Equation Solving Smack Down! cubic equation edition

1. Equation Solving Smack Down!cubic equation edition Lodovico Ferrari  (1522-1565) Nicolo Fontana Tartaglia (1499-1557)

2. Early Algebra • Egypt: The Rind Papyrus (1650 B.C.) solved linear equations. The Cairo Papyrus (300 B. C) solved simple quadratic equations. • Babylonia: Knew the quadratic formula by 1600 B.C. • Greek algebra before 250 A.D. was based on geometry • All of this early algebra was “rhetorical” – it used no symbols.

3. Indian and Arabic Mathematics • India took Greek mathematics and developed early symbolic methods • Arabic mathematics Indian algebra and spread it to Europe • The word "algebra" is named after the Arabic word "al-jabr" from the title of the book al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala, meaning The book of Summary Concerning Calculating by Transposition and Reduction, a book written by the Persian mathematician Muhammad ibn Mūsā al-khwārizmī in 820. The word Al-Jabr means "reunion".

4. Algebra Flourishes in Europe • As Europe awoke from the dark ages, what we know as algebra began to develop • It took hundreds of years for algebra’s modern symbols to evolve

5. Nicole Oresme (c. 1323-1382) • French economist, mathematician, physicist, astronomer, philosopher, psychologist, musicologist, and theologian • Advisor, chaplain, and chief secretary to King Charles V • First to use fraction exponents • May have discovered the rules(but he didn’t use modern notation)

6. Solving The Cubic 3x3 + 7x2 - 3x + 4 • Tartaglia and Ferrari each wanted to solve the general cubic equation first • They feuded publically with insults and accusations of plagarism • Their famous competition was held in Milan in 1548 with the governor as judge • All the important Milanese were present

7. François Viète (1540-1603) • A lawyer by training, he was an amateur mathematician in the court of King Henry IV of France • Known as “The Father of Algebra” • Introduced the first systematic algebraic notation in 1591 • Used letters for constants and unknowns

8. René Descartes (1596-1650) • Descartes used algebra to describe points, lines, and circles geometrically using a coordinate system

9. Evariste Galois (1811-1832) • A gifted mathematician from an early age • At age 16 his school wrote of him:It is the passion for mathematics which dominates him, I think it would be best for him if his parents would allow him to study nothing but this, he is wasting his time here and does nothing but torment his teachers and overwhelm himself with punishments. and described him as “singular, bizarre, original and closed” • Opposed the Royalists during the French Revolution • Was killed in a duel at age 21 • Some say the duel was over a woman (Stephanie-Felice du Motel), other say it was fought for the cause of the revolution

10. Galois’ Mathematics • The night before he died, he tried to put together his theories for posterity and wrote in the margin:There is something to complete in thisdemonstration. I do not have the time. • His work on Group Theory was scorned during his lifetime but it became the basis of modern mathematics • Group theory expresses the formal rules of algebra • Galois used it to prove that there is no general solution to the quintic equation

11. Fermat’s Last Theorem Galois’ work paved the way to proving Fermat’s Last Theorem: If an integer n is greater than 2, then there are no integers a, b, and c that solve an + bn = cn

12. Fermat’s Last Theorem MATH RIOTS PROVE FUN INCALCULABLE by Eric Zorn • News Item (June 23) -- Mathematicians worldwide were excited and pleased today by the announcement that Princeton University professor Andrew Wiles had finally proved Fermat's Last Theorem, a 365-year-old problem said to be the most famous in the field. • Yes, admittedly, there was rioting and vandalism last week during the celebration. A few bookstores had windows smashed and shelves stripped, and vacant lots glowed with burning piles of old dissertations. But overall we can feel relief that it was nothing -- nothing -- compared to the outbreak of exuberant thuggery that occurred in 1984 after Louis DeBranges finally proved the Bieberbach Conjecture. • "Math hooligans are the worst," said a Chicago Police Department spokesman. "But the city learned from the Bieberbach riots. We were ready for them this time." • When word hit Wednesday that Fermat's Last Theorem had fallen, a massive show of force from law enforcement at universities all around the country headed off a repeat of the festive looting sprees that have become the traditional accompaniment to triumphant breakthroughs in higher mathematics. • Mounted police throughout Hyde Park kept crowds of delirious wizards at the University of Chicago from tipping over cars on the midway as they first did in 1976 when Wolfgang Haken and Kenneth Appel cracked the long-vexing Four-Color Problem. Incidents of textbook-throwing and citizens being pulled from their cars and humiliated with difficult story problems last week were described by the university's math department chairman Bob Zimmer as "isolated."

13. Algebra Can Prove Anything? • By the end of the 19th century, mathematicians began to believe that by combining logic and algebra, they could prove any algebraic question that can be posed • Kurt Gödel proved in 1931 that some questions can’t be answered