al’Khwarizmi, whose full name is Abu Abd-Allah ibn Musa al’Khwarizmi, was born about AD 790 near Baghdad, and died about 850. His most important contribution, written in 830, was Hisab al-jabr w’al-muqabala. From the al-jabr in the title we get algebra. The treatise develops a system for the solutions of quadratic expressions including geometric principles for completing the square.

Many say that the Babylonians first developed systems of quadratic equations. This calls for over simplification, because the Babylonians had no concept of an equation. Also, all solutions to Babylonian problems were positive because they were solutions to problems involving lengths.

al’Khwarizmi also gave a classification system for quadratics. He devotes a chapter to each chapter in his treatise and gives methods in solving each differently.

Six Types of Quadratics

  • Squares equal to roots (x^2 = square root of 2)
  • Squares equal to numbers (x^2 = 2)
  • Roots equal to numbers (square root of x = 2)
  • Squares and roots equal to numbers (x^2 + 3x = 25)
  • Squares and numbers equal to roots (x^2 + 1 = 9)
  • Roots and numbers equal to squares (3x + 4 = x^2)

al’Khwarizmi also gives geometric proofs of these methods. In 1145, Savasorda published Liber embadorum that gave the complete solution of the quadratic equations. In 1494, the first edition of Summa de arithmetica, geometrica, proportioni et proportionalita was published. It sets out equation systematically and algebraically.

Scipione dal Ferro is the first credited with solving cubic equations algebraically, around 1515. However, he could only solve cubic equations with the form

x^3 + mx = n

He kept this work a secret until 1526 when he revealed it to his student Antonio Fior. Soon, the work was common knowledge around Bologna, where dal Ferro taught at the University of Bologna.

Other observations in the field of complex equations were also made, primarily that of Harriot. He observed that if

x = b, x = c, x = d then

(x - b)(x - c)(x - d) = 0

which allowed more uses for the cubic equations. Many proofs after this followed, including ones which first proved these principles algebraically, instead of geometrically. The further use of algebra supplemented modern mathematics in a very important way.

The 'al jabr' treatise of al’Khwarizmi was translated, and introduced to europe by Fibonacci

Omar Khayyam (more often recognised fo his 'Rubaiyat') made major contributions in Mathematics, particularly in Algebra. His book Maqalat fi al-Jabr wa al-Muqabila on Algebra provided great advancement in the field. He classified many algebraic equations based on their complexity and recognized thirteen different forms of cubic equation. Omar Khayyam developed a geometrical approach to solving equations, which involved an ingenious selection of proper conics. He solved cubic equations by intersecting a parabola with a circle. Omar Khayyam was the first to develop the binomial theorem and determine binomial coefficients. He developed the binomial expansion for the case when the exponent is a positive integer. Omar Khayyam refers in his Algebra book to another work on what we now know as Pascal's triangle. This work is now lost. He extended Euclid's work giving a new definition of ratios and included the multiplication of ratios. He contributed to the theory of parallel lines.

related; MathWorld

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