13.9: Freezing Point Depression and Boiling Point Elevation: Making Water Freeze Colder and Boil Hotter
Alcuino de York was born in Great Britain in the city of Northumbria in 735 and died on May 19, 804 in Tours, France. He studied in Italy and also at York Cathedral School, where he taught for about 15 years. It was there that he created one of the best libraries in Europe at the time and made the school one of the largest centers of knowledge.
Abraham Bar Hiyya was born in 1092 and died in 1167. He was a Jewish mathematician and astronomer who lived in Spain. It was also known by its Latin name: Savarsoda, which means governor of the city. He was educated in one of the Arab principalities of the Caliphate of Cordoba, but it was in Barcelona that Abraham wrote his original works in Hebrew.
Let S be the sum of the infinite terms of a PG of strictly positive numbers with ratio 2 and 1 = 1. S = (1 + 2 + 4 + 8 + 16 + 32 +…) => from a to 2, all terms are multiples of 2. If we highlight 2, we have: S = 1 + 2. (1 + 2 + 4 + 8 + 16 + 32 +…) As S = (1 + 2 + 4 + 8 + 16 + 32 +…), we have: S = 1 + 2.
We start with the following equality, which we assume to be true: a + b = c We can write equality as follows: (8a-7a) + (8b-7b) = (8c-7c) Putting all multiples of 7 on one side and 8 on the other, we have: 8a + 8b-8c = 7a + 7b-7c Putting 7 on one side and 8 on the other, we have: 8 (a + bc) = 7 (a + bc) Dividing both sides by a + bc have: 8 = 7 Obviously this demonstration has an error, because we all know that 8 is not equal to 7 (or does anyone have any questions?
Marina S. Rodrigues Almeida Psychologist, Pedagoa and Psychopedagogue CRP 06 / 41029-6 Mathematical Concept: Science that investigates through deductive reasoning, the relationships between abstract entities, such as numbers, geometric figures, etc., and the properties of these entities. . (Sérgio Ximenes - Dictionary of Portuguese Language) History Mathematics, which arose in ancient times for the necessities of daily life, has become a huge system of varied and extensive disciplines.
Gladis Blumenthal Abstract The teaching of mathematics has, over the years, undergone successive reforms. Even so, the mathematical school failure continues. As Municipal and State Secretariats of Education strive to absorb and adapt to the new rules in force, National Curriculum Parameters (NCPs) play an important role.
Claudia Lisete Oliveira Groenwald Ursula Tatiana Timm Abstract This article was the result of research carried out at the Lutheran University of Brazil, in the Mathematics Degree course. Emphasizes the importance of games and challenges as a teaching methodology in mathematics classes that need, in order to play them, the use of mathematical knowledge.
Benoit Mandelbrot was born in Warsaw, the capital of Poland, on November 20, 1924. His family was Jewish and had originally come from Lithuania. His father worked as a clothing manufacturer. In 1936, when Benoit was 12, Hitler was beginning to threaten Europe, so the family moved to Paris, where his paternal uncle SzoIem taught mathematics at the University.
Isaac Barrow was an English mathematician and theologian, credited for his discoveries in the area of modern calculus. He was born in 1630 in London and died in the same city on May 4, 1677 at the age of 46. He became a religious minister and taught mathematics, but did not gain due recognition for his discoveries in modern calculus.
Girolamo Cardano was born on September 24, 1501 in Pavia, Italy. He died on September 21, 1576 in Rome. Cardano was an Italian physicist and mathematician who devoted himself to mathematics, physics, astronomy, philosophy, medicine and astrology. In mathematics, his masterpiece is the book Artis Magnae Sive by Regulis Algebraicis (The Great Art or the Rules of Algebra), published in 1545, where are: - the method of solving the degree 3 equations, obtained from his friend Tartaglia (1499 - 1557), and grade 4, obtained from his disciple Lovic Ferrari (1522 - 1565); - the rule: "less times less gives more".
Bhaskara Akaria lived from approximately 1114 to 1185 in India. Born into a traditional family of Indian astrologers, he followed the family's professional tradition, but with a scientific orientation, focusing more on mathematical and astronomical aspects (such as calculating the date and time of eclipses or the positions and conjunctions of planets) that supports Astrology.
George Boole was born in 1815 and died in 1864. British Mathematician, was born in Lincoln on 2 November 1815. He was the son of a shoemaker, thus unable to afford a high degree in education. But his determination would lead to this obstacle being overcome. As a child he studied at Lincoln Elementary School, and then at a Commercial School.
Jean Le Rond d'Alembert, was born on November 17, 1717, and died on October 29, 1783. He was a French mathematician and physicist who developed the early phases of the CALCULUS, formalized the new science of mechanics, and was the editor of science of Diderot's Encyclopedia. With DIDEROT and VOLTAIRE, he was one of the leading figures of enlightenment in France.
David Hilbert was born on January 23, 1862, and died on February 14, 1943. He was a German mathematician whose work in geometry had the greatest influence on the field since Euclid. After making a systematic study of the axioms of Euclidean geometry, Hilbert proposed a set of 21 axioms and analyzed their meanings.
By the year 220 BC many people already thought the earth was round, but no one could tell what its circumference was. Disagreeing with this state of affairs, a Greek citizen named Eratosthenes resolved to remedy the failure. But if he was nonconformist, he was also self-indulgent. And, besides being a commodist, an astronomer, he tried to solve the problem without having to leave the house using the sun.
Leonardo Pisano, born in 1170, and died after 1240. Also known as Leonardo de Pisa or Leonardo Fibonacci, was the first great mathematician of medieval Christian Europe. He played an important role in reviving ancient mathematics and making significant contributions. Liber Abacci (Book of Abacus, 1202), his treatise on arithmetic and elementary algebra, introduced the modern Hindu-Arabic system of numbers using ten symbols.
Diophantus is named after the city which was the largest center of mathematical activity in ancient Greece. Little is known about his life, ignorance even prevents us from securely fixing in which century he lived. Distant dates of a century have been suggested before or after the year 250 d. Ç.
Johann Müller de Königsberg was a German mathematician and astronomer who was born on June 6, 1436, near Königsberg (meaning King's mountain in Latin Regiomontanus) in Germany, and died on July 6 in Rome, Italy. Considered a prodigy from a young age, Johann Müller contributed significantly to trigonometry and astronomy.
Leopold Kronecker (1823 - 1891) was born in Germany to Jewish parents although he opted for Protestantism. He was a very prosperous businessman who had strong connections with professors at the University of Berlin, where he accepted a post in 1883. In contact with Weierstrass, Dirichlet, Jacobi, and Steiner obtained his doctorate in 1845 with a thesis on algebraic number theory.
Tommaso Ceva was an Italian Jesuit mathematician, poet and priest. He was born in the city of Milan on December 20, 1648 and died on February 2, 1737. Although less well known than his brother Giovanni Ceva (from which the term "ceviana" comes from), he was also brilliant mathematical and physical.