LP 1 History of Mathematics PDF

 PREFACE History of Mathematics deals with historical context and timeline that led to the present understanding and applications of the different branches of mathematics. Topics included in this course are not very technical and rigid aspects of mathemat...

 PREFACE History of Mathematics deals with historical context and timeline that led to the present understanding and applications of the different branches of mathematics. Topics included in this course are not very technical and rigid aspects of mathematics; rather they are early, interesting, and light developments of the field. They are intended to enrich the background of the students in the hope that the students find value and inspiration in the historical approach to the mathematical concepts. The course begins with an introduction of the development of mathematics in an ancient period and its origin. By exploring these topics, students are encouraged to see how mathematics was developed over time and in various places to develop a deeper understanding of the mathematics they have already studied. The course then continues to study the development of mathematics to the modern period in which mathematics provides an understanding of its growth. It will cover the issues and various aspects such as the concepts and role of the proof, infallibility and certainty in mathematics and integration of technology in mathematics. These aspects will provide opportunities for actually doing mathematics in a broad range of exercises that bring out the various dimensions of mathematics as a way of knowing, and test the students’ understanding and capacity. (CMO No. 20, series of 2013). At the end of this course, the student should be able to: Demonstrate knowledge on the historical facts and landmarks that led to the development of the different branches and schools of thought in mathematics; Analyze popular problems involving foundational concepts in mathematics; Value mathematics as a dynamic field through sharing of personal experiences of enlightenment relative to the evolution of the different branches of mathematics.  Unit 1 – The Development of Mathematics: Ancient Period 1.0 Intended Learning Outcomes a. Discuss the development of mathematics in the ancient period. b. Show the evolution of numeration systems in ancient times. c. Perform the mathematical operations used in this period. 1.1 Introduction Our journeys of learning the history of mathematics now begin. In this area we are dependent on archaeologists and anthropologists for the comparatively small amount of historical information available. This historical information is very useful as we go along with our exploration in understanding the development of mathematics. Symbol-making has been a habit of human beings for thousands of years. The wall paintings on caves in France and Spain are an early example, even though one might be inclined to think of them as pictures rather than symbols. Symbol representations abuse a basic human ability to make correspondences and understand analogies. The root of the term mathematics is in the Greek word mathemata, which was used quite generally in early writings to indicate any subject of instruction of study. The concept of numbers always the first thing that comes to mind when mathematics is mentioned. From the simplest finger counting by pre-school children to the recent sophisticated proof of Fermat’s last theorem, numbers are a fundamental component of the world of mathematics. We shall now elaborate on the origin of mathematics and its components. Since these origins are in some cases far in the past, our knowledge of them is indirect, uncertain, and incomplete. We shall begin by considering the numeration systems of the important Near Eastern civilizations – the Egyptian and the Babylonian – from which sprang the main line of our own mathematical development. For this chapter, you will be dealing with Mathematics in Ancient Period! 1.2 Origins of Mathematics: Egypt and Babylonia 1.2.1 Egyptian Mathematics The writing of history, as we understand it, is a Greek invention; and foremost among the early Greek historians was Herodotus. Herodotus - Herodotus (c. 484 – 425/413 BCE) was a Greek writer who invented the field of study known today as `history'. - He was called `The Father of History' by the Roman writer and orator Cicero for his famous work The Histories but has also been called “The Father of Lies” by critics who claim these `histories' are little more than tall tales. - Born at Halicarnassus, a largely Greek settlement on the southwest coast of Asia Minor. - He came from a wealthy, aristocratic family in Asia Minor who could afford to pay for his education. His skill in writing is thought to be evidence of a thorough course in the best schools of his day. - He wrote in Ionian Greek and was clearly well read. - He served in the army as a Hoplite in that his descriptions of battle are quite precise and always told from the point of view of a foot soldier. - In early life, he was involved in political troubles in his home city and forced to exile to the island of Samos. - From there, he set out on travels and made three (3) principal journeys, perhaps a merchant, collecting materials and recording his impressions. - Herodotus spent his entire life working on just one project: an account of the origins and execution of the Greco-Persian Wars (499–479 B.C.) that he called “The Histories.” (It’s from Herodotus’ work that we get the modern meaning of the word “history.”) Nile River 3100 BCE – The first dynasty to rule both Upper Egypt (the river valley) and Lower Egypt (the delta). The legacy of the first pharaohs (Menes) included elite of officials and priests, a luxurious court, and for the kings themselves, a role as  intermediary between mortals and gods. This role fostered the development of Egypt’s monumental architecture, including the pyramids, built as royal tombs, and the great temples at Luxor and Karnak. - Writing began in Egypt at about this time, and much of the earliest writing concerned accounting, primarily of various types of goods. There were several different systems of measuring, depending on the particular goods being measured. But since there were only a limited number of signs, the same signs meant different things in connection with different measuring systems. There were two (2) styles in writing: 1. Hieroglyphic writing – for monumental inscriptions 2. Hieratic writing (cursive) – done with brush and ink on papyrus. Jean Champollion (1790-1832) – is a historian who was able to begin the process of understanding Egyptian writing early in the nineteenth century through the help of a multilingual inscription – the Rosetta stone – in hieroglyphics and Greek as well as the later demotic writing, a form of the hieratic writing of the papyri. (Fig. 1.2) 1.2.1.1 Number Systems and Computations Hieroglyphic Numbers In the hieroglyphic system, each of the first several powers of 10 was represented by a different symbol. Their counting system was decimal but non-positional and could deal with numbers of great scale. Below are a decimal system using seven different symbols. Numbers are formed by grouping. Put the smaller digits on the left.  For example, to represent 12,643 the Egyptians would write: Example 1. Example 2. Addition Addition is formed by grouping. Multiplication Multiplication is basically binary.  Example: Multiply 47 x 24 Division Division is also binary. Example: Divide 329 ÷12 Hieroglyphic Fractions Egyptian fractions are unit fractions, that is fractions with numerator 1. Unit fractions are written additively. The hieroglyph for “R” was used as the word ‘part’. For example:  All other fractions must be converted to unit fractions. Example 1: Example 2: 2 1 1 3 1 1 15 = 10 + 30 4 = 2 + 4 Hieratic System Hieratic system is an example of a ciphered system. The Hieratic script was invented and developed more or less at the same time as the hieroglyphic script and was used in parallel with it for everyday purposes such as keeping records and accounts and writing letters. Here each number from 1 to 9 had a specific symbol, as did each multiple of 10 from 10 to 90 and each multiple of 100 from 100 to 900, and so on. A simplified and abbreviated form of the hieroglyphic script in which the people, animals and object depicted are no longer easily recognizable Structurally the same as the hieroglyphic script Written almost exclusively from right to left in horizontal lines and mainly in ink on papyrus Written in a number of different styles such as "business hand" and the more elaborate "book hand". 1.2.2 Babylonian Mathematics 3300 BCE – writing of the most basic kind was developed and continued using a more developed form of the original ‘cuneiform’ (wedge-shaped) script for 3000 years, in different languages. The documents have been unusually well preserved because the texts were produced by making impressions on clay tablets, which hardened quickly and were preserved even when thrown away or used as rubble to fill walls (see Fig. 1.3). The Babylonians developed a system of writing from “pictographs” – a kind of picture writing much like hieroglyphics. The Babylonians used first a reed and later a stylus with a triangular end. With this they made impressions (rather than scratches) in moist clay. Clay dries quickly, so documents had to be relatively short and written all at one time, but they were virtually indestructible when baked hard in an oven or by the heat of the sun. 1.2.2.1 Babylonian Positional Number System The Babylonians were the only pre-Grecian people who made even a partial use of a positional number system. Such systems are based on the notion of place value, in which the value of a symbol depends on the position it occupies in the numerical representation. The Babylonian scale of enumeration was not decimal, but sexagesimal (60 as a base), so that every place a “digit” is moved to the left increases its value by a factor of 60.  When whole numbers are represented in the sexagesimal system, the last space is reserved for the numbers from 1 to 59, the next-to-last space for the multiples of 60, preceded by multiples of 602, and so on. For example, the Babylonian 3 25 4 might stand for the number 2 3 3∙60 + 25∙60 + 4 = 12, 304 and not 3∙10 + 25∙10 + 4 = 3, 254 as in our decimal (base 10) system. The sexagesimal system was an ancient system of counting, calculation, and numerical notation that used powers of 60. In the Babylonian number system, a vertical wedge represented 1, a horizontal wedge represented 10 (Fig. 1.4) The simple upright wedge had the value 1 and could be used nine times, while the broad sideways wedge stood for 10 and could be used up to five times. The Babylonians, preceding along the same lines as the Egyptians, made up all other numbers of combinations of these symbols, each represented as often as it was needed. When both symbols were used, those indicating tens appeared to the left of those for ones, as in Appropriate spacing between tight groups of symbols corresponded to descending powers of 60, read from left to right. As an illustration, we have Instead of using a tens symbol followed by nine units: 1.2.3 Mathematics in Ancient Greece Greeks refined analytical methods by introducing deductive reasoning and mathematical rigor in proofs. Rigor was a thoroughness and attention to detail for improving accuracy. Proofs established analytical methods as having a formalized structure. The study of mathematics as a subject begins the Classical Period, 600 - 300 BC, when the Pythagoreans originate the word mathematics. It is primarily derived from the ancient Greek word mathema meaning any study which a person may learn. Greek Methods were not limited to mathematics. Methods are processes, practices and structures defining a framework of something that has been or is being studied. A method describes something of importance and is used to document and inform. Methods are a means to facilitate or enhance understanding and knowledge. As importantly, the ancient Greeks also considered methods as to their purpose, the manner in which a method is used or applies. The contributions made by the ancient Greeks to the field of mathematics gave birth to modern math. The Greeks were mostly focused on Geometry. 1.2.3.1 Socrates Plato and Aristotle In Greek history the importance of Plato is for his inspiring and guiding others, primarily from his academy. His importance is also from having a living relationship with Socrates and Aristotle. Thales of Miletus (c.624-546 BC)  Pythagoras of Samos (c.572-497 BC) Zeno of Elea (c.450 BC) 1.2.3.1 The Greek number  The Greek number system is largely similar to the Roman and Egyptian number systems. Similar to the Egyptian number system because of the large amount of trade done in that region during that time. The Greek number system is base 10 system, with actually two separate systems for naming and denoting of numbers. The first, using similar names as the second, but instead labeling numbers without symbols, but with the first letter of the name for that number. For example, the number 1 in the first system is called “lota” and denoted by the letter ‘’l’’. In the second system symbols are designated to represent numbers. For example, in the second system, the number 1 is called alpha and is represented by an “A’’ or the symbol alpha (α). The Greek alphabet came from the Phoenicians. They borrowed some of the symbols and made up some of their own. When the Phoenicians invented the alphabet, it contained about 600 symbols. Those symbols took up too much room, so they eventually narrowed it down to 22 symbols. Greek Numerals  Greek Numerals are a system of representing numbers using the letters of the Greek alphabet. The ancient Greeks originally had a number system like the Romans, but in the 4th century BC, they started using the system. Around 500 BC, the Greeks developed a numbering system based on ten. This system used a 27-letter Greek alphabet. The first nine letters stood for numbers, 1 through 9. The next nine letters stood for tens, from 10 through to 90. Example: 1. Alpha (α) + Mu (µ) = 1 + 40 = 41 2. Kappa (κ) + Tau (τ) = 90 + 130 = 320 Acrophonic Numbers Acrophonic numbers (AKA attic or herodianic numbers) was the first system of numbers to be used by the ancient Greeks. They were first described in a 2nd century manuscript by Herodian. 1.2.3 Islamic, Hindu and Chinese Mathematics 1.2.3.1 Islamic Mathematics 8th Century onwards – the Islamic Empire established across Persia, the Middle East, Central Asia, North Africa, Iberia and parts of India that made a significant contributions toward mathematics. One consequence of the Islamic prohibition on depicting the human form was the extensive use of complex geometric patterns to decorate their buildings, raising mathematics to the form of an art. In fact, over time, Muslim artists discovered all the different forms of symmetry that can be depicted on a 2-dimensional surface (Fig. 8).  9th to 15th Centuries – the Golden Age of Islamic science and mathematics flourished throughout the medieval period and the Qu’ran itself encouraged the accumulation of knowledge. The House of Wisdom was set up in Baghdad around 810, and work started almost immediately on translating the major Greek and Indian mathematical and astronomy works into Arabic. Muhammad Al-Khwarizmi - An outstanding Persian mathematician and early Director of the House of Wisdom in the 9th Century. - One of the greatest of early Muslim mathematicians. - His most important contribution to mathematics was his strong advocacy of the Hindu numerical system (1-9 and 0), which he recognized as having the power and efficiency needed to revolutionize Islamic (and, later, Western) mathematics, and which was soon adopted by the entire Islamic world, and later by Europe as well. - Other important contribution was algebra, and he introduced the fundamental algebraic methods of “reduction” and “balancing” and provided an exhaustive account of solving polynomial equations up to the second degree. In this way, he helped create the powerful abstract mathematical language still used across the world today, and allowed a much more general way of analyzing problems other than just the specific problems previously considered by the Indians and Chinese. - The first to use the method of proof by mathematical induction to prove his results, by proving that the first statement in an infinite sequence of statements is true, and then proving that, if any one statement in the sequence is true, then so is the next one. ▪ Spherical Trigonometry Nasir Al-Din Al-Tusi - The 13th Century Persian astronomer, scientist and mathematician. - First to treat trigonometry as a separate mathematical discipline, distinct from astronomy. - He gave the first extensive exposition of spherical trigonometry, including the six distinct cases of a right triangle in spherical trigonometry. One of his major mathematical contribution was the formulation of the famous law of 𝑎 𝑏 𝑐 sines for plane triangles, (sin𝑠𝑖𝑛 𝐴) = (sin𝑠𝑖𝑛 𝐵) = (sin𝑠𝑖𝑛 𝐶) , although the sine law for spherical triangles had been discovered earlier by the 10th century Persians Abul Wafa Buzjani and Abu Nasr Mansur.  Other medieval Muslim mathematicians worthy of note include: the 9th Century Arab Thabit ibn Qurra, who developed a general formula by which amicable numbers could be derived, re-discovered much later by both Fermat and Descartes(amicable numbers are pairs of numbers for which the sum of the divisors of one number equals the other number, e.g. the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284; and the proper divisors of 284 are 1, 2, 4, 71, and 142, of which the sum is 220); the 10th Century Arab mathematician Abul Hasan al-Uqlidisi, who wrote the earliest surviving text showing the positional use of Arabic numerals, and particularly the use of decimals instead of fractions (e.g. 7.375 instead of 73⁄8); the 10th Century Arab geometer Ibrahim ibn Sinan, who continued Archimedes’ investigations of areas and volumes, as well as on tangents of a circle; 1.2.3.2 Hindu Mathematics Mantras from the early Vedic period (before 1000 BCE) invoke powers of ten from a hundred all the way up to a trillion, and provide evidence of the use of arithmetic operations such as addition, subtraction, multiplication, fractions, squares, cubes and roots. A 4th Century CE Sanskrit text reports Buddha enumerating numbers up to 1053, as well as describing six more numbering systems over and above these, leading to a number equivalent to 10421. Given that there are an estimated 1080 atoms in the whole universe, this is as close to infinity as any in the ancient world came. It also describes a series of iterations in decreasing size, in order to demonstrate the size of an atom, which comes remarkably close to the actual size of a carbon atom (about 70 trillionths of a meter). Sulba Sutras (Sulva Sutras) – listed several simple Pythagorean triples, as well as a statement of the simplified Pythagorean Theorem for the sides of a square and for a rectangle (indeed, it seems quite likely that Pythagoras learned his basic geometry from the sulba sutras). Contain geometric solutions of linear and quadratic equations in a single unknown, and give a remarkably accurate figure for the square root of 2, obtained by adding 1 1 1 1+ 3 + (3×4) − (3×4×34) , which yields a value of 1.4142156, correct to 5 decimal places.  The Indians early discovered the benefits of a decimal place value number system, and were certainly using it before about the 3rd Century CE. They refined and perfected the system, particularly the written representation of the numerals, creating the ancestors of the nine numerals that (thanks to its dissemination by medieval Arabic mathematicians) we use across the world today, sometimes considered one of the greatest intellectual innovations of all time. Earliest Recorded Usage of a Circle Character as Number Zero Brahmasphutasiddhanta - His treatment of the concept of (then relatively new) the number zero. Although often also attributed to the 7th Century Indian mathematician Bhaskara I, his “Brahmasphutasiddhanta” is probably the earliest known text to treat zero as a number in its own right, rather than as simply a placeholder digit as was done by the Babylonians, or as a symbol for a lack of quantity as was done by the Greeks and Romans. - Brahmagupta established the basic mathematical rules for dealing with zero (1 + 0 = 1; 1 – 0 = 1; and 1 x 0 = 0), although his understanding of division by zero was incomplete (he thought that 1 ÷ 0 = 0). Almost 500 years later, in the 12th Century, another Indian mathematician, Bhaskara II, showed that the answer should be infinity, not zero (on the grounds that 1 can be divided into an infinite number of pieces of size zero), an answer that was considered correct for centuries. However, this logic does not explain why 2 ÷ 0, 7 ÷ 0, etc, should also be zero – the modern view is that a number divided by zero is actually “undefined” (i.e. it doesn’t make sense). Brahmagupta established the basic mathematical rules for dealing with zero: 1 + 0 = 1; 1 – 0 = 1; and 1 x 0 = 0 (the breakthrough which would make sense of the apparently non-sensical operation 1 ÷ 0 would also fall to an Indian, the 12th Century mathematician Bhaskara II). Brahmagupta also established rules for dealing with negative numbers, and pointed out that quadratic equations could in theory have two  possible solutions, one of which could be negative. He even attempted to write down these rather abstract concepts, using the initials of the names of colors to represent unknowns in his equations, one of the earliest intimations of what we now know as algebra. The so-called Golden Age of Indian mathematics can be said to extend from the 5th to 12th Centuries, and many of its mathematical discoveries predated similar discoveries in the West by several centuries, which has led to some claims of plagiarism by later European mathematicians, at least some of whom were probably aware of the earlier Indian work. Certainly, it seems that Indian contributions to mathematics have not been given due acknowledgement until very recently in modern history. Indian astronomers used trigonometry tables Golden Age Indian mathematicians made fundamental advances in the theory of trigonometry, a method of linking geometry and numbers first developed by the Greeks. They used ideas like the sine, cosine and tangent functions (which relate the angles of a triangle to the relative lengths of its sides) to survey the land around them, navigate the seas and even chart the heavens. For instance, Indian astronomers used trigonometry to calculate the relative distances between the Earth and the Moon and the Earth and the Sun. They realized that, when the Moon is half full and directly opposite the Sun, then the Sun, Moon and Earth form a right angled triangle, and were able to accurately measure the angle as 1⁄7°. Their sine tables gave a ratio for the sides of such a triangle as 400:1, indicating that the Sun is 400 times further away from the Earth than the Moon. Aryabhata The great Indian mathematician and astronomer produced categorical definitions of sine, cosine, versine and inverse sine, and specified complete sine and versine tables, in 3. 75° intervals from 0° to90°, to an accuracy of 4 decimal places. He also demonstrated solutions to simultaneous quadratic equations, and produced an approximation for the value of π equivalent to 3.1416, correct to four decimal places. He used this to estimate the circumference of the Earth, arriving at a figure  of 24,835 miles, only 70 miles off its true value. But, perhaps even more astonishing, he seems to have been aware that π is an irrational number, and that any calculation can only ever be an approximation, something not proved in Europe until 1761. Infinity as the Reciprocal of Zero Bhaskara II who lived in the 12th Century One of the most accomplished of all India’s great mathematicians. He is credited with explaining the previously misunderstood operation of division by zero. He noticed that dividing one into two pieces yields a half, so 1 ÷ 1⁄2 = 2. Similarly, 1 ÷ 1⁄3 = 3. So, dividing 1 by smaller and smaller factions yields a larger and larger number of pieces. Ultimately, therefore, dividing one into pieces of zero size would yield infinitely many pieces, indicating that 1 ÷ 0 = ∞ (the symbol for infinity). He also made important contributions to many different areas of mathematics from solutions of quadratic, cubic and quartic equations (including negative and irrational solutions) to solutions of Diophantine equations of the second order to preliminary concepts of infinitesimal calculus and mathematical analysis to spherical trigonometry and other aspects of trigonometry. Some of his findings predate similar discoveries in Europe by several centuries, and he made important contributions in terms of the systemization of (then) current knowledge and improved methods for known solutions. 1.2.3.2 Chinese Mathematics The simple but efficient ancient Chinese numbering system, which dates back to at least the 2nd millennium BCE, used small bamboo rods arranged to represent the numbers 1 to 9, which were then places in columns representing units, tens, hundreds, thousands, etc. It was, therefore, a decimal place value system, very similar to the one we use today – indeed it was the first such number system, adopted by the Chinese over a thousand years before it was adopted in the West – and it made even quite complex calculations very quick and easy. Written numbers, however, employed the slightly less efficient system of using a different symbol for tens, hundreds, thousands, etc. This was largely because there was no concept or symbol of zero, and it had the effect of limiting the usefulness of the written number in Chinese. The use of the abacus is often thought of as a Chinese idea, although some type of abacus was in use in Mesopotamia, Egypt and Greece, probably much earlier than in China (the first Chinese abacus, or “suanpan”, we know of dates to about the 2nd Century BCE). Lo Shu magic square There was a pervasive fascination with numbers and mathematical patterns in ancient China, and different numbers were believed to have cosmic significance. In particular, magic squares – squares of numbers where each row, column and diagonal added up to the same total – were regarded as having great spiritual and religious significance. The Lo Shu Square, an order three square where each row, column and diagonal adds up to 15, is perhaps the earliest of these, dating back to around 650 BCE (the legend of Emperor Yu’s discovery of the square on the back of a turtle is set as taking place in about 2800 BCE). But soon, bigger magic squares were being constructed, with even greater magical and mathematical powers; culminating in the elaborate magic squares, circles and triangles of Yang Hui in the 13th Century (Yang Hui also produced a triangular representation of binomial coefficients identical to the later Pascals’ Triangle, and was perhaps the first to use decimal fractions in the modern form). Early Chinese Method of Solving Equations Jiuzhang Suanshu” or “Nine Chapters on the Mathematical Art” Written over a period of time from about 200 BCE onwards, by variety of authors.  An important tool in the education of such a civil service, covering hundreds of problems in practical areas such as trade, taxation, engineering and payment of wages. It was particularly important as a guide to how to solve equations – the deduction of an unknown number from other known information – using a sophisticated matrix-based method which did not appear in the West until Carl Friedrich Gauss re-discovered it at the beginning of the 19th Century (and which is now known as Gaussian elimination). Liu Hui The greatest mathematicians of ancient China. Produced a detailed commentary on the “Nine Chapters”in 263 CE. One of the first mathematicians known to leave roots unevaluated, giving more exact results instead of approximations. By an approximation using a regular polygon with 192 sides, he also formulated an algorithm which calculated the value of π as 3.14159 (correct to five decimal places), as well as developing a very early form of both integral and differential calculus. Example: If one of plum and three peaches weigh a total of 750g, and two plums and one peach weigh a total of 500g, how much does a single peach and plum weight? First, double the content of the first scale: Subtract from this the contents of the second set of scales: Therefore, a single peach must weight 200g(1, 000÷5). Then, take the peach off the second scale:  Therefore, a single plum must weight 150g(300÷5). The Chinese Remainder Theorem The Chinese went on to solve far more complex equations using far larger numbers than those outlined in the “Nine Chapters”, though. They also started to pursue more abstract mathematical problems (although usually couched in rather artificial practical terms), including what has become known as the Chinese Remainder Theorem. This uses the remainders after dividing an unknown number by a succession of smaller numbers, such as 3, 5 and 7, in order to calculate the smallest value of the unknown number. A technique for solving such problems, initially posed by Sun Tzu in the 3rd Century CE and considered one of the jewels of mathematics, was being used to measure planetary movements by Chinese astronomers in the 6th Century AD, and even today it has practical uses, such as in Internet cryptography. By the 13th Century, the Golden Age of Chinese mathematics, there were over 30 prestigious mathematics schools scattered across China. Perhaps the most brilliant Chinese mathematician of this time was Qin Jiushao, a rather violent and corrupt imperial administrator and warrior, who explored solutions to quadratic and even cubic equations using a method of repeated approximations very similar to that later devised in the West by Sir Isaac Newton in the 17th Century. Qin even extended his technique to solve (albeit approximately) equations involving numbers up to the power of ten, extraordinarily complex mathematics for its time. If a collection of balls are arranged in rows of 3, there is one ball left over If arranged in rows of 5, there are two balls left over If arranged in rows of 7, there are three balls left over  The Chinese Remainder Theorem proves that the smallest number of balls must be 52. 1.3 ASSESSMENT Test I. Multiple Choice 1. One of the first mathematicians known to leave roots unevaluated, giving more exact results instead of approximations. a. Bhaskara II b. Liu Hui c. Arybhata d. Jean Champollion 2. This mathematician was the first to go to Egypt and bring back to Greece to study geometry. a. Bhaskara II b. Liu Hui c. Thales d. Jean Champollion 3. A book that is larger than any Greek prose work. a. History of Thales b. History of Plato c. History of Herodotus d. History of Pythagoras 4. What are the two different number systems developed by the Egyptians? a. Hieroglyphic system and hieratic numeration b. Hieroglyphic system and hieratic numeration c. Hieroglyphic system and ciphered numeral system d. Hieroglyphic system and sexagisimal place-value system 5. Historian who was able to begin the process of understanding Egyptian writing through the help of a multilingual inscription. a. Jean Champollion b. Menes c. Herodotus d. Thales 6. His major contribution was the formulation of the famous law of sines for plane triangles. a. Nasir-Al Din Al-Tusi b. Abul Wafa Buzjami  c. Abu Nasa Mansur d. Al-Karaji 7. He introduced the fundamental algebraic methods of reduction and balancing. a. Nasir-Al Din Al-Tusi b. Abul Wafa Buzjami c. Muhammed Al-Khwarizmi d. Aryabhata 8. This mathematician and astronomer produced categorical definitions of sine, cosine, versine and inverse sine. a. Nasir-Al Din Al-Tusi b. Abul Wafa Buzjami c. Muhammed Al-Khwarizmi d. Aryabhata 9. Developed simple but efficient numbering system that used small bamboo rods. a. Hindu Mathematics b. Egyptian Mathematics c. Chinese Mathematics d. Babylonian Mathematics 10. Developed a system of writing from picture. a. Hindu Mathematics b. Egyptian Mathematics c. Chinese Mathematics d. Babylonian Mathematics Test II. Solving Answer the following questions using the ancient mathematics conversion system. Show your solution. 1. Represent 375 and 4856 in Egyptian hieroglyphics and Babylonian cuneiform. 2. Use Egyptian techniques to multiply 34 by 18 and to divide 93 by 5. 3. Write the number 10,000 in Babylonian notation. 4. Represent 125, 62, 4821, and 23,855 in the Greek alphabetic notation. Test III. True/False Place a T on the line if you think a statement it TRUE. Place an F on the line if you think the statement is FALSE. ___1. Liu Hui formulated an algorithm which calculated the value of phi as 3.14159. ___2. Hieratic writing is used for monumental writing. ___3. The Egyptians were the only pre-Grecian people who made even a partial use of a positional number system.  ___4. Ancient Greeks considered methods in enhancing knowledge. ___5. The Greek numeral ∅ is equivalent to 400. Test IV. Essays Directions: Write at least one paragraph in response to each of the following questions. Please be guided by the rubrics. 1. What do you regard as the four most significant contributions of the Mesopotamia to mathematics? Justify your answer. 2. Discuss the contribution of Thales, Pythagoras and Plato in the development of mathematics. Please be guided with the rubrics for the essays. Needs Unacceptable Satisfactory Good Criteria Improvement Exceptional (5) (1) (3) (4) (2)  Did not answer Answers are Answers are not Answers are Answers are question. partial or comprehensive accurate and comprehensive, incomplete. Key or completely complete. Key accurate and points are not stated. Key points are complete. Key Content clear. Question points are stated and ideas are clearly not adequately addressed, but supported. stated, not well explained, and supported. well supported. Did not answer Organization Inadequate Organization is Well organized, Organization question. and structure organization or mostly clear coherently (Answers are detract from the development. and easy to developed, and clearly thought answer. Structure of the follow. easy to follow. out and answer is not articulated.) easy to follow. Writing Did not answer Displays over Displays three Displays one to Displays no Conventions question. five errors in to five errors in three errors in errors in (Spelling, spelling, spelling, spelling, spelling, punctuation, punctuation, punctuation, punctuation, punctuation, grammar, and grammar, and grammar, and grammar, and grammar, and complete sentence sentence sentence sentence sentences.) structure. structure. structure. structure. 1.4 References Burton, D.M. (2010). The history of mathematics: An Introduction (7th ed). McGraw-Hill Education Katz, V.J. (2009). A History of Mathematics: An Introduction. 3rd ed. Mastin, L. (2020). Story of Mathematics: Islamic Mathematics Mastin, L. (2020). Story of Mathematics: Chinese Mathematics Mastin, L. (2020). Story of Mathematics: Indian Mathematics Millmore, M. (2022). Discovering Ancient Egypt 1.5 Acknowledgment The images, tables, figures and information contained in this module were taken from the references cited above. DISCLAIMER: This module is not for commercial use and solely for educational purposes only. Some technical terminologies and uses were not changed but the author of this unit ensures that all in-text citations are in the references section. Photos, figures, images, and tables included here belongs to their respective and their copyright.