theoretical mathematical propositions posed by Yang that were strikingly similar to the Euclidean system. However, the first books of Euclid to be translated into Chinese was by the cooperative effort of Matteo Ricci and Xu Guangqi in the early 17th century. Yang writing represents the first in which quadratics Eqations with negative coefficients of 'x' appear, although he attributes this to the earlier liu yi. Yang was also well known for his ability to manipulate decimal fractions. When he wished to multiply the figures in a rectangular field with a breadth of 24 paces 3 4/10ths ft. and length of 36 paces 2 8/10'ths, Yang expressed them in decimal parts of the pace, as 24.68 X 36.56 = 902.3008.

Phase 2 My Number Was 2900 My next Number Was 320 My Next Number is 2500 Phase 3 Phase 4