math through the ages

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Math through the Ages

Quiz 1 Mathematics is an ongoing human endeavor, like literature, physics, art, economics, or music. It has a past and a future, as well as a present. The math we use today is different than the math that were used in the past, and the math well be using in the future will be different as well. Each stage in the development of mathematics builds on what has come before. Each contributor to that development was a person with a past and a point of view. Good way to use history in the mathematics classroom is by using storytelling. For example, when introducing the idea of how to sum an arithmetic progression, the teacher tells a story about Carl Friedrich Gauss. Gausss teacher gave the class a long assignment to take up time, telling the kids to add up all the numbers from 1 to 100. Gauss simply wrote down 5050 and was correct. Telling an interesting story will alow students a better chance to remember it. The story also reminds students that there are real people behind the mathematics that they learn, that someone had to discover the formulas and come up with the ideas It also can lead the class towards discovering the formula for themselves. There are many variations to Gausss story, causing some doubts about the story. So it may be necessary to warn that what they hear may not be the strict historical truth The main limitation of using historical and biographical anecdotes is that too often they are only distantly connected to the mathematics. One way to tightly intertwine the history with mathematics is to provide a broad overview. Historical information often allows us to share the big picture of math. It also explains why certain ideas were developed. History often helps by adding context. Mathematics is created by people in a particular time and place, and is often affected by that context. Knowing more about this context helps us understand how mathematics fits in with other human activities. Knowing the history of an idea can often lead to deeper understanding, both for us and for our students. Ex: Negative numbers. Mathematicians found them difficult to deal with, as they ahad trouble with the concept itself. Understanding this helps us empathize/understand students difficulties. History can also be a good source of student activities. The History of Mathematics in a Large Nutshell The story of mathematics spans several thousand years. Much of the math we now learn in school is very old. It belongs to a tradition that began in the Ancient Near East, then developed and grew in Ancient Greece, India, and the medieval Islamaic empire. Then late-Medieval and Renaissance Europe, and eventually became mathematics as it is now understood as. Other traditions such as the Chinese receive less attention because they had much less influence on the math we now teach. The study of the history of mathematics is based on sources. Before the 20th century, most cultures of Western civilization denied women access to significant formal education, particularly in the sciences. Even when a woman was educated enough to contribute to mathematics, she often struggled to get recognized. Beginnings No one quite knows how when and how math began. We do know that in every civilization that developed writing we also find evidence for some level of mathematical knowledge. Anthropologists have found many prehistoric artifacts that can, perhaps, be interpreted as mathematical. The oldest of which were found in Africa, dating as far back as 37,000 years. They show men and women engaging in mathematical activities for a long time. By about 5000 B.C., mathematics began to emerge as a specific activity. As societies adopted various forms of centralized government, they needed ways of keeping track of what was produced, how much was owed in taxes, etc. It became important to know the size of fields, volume of baskets, etc. Mathematics as a subject was born in the scribal traditions and the scribal schools. Most of the evidence we have for this period comes from Mesopotamia and Egypt. Ancient Egyptians wrote with ink on papyrus, a material that doesnt survive thousands of years. Also, most of the Egyptian archeological digs have been near stone temples and tombs, rather than at the sites of the ancient cities where mathematical documents are most likely to have been produced. We have few documents that hint at what Egyptian mathematics was like. Mesopotamians wrote on clay tablets with a wooden stylus, many of which have survived. This allowed an understanding of their mathematics. The two mathematical styles are quite different. The most extensive source of information about Egyptian mathematics is the Rhind Papyrus. Named after A. Henry Rhind, the 19th century archaeologist who brought it to England, it dates back to 1650 B.C. It contains on one side, extensive tables that were used as aids to computation. On the other side it contains a collection of problems probably used in the training of scribes. The example covers a wide range of mathematical ideas but stay close to the sorts of techniques that would be needed by the scribe to fulfill his duties. Some of the basic features of ancient Egyptian mathematics: They used two numeration system. One for writing on stone, the other for writing on paper. Both were based on grouping by tens One system used different symbols for various powers of ten. Multiples of a particular power were show by repeating the symbol. Like roman numerals. Their basic arithmetic operations were adding and doubling. They used an ingenious method of doubling to multiply or divide. Rather than working with fractions, they worked only with the idea of an nth part. The third 1/3 For other fractions they would express as sums of such parts. They could solve simple linear equations They knew how to compute or approximate the areas and volume of several geometric shapes. Egyptian Mathematics was taught by means of problems that were intended as examples to be imitated. Most of the problems seem to have their roots in the actual practice of the scribes. Some however, seem designed to give young scribes a chance to show their prowess at difficult or complicated computations. Most of our information about the mathematics of Mesopotamia comes from tablets produced between 1900 and 1600 B.C. The mathematics of this region is often referred to as the Babylonian mathematics Many tablets have been discovered, allowing for a clearer picture of what Mesopotamian math was like. The mathematical activity of the Babylonian scribes seems to have arisen from the daily necessities of running a central government. People became interested in the subject for its own sake, pushing the problems and techniques beyond what was strictly practical. The goal was to be a mathematical virtuoso, able to handle impressive and complex problems. Things we can say about Mesopotamian mathematics with a high degree of certainty: Numbers represented using a place-value system based on sixty. Repetitions of a ones symbol and a tens symbol were used to denote the numbers 1 through 59. Used tables of products, reciprocals, and conversion coefficients. Fractions were often expressed in sexagesimal format. They could handle linear equations. Also solve problems that we would describe as leading to quadratic equations. Geometry was devoted to measurement. Appear to have known and applied instances of Pythagorean Theorem. One interesting aspect of Babylonian mathematics is the occurrence of problems that do not even attempt to be practical, but instead have a recreational flavor. Social change of scribes being trained at home and scribal arts becoming family tradition stopped the rapid advancements in math. Overall, Babylonian mathematics was driven by methods Dont know too much about Chinese mathematics. Very few mathematical texts from China due to fragile material (bamboo) and mass book burnings. Texts that we have suggests civil servants that could solve simple mathematical problems. Most important text is called The Nine Chapters on the Mathematical Art. Annotated and supplemented by Liu Hui in 263 A.D Says the material in the book dates back to 11th century B.C but the actual text was put together around 100 B.C. Topics in it are varied. Problems arise from practical situations, but they have already been formalized. Some have recreational flavor. Geometric problems analyzed by cut and paste methods. Original Nine Chapters only contains problems and solutions. Chinese mathematics would grow for centuries but didnt influence the west. Even less is known about early Indian mathematics Evidence of workable number system and practical interest in elementary geometry. Vedas (600 B.C.) contains mathematical material mostly in the context of building altars. In it contains statement of Pythagorean theorem, methods for approximating the diagonal of a square, and surface areas and volumes of solids. Greek Mathematics Greek mathematicians unique in focusing on logical reasoning and proof. Earliest mathematical arguments go back to 600 B.C. Earliest witness to Greek mathematics is Euclids Elements (300 B.C.) Greek was one of the common languages. Not all Greek mathematicians were born in Greece. Greek mathematics refer to the language. Ex: Archimedes was from Syracuse and Euclid was from Alexandria. Had a common tradition. Mathematics was a pursuit for those who had the means and the time. Worked mostly alone. Greek mathematicians focused on geometry. Little interest in practical arithmetic. First Greek mathematicians were Thales (600B.C.) and Pythagoras (500 B.C.) Both men are said to have learned math in Egypt and Babylon. Thales the first to prove some geometrical theorems usch as the usm of angles in any triangle is equal to two right angles, the sides of similar triangles are proportional, and a circle is bisected by any of its diameters. Many stories on Pythagoras Semi-religious society called Pythagorean Brotherhood Home base Crotona. Dedicated to learning They are known for Pythagorean Theorem and incommensurable ratios. Ratios are very important for Greek mathematics since Greek geometers did not directly attach numbers to the objects they studied. Length of a line was not talked about. They used ratios to compare quantities. A ratio that cannot be a ratio of any two whole numbers is irrational. And the segment of the kind is incommensurable. Plato listed mathematics as a fundamental part of the ideal education in his book Republic. Aristotle uses mathematical examples when discussing correct reasoning. This suggests that by this time mathematicians were engaged in working out formal proofs. They began to understand that to prove theorems you needed to start with unproved assumptions. These postulates were accepted as true. Alexander the Great he spread the Greek language, culture and learning to other parts of the world. Greek learning spread to Alexandria and flourished, as the mathematical tradition became strong. Alexandria became the center of Greek mathematics by the end of the 4th century. Elements is the collection of the most important mathematical results of the Greek tradition. Its style and content were very influential to Greek mathematics and Western mathematical tradition. Archimedes (250 B.C.) wrote about areas and volumes of various curved figures Apollonius (200 B.C.) wrote a treatise on conic sections One important component of Elements is the tradition of mathematical problems. Greeks did not measure area by assigning numbers to them, they attacked the measurement of areas by trying to construct a rectangle whose area is the same as the area of a given figure. Problematic when dealing with circles. Problem was not solved Two other problems from Greek times Trisecting the angle constructing an angle that is one-third of a given angle Duplicating the cube constructing a cube whose volume is twice the volume of a given cube. They did solve both but their solutions involved using some sort of auxiliary device. Some Greek geometric problems were so difficult that they served as motivation for the development of coordinate geometry Greek also started using numbers to measure angles Most famous Greek astronomer was Claudius Ptolemy, who livei n Alexandria in 120 A.D His most famous work is the Syntaxis. Provided a workable practical description of astronomical phenomena. Diophantus was one of the most original Greek mathematicians. His Arithmetica focuses one solving algebraic problems. It is a simple list of problems and solutions. In its time had no impact but had a deep influence on European algebraists in 16th and 17th century. Greek mathematics lost some of its creative touch at 300 A.D. Began to emphasize producing editions and commentaries on older works. This created the problem of distinguishing between their additions and comments and the original texts. Pappus is known for his work Collections. He talks about the method of analysis but is vague in doing so. This vagueness led to the mathematicians of the Renaissance to attempt to find out more and ended up to new ideas and discoveries. Theon prepared new editions of Elements. Proclus (450 A.D.) was the last important writer in the Greek tradition. Wrote commentary on parts of Euclids elements. 5th century A.D. marks the end of the Greek mathematical tradition in its classical form. Quiz 2 For the next four hundred years, Europe and North Africa saw very little mathematical activity. Western Europe being invaded led to small intellectual activity. Eastern Europe were not interested in math compared to other subjects. After the Islamic Empire began to settle down and then mathematical research was found. During this quiet period in Europe and North Africa, the mathematical tradition of India grew. There was already some math tradition in India, but received some influence from late Babylonian astronomers and Greek astronomical texts. Astronomy was one of the main reasons for the study of mathematics in India. It started at astronomy but they became interested in math for its own sake Aryabhata, who did most of his mathematical work in the 6th century. Brahmagupta and Bhaskara from the 7th century. Another Bhaskara (Bhaskara II) from the 12th century. The texts we have about them are extended books on astronomy. Indian mathematicians invented the decimal numeration system. Introduced place value and created a symbol to denote an empty space. Some influence from China but were using place-value system based on powers of ten by 600. Also developed arithmetic with the numbers. Also contributed to trigonometry. Focused on sines instead of chords. Also interested in algebra and some combinatorics. Sum of arithmetic progressions. Quadratic equations. Several variables. Indian mathematicians produced a wide range of interesting mathematics. They didnt not give proofs or derivations, so no explanations are found on how they found an answer. Europe did not directly learn from India. But the results found in India made its way to Baghdad and the Arabic mathematic traditions. Islamic Empire stretched all the way to western edge of India to parts of Spain. Abbasids, has just come into power. They wanted to establish a new imperial capital. This new city called Baghdad became the cultural center of the Empire. Located on the Tigris River. Became the place where East and West could meet. First scientific works brought to Baghdad were books on Astronomy from India. Abbasid founded the House of Wisdom, an academy of science. Began to gather scholarly manuscripts in Greek and Sanskrit and scholars who could read them. They studied many Greek and Indian mathematical books. Began the new era of mathematical creativity One of the first Greek texts to be translated was Elements, which had a huge impact. Arabics adopted the Euclidean approach. Formulated theorems in his style. Arabic mathematical tradition marked by the use of a common language. Muhammad Ibn Musa AL-Khawrizmi fom the middle of the 9th century wrote many books. Explained decimal place value system, quadratic equations Algebra became important in Arabic math. Omar Khayyam (1048-1131) wrote a book to find a way to solve equations of degree 3, which he was unable to do algebraically but did geometrically. Only positive numbers made sense to Arabics. More willing to consider lengths of linesegments as numbers. Investigated foundations of geometry. Combinatorics also shows up. Practical math was involving. Only a small part of the Arabic tradition was transmitted to Europe. Which meant that many of the results had to be rediscovered. Around the 10th century, things were stable enough in Western Europe for people to focus on education again. People could go to places under Islamic control, such as Spain, to learn more about math. In the following centuries, European scholars spent time in Spain translating Arabic treatises on all sorts of subjects. Oldest and easiest texts were likely to be found in Spanish libraries because they were not a center of cultural activity. Aristotles work led to the interest in kinematics. Nicole Oresme came up with a graphical method for representing changing quantities that anticipates the modern idea of graphing a function. Trade was another source of contact for Europeans and the Islamic empire. You could travel and learn mathematics, just like Leonardo of Pisa did with his dad, who was a trader. His first book was the Liber Abbaci, which explained Hindu-Arabic numeration and geometric explanation of the rules for solving quadratic equations and a few other theoretical passages. He also published Practica Geometriae, which is a manual of practical geometryAs Italian merchants developed a need for calculations, many Italian abbacists tried to write books on arithmetics and algebra to meet this need. Luca Paciaoli wrote a book on the practical mathematics. His Summa was one of the first books to be printed. Around the end of the 14th century, many cultures around the world became interested in mathematics. Mayans developed a base-twenty place-value system China develops a sophisticated method for solving many problems. Indian and Arabic math continues to grow There were some contacts but little mathematical knowledge Changed when Europeans began to develop the art of navigation and to travel to distant countries and spread their culture. Jesuit cultural network extended throughout the world and European mathematics was taught and studied everywhere and eventually became the dominant form of mathematics worldwide. Astrology was an important part of long-range navigation and making star charts depended on trigonometry. Because of this trigonometry became an important theme in the 15th and 16th centuries. Also a growing interest in arithmetic and algebra with the rise of the merchant class Johannes Muller wrote about wrote on both trig and algebra, also writing on solely trig. At this time the list of trig functions became standardized. New formulas and new applications were discovered. Sines and cosines continued to be thought of as lengths of particular line segments. No one thought of them as ratios or lengths on a unit circle. Albrecht Durer wrote first printed work dealing with higher plane curves. In the 16th and early 17th centuries, algebra that takes center stage. Equations and operations were expressed in words, written out in full. Italian algebraists used the word cosa, meaning thing for the unknown quantity in their equations. Other countries used coss. As a result they were called the cossists, and algebra became known as the cossic arts. They did not use symbols for quantities other than the unknown. Several algebraists attemted to find amethod to solve cubic equations. First breakthrough: Ferro and then Tartaglia. Both discovered how to solve certain kinds of cubic equations, both kept it secret. Cardano convinced Tartaglia to share the method of the cubic and generalized it to a way of solving any cubic equation and shared it. Cardanos notation was of the old one: x^3 = 15x + 4 was cubus.aeq.15.cos.p.4 His method didnt work in the appearance of a negative root. Bombelli fixed this. Bombellis method can be described as the beginning of complex numbers. He also linked algebra and geometry in a more direct way. Algebra began to look more like it does today towards the end of the 16th century. Viete was a cryptographer, he introduced the notion to use letters to stand for numbers in equations. The most important thing Viete did was to promote algebra as an important part of mathematics Rene Descartes suggested the use of lowercase letters from the end of the alphabet for unknown quantities and lowercase letters from the beginning of the alphabet for known quantities. Three innovations that were important No one could figure out how to solve the fifth-degree equation led algebraists to ask deeper questions. Led to theory about polynomials and their roots evolved Descartes and Pieere de Ferman linked algebra and geometry. Using algebra to solve geometric questions. Fermat introduced a whole new category of algebraic problems. Fermat never published proofs but instead wrote letters to his friends. Other people were trying to use math to understand the universe Galileo Galilei studied astronomy and the physics of moving bodies. He insisted to use mathematics in order to have a chance of understanding the world. Johannes Kepler used the old Greek geometry of conic sections to describe the solar system. Studying motion led to difficult questions related to the infinite divisibility of space and time.Pg 60-95Keeping Count: Writing Whole Numbers How to write numbers efficiently was a problem for a long time. Simplest way to do this was by tallying making single mark for each thing counted. Still used in scorekeeping Tally system is too simple, its weakness is that it is too simple so it becomes difficult to use for large numbers. Cultures improved on this method. From 3000B.C. to 1000B.C. ancient Egyptians improved on the tallying system by choosing a few more number symbols and stringing them together until the values added to the number they wanted. These were hieroglyphics When Egyptians used ink, they developed better way to write numbers. New system used distinct basic symbols for each unit value from 1 to 9 for each ten multiple from 10 to 90 for each hundred multiple from 100 to 900, and for each thousand multiple from 1000 to 9000 Babylonian numeration system was based on two wedge-shaped symbols that were easily formed in tablets. Mesopotamians had several different numeration systems. Used place-value system. Multiplied successive groups of symbols by increasing powers of sixty. Their system called the sexagesimal system. Numbers from 60 to 3599 were represented by two groups of symbols. They wrote 3600 or greater by using more combinations of the two basic wedge shapes, placed further to the left. Major difficulty of Babylonian system is the ambiguity of the spacing between symbol groups. Mayans had similar numeration system. Two basic symbols, dot for number one and a bar for ten. These groups were arranged vertically. Lowest group represented singlue units, the values of the second multiplied by 20, the value of third multiplied by by 18*20, and so on. Spacing difficulty was dealth with by using a symbol for zero to show when a grouping was skipped. Mayan culture was not known to Europeans until later, so their system had no influence. Both the ancient Greeks and Romans were more primitive than the relatively simple and efficient Babylonian system. Main Greek system used 25 letters of their alphabet and two extra symbols. Nine symbols for the units Nine for the tens Nine for the hundreds. Numbers larger than one thousand had a special mark to indicate that the numeral shown was to be multiplied by 1000 Dominance of Roman empire made Roman numeration the commonly accepted European way of writing numbers for many centuries afterwards. Additive not positional. Values of the basic symbols added to determine the value of the entire numeral. Larger numbers were written by putting a bar over a set of symbols to indicate multiplication by 1000. Writing a smaller value before a bigger meant subtraction. Ambiguity was avoided by requiring that only symbols representing powers of ten may be subtracted, and they may only be paired with the next two larger values. One current method for writing numbers is called the Hindu-Arabic system. Invented by Hindus and picked up by Arabs. Uses place value and is based on powers of ten. Basic symbols called digits, represent the numbers zero through nine. Did not replace Roman numeration due to old habits. Roman numeration not good for computationReading and Writing Arithmetic: Where the Symbols Came From Symbols of arithmetics became universal. But wasnt always the case Ancient Greeks and Arabs didnt use any symbols for arithmetics. Wrote out the problems and solutions in words. Arithmetic and algebra statements were written only in words by most people for many, many centuries, right through the Middle Ages. Here are some ways in which (5+6) 7 = 4 would have appeared during the centuries from the Renaissance to now.Nothing Becomes a Number: The Story of Zero. Many think of zero as nothing. The fact that it is not nothing lies at the root of at least two important advances in mathematics. First, in Mesopotamia sometime before 1600B.C. By then Babylonians had a well-developed place-value system for writing numbers. Based on grouping by 60. Had two basic wedge-shaped symbols. Spacing between marks were not consistent since they were writing into soft clay tablets. For example, 72 and 3612 were written with the same symbols but 3612 had more space between the first mark. Babylonians started using their end-of-sentence symbol to show that a place was being skipped. Zero began its life as a place holder. Credit for base-ten place value system belongs to Hindus. They used a small circle as the placeholder symbol. Arabs learned this system. The symbols for single digits changed a bit but the principles were the same. Indian word for this absence of quantity, sunya, became the Arabic sifr, then the Latin zephirum, and these words in turn evolved into the English words zero and cipher. Today zero still indicates that a certain power of ten is not being used. By the 9th century A.D the Hindus made a conceptual leap that ranks as one of the most important mathematical events of all time. They began to recognize sunya, the absence of quantity, as a quantity itself. Mahavira wrote that a number multiplied by zero results in zero, and that zero subtracted from a number leaves the number unchanged. He also claimed that a number divided by zero remains unchanged. Bhaskara declared a number divided by zero to be an infinite quantity. Main important part is that they recognized zero as a number. To compute with zero, you must first recognize it as an abstraction on part with one, two, three. Move from counting one or two goats to ideas that exist even if they arent counting anything at all. Then it makes sense to treat zero as a number. Greeks didnt take this step into abstraction. Hindu recognition of 0 as a number was key for unlocking the door of algebra. Zero found its way to the West through the writing of Al-Khawrizmi Zero is not yet thought of as a number, just a placeholder in Khawrizmis writings. Many Europeans learned about the decimal place system and the essential role of zero from the translations. As the new system spread, it became necessary to explain how to add and multiply when one of the digits was zero. Helped make it seem more like a number. The Hindu idea that one should treat zero as a number in its own right took a long time to get to Europe. Many mathematicians in the 16th and 17th centuries were unwilling to accept zero as a root of equations. Two of those mathematicians used zero in a way that formed the theory of equations. Thomas Harriot, a geographer and the first surveyor of the Virginia colony, proposed a simple technique for solving algebraic equations. Move all the terms of the equation to one side of the equal sign so that the equation takes the form [Polynomial] = 0. Called the Harriots Principle This was popularized by Descartes in his book on analytic geometry. Harriots Principle became more powerful when linked with coordinate geometry of Descartes. By the 18th century, the status of zero had grown from place holder to number to algebraic tool. As 19th century mathematicians generalized the structure of the number systems to form the rings and fields of modern algebra, zero became the prototype for a special element. Fact that 0 plus or times a number leaves that number unchanged became the defining properties of the additive identity Harriots principle characterized a particularly important type of system called an integral domain.Something Less Than Nothing? Negative Numbers Negative numbers were not generally accepted even by mathematicians, until a few years ago They werent accepted until the middle of the 19th century. Numbers arose from counting and measuring. Fractions from a refined form of counting using smaller units. The smallest number possible was zero, so a concept of negative numbers was difficult to understand. Negative numbers first appeared when people began to solve equations that led to x being equal to a negative number Scribes of Egypt and Mesopotamia could solve such equations more than three thousand years ago, but never considered the possibility of negative solutions. Chinese mathematicians seem to have been able to handle negative numbers in intermediate steps towards solving equations. Our mathematics is rooted mainly in the work of ancient Greek scholars. Greeks ignored negative numbers completely. Greeks considered numbers as being positive whole numbers and thought of line segments, areas, and volume as different kinds of magnitudes. Even Diophantus never considered anything but rational numbers. Indian mathematician, Brahmagupta, recognized and worked with negative numbers as early as the 7th century. Treated positive numbers as possessions and negative numbers as debts. Later Indian mathematicians continued in this tradition. But they regarded negative quantities with suspicion for a very long time. Early European understanding of negative numbers was not directly influenced by this work. Indian mathematics came to Europe through the Arabic mathematical tradition However, the Arab mathematicians didnt use negative numbers. This could have been because algebraic symbols didnt exist back then. One thing Arabs understood was how to expand products of the for (x-a)(x-b) They knew that in this situation negative times negative is positive and negative times positive is negative. But they only applied this to problems involving subtractions whose answers are positive. Europeans learned from their predecessors a kind of math that dealt with only positive numbers. They were left to deal with negative numbers on their own. European math made huge leaps after the Renaissance, motivated by astronomy, navigation, physical science, warfare, commerce. There was continued resistance to negative numbers. In 16th century even prominent mathematicians such as Cardano, Viete, and Stifel rejected negative numbers. By the early 17th century, the usefulness of negative numbers became too obvious to ignore and some European mathematicians began to use negative numbers. But misunderstanding and skepticism about negative numbers persisted. If negatives are accepted as numbers, then rules for solving equations can lead to square roots of negatives. But if negative numbers make sense, then the necessary rules of their arithmetic require that squares of negatives be positive. Since squares of positive must also be positive, this means that sqrt(-1) can be neither positive nor negative. Even those 17th century mathematicians who accepted negative numbers were unsure of where to put them in relation to the positive numbers. They could operate with negative numbers but had difficulties with the concept itself. Antoine Arnauld argued that if -1 is less than 1 then the proportion -1 : 1 = 1 : -1, which says that a smaller number is to a larger as the larger number is to the smaller, is absurd. John Wallis claimed negative numbers to be larger than infinity. Isaac Newton said that quantities are greater than nothing or less than nothing. This definition was taken seriously due to his reputation. Questions lingered on how a quantity could be less than nothing. By the middle of the 18th century, negatives had become accepted as numbers. Many reputable scholars still had misgivings about them. Euler seemed comfortable with negative quantities. French not so much. When Euler had to explain why the product of two negative numbers were positive, he dropped the interpretation of negative numbers as debts and argues in a formal way, saying that a times b should be the opposite of a times b Half a century later, there were still doubters. Such as Augustus De Morgan Represents the last gasp of a fading tradition in the face of the emergence of a much more abstract approach to algebra. With the work of Gauss, Galois, Abel and others, study of algebraic equations evolved into a study of algebraic systems. In this more abstract setting, the real meaning of numbers became less important than their operational relationships to each other. In this setting negative numbers became important and their doubts disappeared.By Tens and Tenths: Metric Measurement It became important to measure as humans began to trade. Systems of measurements had to be based on some agreed-upon unit of measure. Some of the earliest standards were part of the human body Span the distance from the tip of the thumb to the tip of the little finger with the fingers spread out Palm The breadth of the four fingers held close together Digit The breadth of the first finger or the middle finger Foot Problem with these is that the sizes of the human body parts vary from person to person. They used a king for the standard units to be based. King Henry I declared the distance from the tip of his nose to the tip of his thumb with his arm stretched out as a yard. This became the basis for length in the English system of measurement. Main difficulty with using this system is calculating with the peculiar relationships among the various sizes of its units Many different measurement systems were used in different countries throughout the world until the late 18th century. The need for a single, universal unit of measure increased with international trade. French Academy of Sciences studied about a system based on the length of a pendulum that would make one full swing per second. Decided that the variations in temperature and gravity in different parts of the world would make this length unreliable. Proposed a new system baed on the length of sea-level meridian arc from the Equator to the North Pole. They called one ten-millionth of this arc a meter. Utility of the metric system comes from the fact that all smaller and larger units of length are power-of-ten multiples of the meter with prefixes that tell you which power of ten to use. Units of area and volume come from the same basic unit. Units of mass also based on the meter. Kilogram was defined to be the mass of pure water contained in a cube one decimeter on a side. Volume measure called liter French surveyors defined a new unit of angular measure, the grade, which was one hundredth of a right angle. This new unit was used to determine the precise length of one meter. French Academy accepted the finding but rejected the angular unit. It instead chose the radians as the standard metric unit of angular measure. Academy wanted to use radians to preserve the unique relationship between an angular measure a in radians and the linear measures of its arc, which is given by a = s/r Thus the grade is not really a metric unit but was used commonly in the 19th century Republic of France officially adopted the French Academys system in 1795. Public acceptance of the system came slow. Discarded by Napolean but restored later. International implementation when seventeen countries signed the Treaty of the Meter in 1875 New system called the International System of Units, based on meters and kilograms and five other basic units made for modern technologys increasing demands for precision. US passed a law in 1866 making it legal to use the metric system in commerce. Also the only English-speaking country to sign the Treaty of the Meter in 1875 Transition from English system to metric has been slow and grudging.Measuring the Circle: The Story of Pi Pi originally was a greek letter that corresponds to our letter p. The ratio of the circumference to the diameter of a circle is always the same. Circle shape is important for humans practically, the constant in the formulas for area and circumference is important. Finding the value of the constant pi has been a mystery for a long time. A timeline 1650 B.C Rhind Papyrus gave a procedure for computing the area of a circle that used the constant 4(8/9)^2 240B.C Archimedes showed it is between 3 10/71 and 3 10/70 150 A.D Ptolemy used 377/120 480- Zu Chopngzhi used 35/113 1600 Decimal value for it was computed to 35 places. 1706 William Jones first used the greek letter as the name of this number. Adopted by Euler and by the end of 18th century became the common name. 1873 William Shanks of England computed by hand a decimal value for pi to 607 places. Took more than 15 years. Digits after the 527th are incorrect. 1949 - John von Neumann used ENIAC computer to work out pi to 2035 places. Around 1765, Johann Lambert proved that pi is an irrational number. Just a few decimal places are good for practical purposes.The Cossic Art Writing Algebra with Symbols X and other letters now used to represent unknown numbers are relatively new. Standardization of notation was a critical step in the use and progress of algebra Good mathematical notation should be a universal language that clarifies ideas, reveals patterns, and suggests generalizations. Used to use words for equations in early years around 1200s such as the cube and seven things less five squares is equal to the root of six more than the thing Called a rhetorical approach Late 15th century some started to use symbols such as cu.m.5.ce.p.7.co------Rv.co.p.6 Co is the cosa, the unknown quantity. Ce and cu are for censo and cubo used for the square and the cube of the unkown. Long dash for equal sign R to denote square root Early 16th century Germany used some symbols that we use today such as + and and square root signs.

Higher powers written by combining square and cube symbols multiplicatively.

Chuquet wrote successive powers of the unknown by putting numerical superscripts on the coefficients. 5x^4 would be 5^4 1^3.m.5^2.p.7^1.montent.R^2.1^1.p.6^0 Not published Bombelli reintroduces the concept of placing exponents. Viete introduced using letters for both constants and unknowns. Harriot propses 5aaa+7ee way of writing Pierre Herigone proposes 5a3+7e2 Descartes proposes 5a^3+7e^2 Descartess method became standard. Used lowercase letters from the end of the alphabet for unknowns and lowercase letters from the beginning of the alphabet for constants. Introduced reverse 8 with end cut for equal sign Rover Recorde introduced the = sign in 1557 became widely used in England but not yet in other parts of Europe. Leibniz and Newton both adopted the equal sign. Leibnizs superior calculus notation of Leibniz gradually superseded that of Newton.Linear Thinking: Solving First Degree Equations Almost everyone who studied mathematics developed techniques for solving such problems. Rhind Papyrus contains several problems with first degree equations. Some simple and some complicated Used the method of false position. Assume a quantity is a certain number, one that may be incorrect but one that makes computation easy, and use the incorrect result of the guess to find the number by which needs to be multiplied to get the correct answer. Can only be applied to equations of the form Ax = B. For equations of the form Ax + C = B, we use the double false position. Guess once, find error, guess second time, find error. Cross multiply the guesses with the errors. Take the difference and divide by the difference of the errorsA Square and Things: Quadratic Equations Al-Khwarizmi starts out his book with a discussion of quadratic equations. Had to use words to describe the quadratic formula. Some differences between his version and modern version: Modern day we put the b at the beginning. Khwarizmi put the c on the right-hand side Modern day we care about both positive and negative square roots, but back then they only wanted the positive roots. He showed geometrically why his formula was true. Early in the 17th century, mathematicians came up with the idea of using letters to represent numbers. Thomas Harriot and Descartes noticed that its much easier if we make an equation = 0.A Cheerful Fact: The Pythagorean Theorem a and be are the lengths of the shorter side and c is the hypotenuse. Origins of the theorems are hard to trace. Greek tradition associates the theorem with Pythagoras. But we hear this from authors many centuries after Pythagoras. Pythagoras was a legend but little evidence that he was interested in mathematics. We find the theorem in one form or another in ancient world. In Mesopotamia, Egypt, India, China, and Greece. Oldest reference from India in the Sulbasutras, dating back to the first millennium B.C. Says that the diagonal of a rectangle produces as much as is produced individually by the two sides. Similar statements found in all of ancient cultures. Also in ancient cultures, we find triples of whole numbers that work as asides of right triangles. Most famous is (3, 4, 5). 9+16 = 25 Such triples arent easy to find, but are found in most ancient cultures Evidence suggests that the Pythagorean Theorem was actually known before Pythagoras himself. Two competing explanations One postulates a common discovery, which would have happened in prehistoric times Other argues that the theorem is so natural that it was discovered by many different cultures. Discovering theorem to be true is different than finding a proof of it. Earliest proofs used square in square pictures Arrange four identical triangles around a square whose side is their hypotenuse Many ways to prove Pythagorean theorem Most famous of all proofs from Euclids Elements His 47th proposition says that in right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle. His proof drops a perpendicular from the upper vertex of the right triangle, splitting the bottom square into two pieces. Then he proves that each piece of the bottom square is equal to the corresponding smaller square. Euclid proves the converse If in a triangle the square on one of the sides equals the sum of the squares on the remaining two sides of the triangle, then the angle contained by the remaining two sides of the triangle is right. Also proves kc^2 = k(a^2+b^2) = ka^2 + kb^2 Neatest proof of the theorem: Start with right triangle, and drop a perpendicular from the right angle onto the opposite side. Then you have three similar triangles. Obvious that two small triangles add up to the big one. Pythagorean Theorem remains important in multiple ways. Distance between two points formula comes from Pythagorean TheoremOn Beauty Bare: Euclids Plane Geometry 2300 years ago in Alexandria, a teacher named Euclid wrote the worlds most famous axiomatic system. His system studied by Greek and Roman scholars for thousand years Translated by Arabs too Became standard for logical thinking That system is Euclids description of plane geometry Geometry as a logical discipline began with Thales. First Greek philosopher and father of geometry as a deductive study. Began the search for unifying rational explanations of reality. Investigated logical ways to derive some geometric statements from others. By Euclids time, Greeks had developed a lot of mathematics, virtually all of it related to geometry or number theory. Platos philosophy and Aristotles logic were firmly established by then. Many mathematical results proven from apparently basic ideas Euclid organized and extended a large portion of what the Greek mathematics had learned. He wanted to put Greek math on a unified, logical foundation. He wrote an encyclopedic work called Elements separated into thirteen parts called books Books 1-4 and 6 are about plane geometry Books 11-13 are about solid geometry Books 5 and 10 are about magnitudes and ratios Books 7-9 are about whole numbers. Contains 465 theorems Very formal and dry Proofs end with a restatement of the proposition which was to be proved Paid special attention to geometry. He specified some basic statements that captured the essential properties of points, lines, angles, etc. Wanted to systemize observable relationships among spatial figures. Book V important because it contains a detailed theory of raitos among quantities of various types Euclid connects his work to Platos philosophy. Book I begins with ten basic assumptions Things equal to the same thing are also equal to each other If equals are added to equals, the results are equals If equals are subtracted from equals, remainders are equal Things that coincide with one another are equal to one another. The whole is greater than the part A straight line cane be drawn from any point to any point A finite straight line can be extended continuously in a straight line A circle can be formed with any center and distance All right angles are equal to one another If a straight line falling on two straight lines makes the sum of the interior angles on the same side less than two right angles, then the two straight lines, if extended indefinitely, meet on that side on which the angle sum is less than the two right angles. First five are general statements about quantities that were obviously true Last five are specifically geometric From the simple beginning, Euclid reconstructed the entire theory of plane geometry Elements is not just about shapes and numbers, it is about how to think. Was very influential. Descartes based part of his philosophical method on the long chains of reasoning used in Euclid Isaac Newton and Baruch Spinoza used the form of Euclids Elements to present their ideas Euclid was studied for many centuries as a model of precise. Students has burial tradition for Euclids book because it was so difficult to study. In modern day Euclids logical structure has been deemphasized.