Methods for solving Equations

1. Methods for solvingEquations by: Rita Barrocas 11ºB

2. Scipione del Ferro • Scipione del Ferro was the first mathematician to find a solution to the depressed cubic equation which is the equation of the form:ax3 + bx = c • He was born on 6th of February of 1465 in Bologna, Italy. His father was a papermaker which was a rising profession due the invention of the printing with moveable types. He attended University of Bologna which had been founded in the 11th century. by: Rita Barrocas 11ºB

3. In 1496, del Ferro became a lecturer in mathematics at the University of Bologna. It’s not known about any book that del Ferro had wroten, and it is belived that it is because he was very reserved about his dicoveries and only share them with close friends and selected students. He kept his discoveries in a notebook. This notebook was later the prove that del Ferro had indeed discovered the solution to the depressed cubic equation. • Del Ferro died on November 5, 1526. When he died, the notebook was given to is son-in-law Hannibal Nave. The discovery of this notebook was what influenced Cardano to release his famous math book Ars Magna. • Cardano had promised Tartaglia that he would not reveal the method but he felt that del Ferro's earlier discovery of the method excused him from this promise. by: Rita Barrocas 11ºB

4. NicolòTartalia • On the 16th Century in the Renassaince in italy, Bologna University was known by the intense mathematics competitions. • In 1535 There was such a competition that the improbable figure of NicolòTartaglia discovered a matematical finding that was consideres impossible that had preplexed the best matematicians of China, India and the islamic world. • Niccolò Fontana became known as Tartaglia, that means “the stamerer”, for a communication defect that he suffered due to an damage he received in a battle during the invasion of the French army. • He was a poor engineer known for designing fortifications, a evaluator of topography and a bookkeeper in the Republic of Venice. by: Rita Barrocas 11ºB

5. But he was also a self-taught, but wildly ambitious, mathematician. • He was distinguished by producing, the first Italian translations of works by Archimedes and Euclid from untouched Greek texts, as well as an acclaimed compilation of mathematics of his own. by: Rita Barrocas 11ºB

6. Tartaglia'sgreatest contribution to mathematical history, occurred in 1535 when he won the Bologna University mathematics competition by demonstrating a general algebraic formula to solve cubic equations , equations with terms including x3, something that was an impossibility in that time, requiring an understanding of the square roots of negative numbers. In the competition, he beat Scipione del Ferro, who had coincidentally produced his own partial solution to the cubic equation wich never had happened before. However Scipione'ssolution perhaps predated Tartaglia’s, it was much more limited, and Tartaglia is usually credited with the first general solution. In the highly competitive and cut-throat environment of 16th Century Italy, Tartaglia even hided his solution in the form of a poem in an attempt to make it more difficult for other mathematicians to steal it. by: Rita Barrocas 11ºB

7. Tartaglia’sdefinitive method was, however, leaked to GerolamoCardano, a rather eccentric and confrontational mathematician, doctor and Renaissance man, and author throughout his lifetime of some 131 books. Cardano published it himself in his 1545 book "Ars Magna" (despite having promised Tartaglia that he would not), along with the work of his own brilliant student LodovicoFerrari. Ferrari, on seeing Tartaglia's cubic solution, had realized that he could use a similar method to solve quartic equations . • Tartaglia, Cardano and Ferrari between them demonstrated the first uses of what is now known as complex numbers, combinations of real and imaginary numbers of the type a + bi, where i is the imaginary unit √-1. It fell to another Bologna resident, Rafael Bombelli, to explain, at the end of the • An in the end of the 1560's, Rafael Bombelli that was also from bologna was the one thadeterminated exactly what imaginary numbers really were and how they could be used. by: Rita Barrocas 11ºB

8. Tartaglia died ruined and unknown, despite having produced the cubic equation’s solution, the first translation of Euclid’s “Elements” in a modern European language, formulated Tartaglia's Formula for the volume of a tetrahedron, devised a method to obtain binomial coefficients called Tartaglia's Triangle, that was then, developed to the pascal’s triangle and become the first to apply mathematics to the investigation of the paths of cannonballs. Even today, the solution to cubic equations is not known as Tartgalia’s Formula but as Cardano’s. by: Rita Barrocas 11ºB

9. Horner's rule • The Horner’s rule, which has this name due to after William George Horner, is an algorithm for the efficient evaluation of polynomials in monomial form. It is a manual process by which one may approximate the roots of a polynomial equation and it is also viewed as a fast algorithm to divide a polynomial by a linear polynomial with Ruffini's rule. by: Rita Barrocas 11ºB

10. Given the polynomial: • where are real numbers, we need to evaluate the polynomial with a value of x, for example x0. • To do this, we delineate a new succession of constants : by: Rita Barrocas 11ºB

11. Then b0 is the value of p(x0). • the polynomial can be written in the form • so, by iteratively substituting the bi into the expression: by: Rita Barrocas 11ºB

12. FrançoisViete • François Viete is considered for some to be the father of algebra as it’s knowntoday; • his innovative look at how equations are used to solve problems changed themethods behind analysis and that was what allowed algebra to evolve . • François Viete’sway of implicating new notation and theory behind equations permitted for the ancientGreek method to be absolutely rebuilt, allowing others to investigateproblems with a new view. by: Rita Barrocas 11ºB

13. During the time that men have tried to discover the extent π, they have come up with many ideas of figuring out as many digits as possible. • The earliest attempts at unraveling the mysteries of π were really just "guess and check" figures. They included everything from 22/7 to 211875/67441. It was enough to satisfy the needs during the centuries, however, the mathematicians continued the research. by: Rita Barrocas 11ºB

14. Next, they evolved to the next and latest phase in the calculation of pi: infinite products and sums. ThistrendbeganwithFrancoisViete’s formula: • This type of equation allows one to calculate one term at a time, consequently allowing one mathematician to work on one term and another to pick up where the other one left off. • Viete's method was extremely slow and clumsy, however it created a base for almost all advancements in π that had been done after that. by: Rita Barrocas 11ºB

15. Gauss Gauss developed the Gaussian function and it is a function of the form: for some real constants a, b, c > 0, and e ≈ 2.718281828 (Euler’s number) by: Rita Barrocas 11ºB

16. The graph of a Gaussian function has the characteristic of having a symmetric "bell curve" shape that quickly falls off towards plus or minus infinite. The parameter a is the height of the curve's culmination, b is the position of the centre of the culmination and c controls the thickness of the shape. Gaussian functions are usually used in statistics where they describe the normal distributions in signal processing where they serve to define Gaussian filters, in image processing where two-dimensional Gaussians are used for Gaussian blurs, and in mathematics where they are used to solve heat equations and diffusion by: Rita Barrocas 11ºB

17. John Atanasoff • John Atanasoffwas a bulgarianmatematicianwhohasbornin 1903 andhewasthe inventor of the first general purpose electronic computer. • Atanasoffdidn’tinventedit for fameorglory, heinvented ir for necessity • At that time, the Monroe Calculator, a mechanical machine, was the easiest method of performing mathematics automatically, but Atanasoff wanted to improve upon this device using various methods. • His initial research involved using combinations of IBM tabulators and Monroe calculators. by: Rita Barrocas 11ºB

18. Later, in 1936, he invented an analog calculator that could be used to analyze surface geometry. • However, He find these solutions were still relatively rudimentary and what he really desired was an electronic calculator. • In late 1937, he discovered a solution that would work and, with a $650 donation, he worked on building a prototype. • In November of 1939, he and a graduate student, Clifford Berry, constructed the Atanasoff-Berry Computer. The computer was able to solve up to 29 linear equations simultaneously. Its calculating power came principally from vacuum tubes and regenerative capacitor memory by: Rita Barrocas 11ºB

19. In 1940, Atanasoff and Berry performed research that suggested that their machine was the first of its kind. • On January 15, 1941, was told to the world about the existence of the device by a news article in the Des Moines Register . • In June of 1941, a scientist, called John Mauchly, vwent to Atanasoff's home and examined the machine during four days. • In September, Atanasoff took a job during the war as Chief of the Acoustic Division at the Naval Ordnance Laboratory. by: Rita Barrocas 11ºB

20. During this time, Atanasoff believed that the Iowa State College was applying for a patent on the computer., however, they were not. • In 1943, Mauchly, the man who had examined Atanasoff's computer, began working on ENIAC, a general purpose electronic computer. • In 1945, the US Navy decided to start working on a project known as the NOL computer, project and putted Atanasoff in charge . But the project was canceled since Atanasoff's work on acoustic research was considered more important. by: Rita Barrocas 11ºB

21. In 1947, Mauchly and John Eckert, his partner, applied for a patent on their ENIAC , which was only granted in 1964. The rights to the patent were sold to Remington Rand, which began manufacturing commercial computer systems. • On the 26th of May of 1967 ,the patent was contested and the trial lasted until the 13rd, of March of 1972, with more than 30,000 evidence exhibits and 77 witnesses. • On the 19th of October of 1973, the United States District Court Judge Earl Larson ruled that the patent was invalid while the ENIAC was derived from the Atanasoff-Berry Computer.Was proved that Atanasoff and Berry were the rightful owners of the patent and the creators of the first general purpose electronic computer, but the issue was neglected until much later. by: Rita Barrocas 11ºB

22. In 1952, Atanasoff founded the Ordnance Engineering Corporation, but sold the company to Aerojet General Corporation in 1956. • In 1961, he started Cybernetics Incorporated company.In 1981, Atanasoff was awarded the Computer Pioneer Medal from the IEEE. • In 1990, had been given to him the United States National Medal of Technology from President George Bush. • On the 15th of June of 1995 he died from natural causes. Today, by: Rita Barrocas 11ºB