A Brief History of Mathematics

Transcription

1 A Brief History of Mathematics I. Ancient Period Prehistoric people probably first counted with their fingers. They also had various methods for recording such quantities as the number of animals in a herd or the days since the full moon. To represent such amounts, they used a corresponding number of pebbles, knots in a cord, or marks on wood, bone, or stone. They also learned to use regular shapes when they molded pottery or carved arrowheads. As civilizations developed, the need for numeration systems, measurement techniques and arithmetic procedures also arose. By about 3000 B.C., mathematicians of ancient Egypt used an additive base ten system that was without place values. The Egyptians developed geometric formulas for finding the area and volume of simple figures. Their mathematics had many practical applications, ranging from surveying fields after the annual floods to making the intricate calculations necessary to build the pyramids. By 2100 B.C., the people of ancient Babylonia had developed a sexagesimal numeration system; a system based on groups of sixty. The system had important uses in astronomy and also in commerce, because sixty can be divided easily and works well with a calendar. It was also notable for the use of place value to represent numbers of any size. The system survives today in the way we measure time and angles. The Babylonians also went beyond the Egyptians in algebra and geometry. They found solutions to quadratic equations and developed techniques for calculating square roots. Chinese mathematics originally developed to aid record keeping, land surveying, and building. By the 100's B.C., the Chinese had devised a decimal system of numbers that included fractions, zero, and negative numbers. They solved arithmetic problems with the aid of special sticks called counting rods. The Chinese also used these devices to solve equations even groups of simultaneous equations in several unknowns. Perhaps the best-known early Chinese mathematical work is the Chiu Chang Suan Shu (Nine Chapters on the Mathematical Art, a handbook of practical problems that was compiled in the first two centuries B.C. In 263 A.D., the Chinese mathematician Liu Hui wrote a commentary on the book. Among Liu Hui's greatest achievements was his analysis of a mathematical statement called the Gou-Gu theorem. The theorem, known as the Pythagorean theorem in the West, describes a special relationship that exists between the sides of a right triangle. Liu Hui also calculated the value of pi more accurately than ever before. He did so by using a figure with 3,072 equal sides to approximate a circle. II. Greek Period (600 B.C. to 400 A.D.) Ancient Greek scholars introduced the concepts of logical deduction and proof to create a systematic theory of mathematics. According to tradition, one of the first to provide mathematical proofs based on deduction was the philosopher Thales, who worked in geometry about 600 B.C. He was a Greek merchant whose travels brought him in contact with the mathematics of Babylonia and Egypt. Until this time, geometry had consisted strictly of measuring techniques (in fact the word geometry means earth measurement ). However, Thales made abstract, general statements, such as when two lines intersect, they create pairs of equal angles, and attempted to justify those statements logically. The Greek philosopher Pythagoras, who lived about 550 B.C., explored the nature of numbers, believing that everything could be understood in terms of whole numbers or their ratios. His followers explored number patterns and discovered irrational numbers. They had a significant influence on Greek philosophy and the promotion of mathematics for its own sake. Around 300 B.C, Euclid, one of the foremost Greek mathematicians, organized the geometrical ideas of the previous three hundred years into a systematic, logical structure in his work The Elements of Geometry. In this book an entire system of geometry is constructed by means of abstract definitions,

2 accepted facts (postulates) and logical deductions. It had an enormous impact on mathematical thought and became the model for the development of a mathematical system. During the 200's B.C., the Greek mathematician and physicist Archimedes used the method of exhaustion to find many formulas for the volume and surface areas of solids and to calculate a highly accurate value for pi (the ratio of a circle's circumference to its diameter). He was also famous for creating many engineering devices, such as Archimedes screw and discovering some of the fundamental laws of physics. Working at about the same time, the Greek mathematician Appolonius of Perga, known as the Great Geometer wrote an eight volume work in which he investigated the curves obtained by taking cross sections of a double cone. These curves, the circle, ellipse, parabola and hyperbola, are called conic sections. One of the last great Greek scholars, Ptolemy, applied geometry and trigonometry to astronomy about A.D In a book, known as the Almagest, he presented a scheme for the motions of the heavenly bodies. He claimed that the earth was stationary and that it was in the center of a larger sphere around which the sun, stars, moon and planets revolved at uniform rates of speed. This model became the accepted theory of the solar system throughout the middle ages, both in the European and the Islamic worlds. Pythagoras of Crotona Thales of Miletus Apollonius of Perga Archimedes of Syracuse Euclid and Ptolemy of Alexandria Eratosthenes of Cyrene The World of Greek Mathematics

3 III. Hindu-Arabian Period (200 B.C. to 1200 A.D. ) Mathematics and the sciences entered a long period of stagnation with the decline of the Greek and Roman civilizations. This inactivity was uninterrupted until after the Islamic religion and the resulting Islamic culture were founded by the prophet Muhammad in A.D Within a century, the Islamic empire stretched from Spain, Sicily, and Northern Africa to India. Islamic culture encouraged the development of the sciences as well as the arts. Arab scholars translated many Greek and Hindu works in mathematics and the sciences, including Apollonius's work on conic sections. It is likely that much of the Greeks' work in science and mathematics would have been lost if not for these Arab scholars. The Arab mathematician Mohammed ibn Musa al-khowarizmi wrote two important books around A.D. 830, each of which was translated into Latin in the twelfth century. Much of the mathematical knowledge of medieval Europe was derived from the Latin translations of al-khowarizmi's two works. Al-Khowarizmi's first book, on arithmetic, was titled Algorithmi de numero Indorum (or al- Khowarizmi on Indian Numbers). The Latin translation of this book introduced to Europe the Hindu number system and the simpler calculation techniques (such as the procedures for multiplication and long division) that system allows. This system is now called the Hindu-Arabic number system. The book's title is the origin of the word algorithm, which means a procedure for solving a certain type of problem, such as the procedure for long division. Al-Khowarizmi's second book, Al-Jabr w'al Muqabatah, discussed linear and quadratic equations. In fact, the word algebra comes from the title of this second book. This title, which translates literally as Restoration and Opposition, refers to the solving of an equation by adding the same thing to each side of the equation (which "restores the balance" of the equation) and simplifying the result by canceling opposite terms (which is the title's "opposition"). For example (using modern symbolic algebra): 6x = 5x+ 11 6x + -5x = 5x x "al-jabr" or restoration of balance x = 11 al-muqabalah" or opposition The quote below, from a translation of Al-Jabr w'al Muqabalah, demonstrates several important features of al-khowarizmi 's algebra. First, it is entirely verbal, as was the algebra of Apollonius there is no symbolic algebra at all. Second, this algebra differs from that of Apollonius in that it is not based on proportions. Third, the terminology betrays the algebra's connections with geometry. When al- Khowarizmi refers to "a square," he is actually referring to the area of a square; when he refers to "a root," he is actually referring to the length of one side of the square (hence the modern phrase "square root"). Modern symbolic algebra uses the notations x 2 and x in place of "a square" and "a root." The quote from Al-Jabr w'al Muqabalah is on the left; a modern version of

4 the same instructions is on the right. You might recognize this modern version from intermediate algebra, where it is called "completing the square." The following is an example of squares and roots equal to numbers: a square and 10 roots are equal to 39 units. The question therefore in this type of equation is about as follows: what is the square which combined with ten of its roots will give a sum total of 39? The manner of solving this type of equation is to take one half of the roots just mentioned. Now the roots in the problem before us are 10. Therefore take 5, which multiplied by itself gives 25, an amount which you add to 39, giving 64. Having taken then the square root of this which is 8, subtract from it the half of the roots, 5, leaving 3. The number three therefore represents one root of this square, which itself, of course, is 9. Nine therefore gives that square. x x = 39 Solve for x 2 ½ 10 = = 25 x x + 25 = x x + 25 = 64 (x + 5) 2 = 64 x + 5 = 8 x = 8-3 x = 3 x 2 = 9 Al-Khowarizmi, like Apollonius, understood numbers to be lengths of line segments, areas, and volumes. He did not recognize negative numbers, because neither a line nor an area nor a volume can be represented by a negative number. Arab astronomers of the 900's made major contributions to trigonometry. During the 1000s, an Arab physicist known as Alhazen applied geometry to optics. The Persian poet and astronomer Omar Khayyam wrote an important book on algebra about In the 1200s, Nasiral-Din al-tusi, a Persian mathematician, created ingenious mathematical models for use in astronomy. IV. Period of Transmission (1000 AD 1500 AD) Contest between abacus and newer methods The Moors (Moslems from North Africa) entered Spain in A.D. 711 and built universities in Toledo, Cordoba, and Seville. This culture was the only major exception to the mathematical and scientific stagnation in Europe that started with the end of the Roman Empire and continued through the Middle Ages. By the twelfth century, many Arabic mathematical and scientific works (including al-khowarizmi's two books), as well as Greek and Hindu works, were translated into Latin, often by Jewish scholars in Spain. Greek works of literature and philosophy were also translated. In 1202, Leonardo of Pisa (Fibonacci), an Italian mathematician, published a book on algebra, the Arab number system, and pen and paper arithmetic techniques that helped promote this system. Hindu-Arabic numerals gradually replaced Roman numerals and the use of an abacus in Europe. During the 1400s and 1500's, European explorers sought new overseas trade routes, stimulating the application of mathematics to navigation and commerce. As trade expanded, Arab and Greek knowledge was transmitted throughout Europe. In 1453 the Turks conquered Constantinople, the last remaining center of Greek culture. Many Eastern scholars moved from Constantinople to Europe, bringing Greek knowledge and manuscripts with them. Around the same time, Gutenberg invented the movable-type printing press, which greatly increased the availability of scientific information in the form of both new works and translations of ancient works. Translations of Euclid's work, Ptolemy s Almagest and some of Apollonius's work on geometry were printed, as was the Franciscan monk Luca Pacioli's Summa de arithmetica, geometrica, proportioni et proportionalita, which was a summary of the arithmetic, geometry, algebra, and double-entry bookkeeping known at that time.

5 V. Early Modern Period (1500 AD 1800 AD) Development of Algebra and Analytic Geometry For the Arab mathematicians, algebra was a set of specific techniques that could be used to solve specific equations. There was little generalization, and there was no way to write an equation to represent an entire class of equations, as we would now write x 2 + bx + c = 0 to represent all quadratic equations. There were only ways to write specific equations such as 3x 2 + 5x +7 = 0. Thus, it was impossible to write a formula like the quadratic formula [if ax 2 + bx + c = 0, then x = (-b ± )/2a]. It was only possible to give an example, such as al-khowarizmi's example of completing the square. In the late sixteenth century, algebra matured into a much more powerful tool. It became more symbolic. Exponents were introduced; what had been written as "cubus," "A cubus" or "AAA" could now be written as "A 3." The symbols +. -, and = were also introduced. Francois Viete, a French lawyer who studied mathematics as a hobby, began using vowels to represent variables and consonants to represent constants. This allowed mathematicians to represent the entire class of quadratic equations by writing "A 2 + BA = C" (where the vowel A is the variable and the consonants B and C are the constants) and made it possible to discuss general techniques that could be used to solve classes of equations. All these notational changes were slow to gain acceptance. No one mathematician adopted all the new notations. Viete's algebra was quite verbal. He did not even adopt the symbol + until late in his life. In 1637, the famous French philosopher and mathematician Rene Descartes published La Geometrie, a work that explored the relationship between algebra and geometry in a way unforeseen by Apollonius and al-khowarizmi. Descartes showed how to interpret algebraic operations and solve quadratic equations geometrically. He also showed that algebra could be applied to geometric problems. This approach is now called analytic geometry. To the readers of Descartes it was an amazing method that combined algebra and geometry in new and unique ways. However, it did not especially resemble our modern analytic geometry, which consists of ordered pairs, x and y axes, and a correspondence between algebraic equations and their graphs. Descartes used an x axis, but he did not have a y axis. Although he knew that an equation in two unknowns determines a curve, he had very little interest in sketching curves: he never plotted a new curve directly from its equation. In 1629, eight years before Descartes's La Geometrie, the French lawyer and amateur mathematician Pierre de Fermat attempted to recreate one of the lost works of Apollonius on conic sections using references to that work made by other Greek mathematicians. Fermat applied Viete's algebra to Apollonius's work and created an analytic geometry much more similar to the modern one than was Descartes's. Fermat emphasized the sketching of graphs of equations. He showed a parallelism between certain types of equations and certain types of graphs. For example, he showed that the graph of "d planum p. a planum aequetur b in e (d 2 + a 2 = be) is always a parabola. Modern analytic geometry is thus considered to be an invention of both Descartes and Fermat. Descartes's algebra was more modern and sophisticated than Fermat s or any of his contemporaries. Fermat, on the other hand, developed the important relation between geometric shapes and a coordinate graph. Together they are credited with developing analytic geometry to the point where calculus could be invented.

6 Problems: 1. What type of mathematics are the Greeks known for? 2. What are the conic sections? 3. Which Greek mathematician is known for his work on conic sections? 4. What influential book did Ptolemy write? Why is it important? 5. How did Thales influence Greek mathematics? 6. The mathematics and science of the Greeks could well have been lost if it were not for a certain culture. Which culture saved this Greek knowledge, expanded it, and reintroduced it to Europe? 7. What were the subjects of al-khowarizmi's two books? 8. Why is our modern number system called the Hindu-Arabic number system? What is important about this system? 9. Al-Khowarizmi described his method of solving quadratic equations with an example; he did not generalize his method into a formula. What characteristic of the mathematics of his time limited him to this form of a description? What change in mathematics lifted this limitation? To whom is that change due? Approximately how many years after al- Khowarizmi did this change occur? 10. What is analytic geometry? 11. What did Descartes contribute to analytic geometry? 12. What did Fermat contribute to analytic geometry? 13. What did Viete contribute to algebra?