The History of Mathematics

1. The History of Mathematics http://www.math.wichita.edu/~richardson/timeline.html http://www-groups.dcs.st-and.ac.uk/~history/Indexes/HistoryTopics.html http://www-groups.dcs.st-and.ac.uk/~history/BiogIndex.html

2. Completing a SquareSolving a Quadratic Equation al-Khwarizmi Iraq (ca. 780-850) x2 + 10 x = 39 x2 + 10 x + 4·25/4 = 39+25 (x+5)2 = 64 x + 5 = 8 x = 3

3. The Bridges of KonigsbergTopology Leonhard Euler Switzerland 1707 - 1783 • In Konigsberg, Germany, a river ran through the city such that in its centre was an island, and after passing the island, the river broke into two parts. Seven bridges were built so that the people of the city could get from one part to another. • The people wondered whether or not one could walk around the city in a way that would involve crossing each bridge exactly once

4. The city

5. Infinite Prime Numbers Euclid Greece 325 – 265BC Theorem: There are infinitely many prime numbers. Proof:Suppose the opposite, that is, that there are a finite number of prime numbers. Call them p1, p2, p3, p4,....,pn. Now consider the number • (p1*p2*p3*...*pn)+1 • Every prime number, when divided into this number, leaves a remainder of one. So this number has no prime factors (remember, by assumption, it's not prime itself). • This is a contradiction. Thus there must, in fact, be infinitely many primes.

6. The Search for Pi

7. The Search for Pi • François Viète (1540-1603) France - determined that: • John Wallis (1616-1703) English - showed that: • While Euler (1707-1783) Switzerland derived his famous formula: • Today Pi is known to more than 10 billion decimal places.

8. Laura T Ancient Babylonia The Sumerians had developed an abstract form of writing based on cuneiform (i.e. wedge-shaped) symbols. Their symbols were written on wet clay tablets which were baked in the hot sun and many thousands of these tablets have survived to this day. It was the use of a stylus on a clay medium that led to the use of cuneiform symbols since curved lines could not be drawn. The later Babylonians adopted the same style of cuneiform writing on clay tablets. The Babylonians had an advanced number system, in some ways more advanced than our present systems. It was a positional system with a base of 60 rather than the system with base 10 in widespread use today.

9. The Four Colour Theorem The Four Colour Conjecture was first stated just over 150 years ago, and finally proved conclusively in 1976. It is an outstanding example of how old ideas combine with new discoveries and techniques in different fields of mathematics to provide new approaches to a problem. It is also an example of how an apparently simple problem was thought to be 'solved' but then became more complex, and it is the first spectacular example where a computer was involved in proving a mathematical theorem.

10. Laura T Ancient Babylonia The Babylonians divided the day into 24 hours, each hour into 60 minutes, each minute into 60 seconds. This form of counting has survived for 4000 years. To write 5h 25' 30", i.e. 5 hours, 25 minutes, 30 seconds, is just to write the sexagesimal fraction, 5 25/60 30/3600. We adopt the notation 5; 25, 30 for this sexagesimal number, for more details regarding this notation see our article on Babylonian numerals. As a base 10 fraction the sexagesimal number 5; 25, 30 is 5 4/10 2/100 5/1000 which is written as 5.425 in decimal notation.

11. Laura T Ancient Babylonia Perhaps the most amazing aspect of the Babylonian's calculating skills was their construction of tables to aid calculation. Two tablets found at Senkerah on the Euphrates in 1854 date from 2000 BC. They give squares of the numbers up to 59 and cubes of the numbers up to 32. The table gives 82 = 1,4 which stands for 82 = 1, 4 = 1 x 60 + 4 x 1 = 64 and so on up to 592 = 58, 1 = 58 x 60 +1 x 1 = 3481). The Babylonians used the formula ab = [(a + b)2 - a2 - b2]/2 to make multiplication easier. Even better is their formula ab = [(a + b)2 - (a - b)2]/4 which shows that a table of squares is all that is necessary to multiply numbers, simply taking the difference of the two squares that were looked up in the table then taking a quarter of the answer.

12. Laura T Egyptian numerals The following hieroglyphs were used to denote powers of ten: or Multiples of these values were expressed by repeating the symbol as many times as needed. For instance, a stone carving from Karnak shows the number 4622 as

13. Chinese mathematics has developed greatly since at least 100 BC. Although the Chinese refer back to their ancient texts, many of which were written on strips of bamboo, they are constantly coming up with ways of working out problems. One of the earliest Chinese mathematicians was a man named Luoxia Hong (130BC – 30BC). He designed a new calendar for the Emperor, which featured 12 months, based on a cycle of 12 years. This inspired many people to design calendars and the one we have today. The Chinese also came up with a rule called the Gougu Rule. This is the Chinese version of Pythagoras. Liu Hui (220AD – 280AD) tried to find pi to the nearest number. He eventually got to 3.14159, which in those days was thought to be an incredible achievement. Chinese Mathematics Jeremy

14. Moscow Papyrus: arithimetic The Moscow Papyrus is located in a museum hence the name. The papyrus was copied by a scribe and was brought to Russia. The papyrus contains 25 maths problems involving simple “equations” and solutions. The problems are not in modern form. The problem that has generated the most interest is the volume of a truncated pyramid (a square based pyramid with the top portion removed). The Egyptians discovered the formula for this even though it was very hard to derive. The Actual author of the equation is unknown. But this is what he/she discovered: The Moscow Papyrus is 15 feet long and about 3 inches wide. Alannah Problem 14: Volume of a truncated Pyramid

15. Johannes Widman was a German mathematician who is best remembered for an early arithmetic book which contains the first appearance of + and – (both adding and subtracting, and positive and negative) signs in 1498. Johannes Widman The plus and minus sign His book was better than anybody else's because he had more and a wider range of examples. The book remained in print until 1526.(28 years after it was first published.) Alannah

16. Moscow Papyrus: Arithmetic As it name may suggest, the Moscow papyrus is located in the Museum of Fine Arts in Moscow. In around 1850BC, the papyrus was copied by an anonymous scribe and was bought to Russia in the 19th Century. It contains 25 problems containing simple equations and solutions. However, the equations are not in modern form. The problem that generates the most interest is the calculation of the volume of a truncated pyramid (a square based pyramid with the top cut off) The Egyptians seemed to know this difficult formula. V= (1/3)(a² + ab + b²)(h) Alex

17. Johannes Widman (1462 – 1498) Widman is best known for a book on arithmetic which he wrote (in German) in 1489AD. This contains the first appearance of + and – signs. This was better than those that had come before it with a wider range of examples. The book continued to be published until 1526AD. Then, Adam Ries – amongst others – produced superior books. Alex

18. Chu Shih-chieh (Zhu Shijie) Zhu Shijie was one of the greatest Chinese mathematicians. He lived during the Yuan Dynasty. Yang worked on magic squares and binomial theorem, and is best known for his contribution of presenting 'Yang Hui's Triangle'. This triangle was the same as Pascal's Triangle, discovered independently by Yang and his predecessor Jia Xian .Yang was also a friend to the other famous mathematician Qin Jiushao. An early form of Pascal’s triangle Calum

19. Magic squares In mathematics, a magic square of order n is an arrangement of n² numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. A normal magic square contains the integers from 1 to n². All non trivial magic squares exist for n≥3. An example of a magic square The equation Calum

20. Johannes Widmann + Johannes Widmann (born c. 1460 in Eger; died after 1498 in Leipzig) was a German mathematician who was the first to use the addition (+) and the subtraction (-) signs. Widmann attended the University of Leipzig in the 1480s, and published Behende und hubsche Rechenung auff allen Kauffmanschafft, his work making use of the signs, in Leipzig in 1489. - Calum

21. Charis Aristarchus Aristarchus of Samos was a Greek mathematician and astronomer. He was born in about 310BC and died at around 230BC. He is the first person to suggest a universe with the Sun at the centre instead of the Earth. He tried to work out the sizes of the Sun and the Moon and how far away they are. He worked out that the Sun was 20 times further away than the moon and 20 times bigger. Both these estimates were too small but the reasoning behind it was right.

22. Blaise Pascal (1623-1662) Charis Pascal was a French mathematician and physicist. His father was a tax official and Pascal made a calculating machine that did addition and subtraction to make his work easier. He wrote about Pascal's triangle. Each number is the sum of the two above it. There are lots of different patterns in the triangle. Some are shown on the diagram. Pascal also worked with Fermat on the theory of probability, and he wrote about projective geometry when he was only 16.

23. Pythagoras and the Mathematikoi - Pythagoras was the leader of a Society which consisted mainly of followers called the mathematikoi. - The mathematikoi owned nothing personal and were vegetarians. - Any mathematical discoveries they made the credit was given to Pythagoras. - Everything we know about Pythagoras and the mathematikoi was only recorded properly 100 years later as they apparently wrote none of their information down. David

24. Trigonometry of Hipparchus -Hipparchus was a Greek mathematician who invented one of the first trigonometry tables which he needed to compute the orbits of the Sun and Moon. -The table on the right represents the Chord function. The chord of an angle is the length between two points on a unit circle separated by that angle. -If one the angles is zero it can be easily related to the sine function. And the used in the half angle formula: David

25. JAPANESE MATHEMATICS The system of Japanese numerals is the system of number names used in the Japanese language. The Japanese numerals in writing are entirely based on the Chinese numerals and the grouping of large numbers follow the Chinese tradition of grouping by 10,000. Like in Chinese numerals, there exists in Japanese a separate set of kanji for numerals called daiji (大字) used in legal and financial documents to prevent unscrupulous individuals from adding a stroke or two, turning a one into a two or a three. The formal numbers are identical to the Chinese formal numbers except for minor stroke variations George http://en.wikipedia.org/wiki/Japanese_numerals

26. John Napier ( 1550-1617) • Napier was a Scottish mathematician who studied math like a hobby as he never had time to spend on calculations between working on theology. • He is best known, along with Joost Burgi, for his invention of logarithms • He is also famous for the invention of two theories: • Napier’s analogy (used in solving spherical triangles) • And Napier’s bones. (used for mechanically multiplying, dividing, taking square roots and cube roots Hannah

27. Pierre de Fermat (1601- 1665) • Fermat was a French mathematician who is best known for his work on number and theory • One of his last theorem’s was proven by Andrew Wiles in 1994. • Whilst in Bordeaux, Fermat produced work on maxima and minima, which was important. His methods of doing this were similar to ours , however as he has not a professional mathematician his work was very awkward. • Fermat’s last theorem was that if you had the equation : xn+yn = zn n in this equation can be no more that two. When n is more than two the equation does not work Hannah

28. Issy Hipparchus Hipparchus is most known for Trigonometry. He did not discover this on his own however. Menelaus and Ptolomy, helped with this. “Even if he did not invent it, Hipparchus is the first person of whose systematic use of trigonometry we have documentary evidence." some historians say. Some even go as far as to say that he invented trigonometry. Not much is known about the life of Hipparchus. But it is believed that he was born at Nicaea in Bithynia, and lived from 190 BC to 120 BC

29. Issy Algebra Algebra is a branch of mathematics concerning the study of structure, relation, and quantity. Together with geometry, analysis, combinatorics, and number theory, algebra is one of the main branches of mathematics. Elementary algebra is provides an introduction to the basic ideas of algebra, including effects of adding and multiplying numbers, along with factorization and determining their roots. Algebra is much broader than elementary algebra and can be generalized. In addition to working directly with numbers, algebra covers working with symbols, variables, and set elements. The history of algebra began in ancient Egypt and Babylon, where people learned to solve linear (ax = b) and quadratic (ax² + bx = c) equations, as well as indeterminate equations such as x² + y² = z², whereby several unknowns are involved. The ancient Babylonians solved arbitrary quadratic equations by essentially the same procedures taught today. They also could solve some indeterminate equations

30. John-Jack The Roman Abacus • The Roman Abacus was devised by Roman traders adapting ideas that had been picked up in Egypt. • The Abacus is made up of grooves in a slate tile with marbles that run in them. • The Abacus was originally made in Babylon using stones and ditched made in the dry soil in 2700 BC. • The Abacus was originally made in Babylon using stones and ditched made in the dry soil in 2700 BC. • Each Abacus used a different scale depending on the user. Traders often used ones with fractions up to 1/12. • This would mean that they could subtract quite accurately eg. To subtract 1/3 you would take a bead from the 1/4 column and one from the 1/12 column.

31. Laura W Buffon’s needle problem Georges-Louis Leclerc, Comte de Buffon lived from in September 7, 1707 to April 16, 178. He had many different careers, as a naturalist, mathematician, biologist, cosmologist and author. The Lycée Buffon in Paris is named after him. The problem he is famous for is: Suppose we have a floor made of parallel strips of wood, each the same width, and we drop a needle onto the floor. What is the probability that the needle will lie across a line between two strips? Or in more mathematical terms … Given a needle of length l dropped on a plane ruled with parallel lines t units apart, what is the probability that the needle will cross a line? For n needles dropped with h of the needles crossing lines, the probability is: This is useful because it can be rearranged to get an estimate for pi

32. Egyptian Numerals Kyle The Egyptians had a writing system based on hieroglyphs from around 3000 BC. Hieroglyphs are little pictures representing words. It is easy to see how they would denote the word "bird" by a little picture of a bird but clearly without further development this system of writing cannot represent many words. The way round this problem adopted by the ancient Egyptians was to use the spoken sounds of words. For example, to illustrate the idea with an English sentence, we can see how "I hear a barking dog" might be represented by: "an eye", "an ear", "bark of tree" + "head with crown", "a dog". Of course the same symbols might mean something different in a different context, so "an eye" might mean "see" while "an ear" might signify "sound". The Egyptians had a bases 10 system of hieroglyphs for numerals. By this we mean that they has separate symbols for one unit, one ten, one hundred, one thousand, one ten thousand, one hundred thousand, and one million.

33. Kyle The Addition & Minus Signs The plus and minus signs are symbols representing positive and negative, the meaning of them has been around since the Egyptian times, but the actual symbols + and – were first published by Johannes Widmann. Minus A Jewish tradition that dated from at least from the 19th century was to write plus using a symbol like an inverted T. This practice was then adopted into Israeli schools (this practice goes back to at least the 1940s) and is still commonplace today in some elementary schools (including secular schools) while fewer secondary schools. It is also used occasionally in books by religious authors, but most books for adults use the international symbol "+". The usual explanation for the origins of this practice is that it avoided the writing of a symbol "+" that looked like a Christian cross. Unicode has this symbol at position U+FB29 "Hebrew letter alternative plus sign" -Addition Jewish Addition Symbol

34. Kyle CHAOTIC BEHAVIOR In mathematics, chaos theory describes the behaviour of certain dynamical systems – that is, systems whose states evolve with time – that may exhibit dynamics that are highly sensitive to initial conditions (popularly referred to as the butterfly effect). As a result of this sensitivity, which manifests itself as an exponential growth of perturbations in the initial conditions, the behaviours of chaotic systems appears to be random. This happens even though these systems are deterministic, meaning that their future dynamics are fully defined by their initial conditions, with no random elements involved. This behaviour is known as deterministic chaos, or simply chaos. Chaotic behaviour is also observed in natural systems, such as the weather. This may be explained by a chaos-theoretical analysis of a mathematical model of such a system, embodying the laws of physics that are relevant for the natural system.

35. Hieroglyphic numerals in Egypt Horus was Egyptian God who fought the forces of darkness (in the form of a boar - a pig) and won. His eye is a symbol for Egyptian Unit Fractions. Each part of the eye is a part of the whole. All the parts of eye, however, don't add up to the whole. This, some Egyptologists think, is the sign that the knowledge can never be total, and that one part of the knowledge is not possible to describe or measure. Hieroglyphs were introduced for numbers in 3000BCE. Their number system was based on units of 10. They used simple grouping to make different numbers. The Egyptians used different images for their hieroglyphs. Nina

36. Nina Pythagoras of Samos • Pythagoras was an ancient Greek mathematician. Pythagoras was born about 569 BC in Samos, Ionia and died about 475 BC. • Pythagoras invented Pythagoras's theorem which is the idea that in a right angled triangle the two shorter sides squared and added equals the longest side (the hypotenuse) squared. • It was thought that the Babylonians 1200 years earlier knew this before but Pythagoras was the one to prove it. It is said that this is the oldest numbertheory document in existence. This theorem works for every right angled triangle

37. Algebra Jack G • While the word "algebra" comes from Arabic word (al-jabr)its origins are from the ancient Babylonians. With this system they were able to discover unknown values for a class of problems typically solved today by using linear equations, quadratic equations, and indeterminate linear equations. • The geometric work of the Greeks, typified in the Elements, provided the framework for finding the formulae beyond the solution of particular problems into more general systems of stating and solving equations. • The Greek mathematicians Hero of Alexandria and Diophantus ("the father of algebra") made algebra into a much higher level. People argye that al-Khwarizmi, who founded the discipline of al-jabr, deserves that title instead.

38. Algebra Jack G • Later, the Indian mathematicians developed algebraic methods to a high degree of sophistication. Al-Khwarizmi produced the "reduction" and "balancing" (the transposition of subtracted terms),He gave an explanation of solving quadratic equations supported by geometric proofs. • The Indian mathematicians Mahavira and Bhaskara II, the Persian mathematician Al-Karaji and the Chinese mathematician Zhu Shijie, solved various cases of cubic, quadratic, quintic and higher-order polynomial equations using numerical methods. • Gottfried Leibniz discovered the solution to simultaneous equations

39. Napier’s bones are basically a big multiplication square. It was used before calculators for multiplication of !HUGE! Numbers. To do a sum using them you arrange the bones in the order of the number to multiply like in the example: the sum is 46785399*7. Then, starting from the left, you just add all the numbers in the row, carrying the tens. Napier’s Bones Paul http://en.wikipedia.org/wiki/Napier%27s_bones

40. Nicole Ahmes Ahmes was the Egyptian scribe who wrote the Rhind Papyrus - one of the oldest known mathematical documents.

41. Born: about 1680 BC in EgyptDied: about 1620 BC in Egypt • The Rhind Papyrus, which came to the British Museum in 1863, is sometimes called the 'Ahmes papyrus' in honour of Ahmes. Nothing is known of Ahmes other than his own comments in the papyrus. • Ahmes claims not to be the author of the work, being, he claims, only a scribe. He says that the material comes from an earlier work of about 2000 BC. • The papyrus is our main source of information on Egyptian mathematics. Nicole

42. Hieroglyphic Numerals Hieroglyphics were used by the Egyptians in around 3000BC. These symbols below are what they would use as numbers. Although they only have to write one symbol for one million and we have to do seven, there is a fault. To write one million take one they would have to write 54 symbols. 999999 = 100000 1million or Infinity 1 10 100 1000 10000 Single stroke Heel bone Coil of rope Water Lily Finger Man with both hands raised Michael Tadpole or frog

43. Xenocrates of Chalcedon Xenorcrates was a Greek philosopher, mathematician and leader of the platonic army from 339BC to 314BC. Xenocrates is known to have written a book On Numbers, and a Theory of Numbers, besides books on geometry. Plutarch writes that Xenocrates once attempted to find the total number of syllables Birth: 396BC, Chalcedon Died: 314BC, Athens Interests: logic, physics, metaphysics, epistemology, mathematics, ethics. Ideas: developed the philosophy of Plato. that could be made from the letters of the alphabet. According to Plutarch, Xenocrates result was 1,002,000,000,000. This possibly represents the first instance that a combinatorial problem involving permutations was attempted. Xenocrates also supported the idea of indivisible lines (and magnitudes) in order to counter Zeno's paradoxes. Michael

44. Abu'l Hasan Ahmad ibn Ibrahim Al-Uqlidisi Abu'l Hasan Ahmad ibn Ibrahim Al-Uqlidisi was an Arab mathematician who was active in Damascus and Baghdad. He wrote the earliest surviving book on the positional use of the Arabic numerals, around 952. It is especially notable for its treatment of decimal fractions, and that it showed how to carry out calculations without deletions. While the Persian mathematician Jamshīd al-Kāshī claimed to have discovered decimal fractions himself in the 15th century, J. Lennart Berggrenn notes that he was mistaken, as decimal fractions were first used five centuries before him by al-Uqlidisi as early as the 10th century. Michael

45. Ollie Zhu Shijie of China Zhu Shijie was born in the 13th century near Beijing. Two of his mathematical works have survived; “Introduction to Computational Studies” and “Jade Mirror of Four Unknowns”. This book brought Chinese algebra to its highest level and it is his most important work. He makes use of the Pascal Triangle centuries before Blaise Pascal brought it to common knowledge.

46. Francesco Pellos 1450 – 1500 AD • Francesco Pellos, from Nice, is the earliest example of the use of the decimal point • He wrote an arithmetic book, called Compendio de lo Abaco, in 1492. • In this book he makes use of a dot to denote the division of a number by a power of ten. This has evolved to what we now call a decimal point.

47. Thales (620-547BC)Discoverer of deductive Geometry Jack S • “Father of deductive geometry” • Credited for five theorems • 1) A circle is bisected by any diameter. • 2) The base angles of an Isosceles Triangle are equal. • 3) The angles between two intersecting straight lines are equal. • 4) Two triangles are congruent if they have two angles and one side equal. • 5) An angle in a semicircle is a right angle.

48. Jack S Bhaskara • Can be called Bhaskaracharya meaning “Bhaskara the teacher”. • Lived in India. • Famous for number systems and solving equations which was not achieved in Europe for several centuries. • More information at http://www.maths.wichita.edu/~richardson/

49. Hypatia of Alexandria (AD 355 or 370 – 415) Rachel Hypatia’s father (Theon) was a mathematician in Alexandria in Egypt and he taught her about mathematics. From about the year 400 onwards she lectured on mathematics and philosophy. She also studied astronomy and astrology and may have invented astrolabes (which can be used to study astronomy) . However, there is no proof that she did this. Although she did not make any discoveries herself, she helped her father Theon with some of his works, and was the first woman to make a significant contribution to the development of mathematics. She was believed by some people to practise magic and was also hated for being a pagan. In AD 415 she was murdered in the street by a group of monks. The Crater Hypatia and Rimae Hypatia (features of the moon) are both named after Hypatia. http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Hypatia.html

50. Julia Hall Bowman Robinson (1919 – 1985) Rachel Julia was born in Missouri in the USA. When she was nine, she caught scarlet fever, which was followed by rheumatic fever. In total she missed two years of school. Over the next year, she had lessons three mornings a week and managed to get through four years of education (fifth to eighth grades). In her last year at school she was the only girl in her maths and physics classes. In 1948 she started work on Hilbert’s tenth problem (Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers) and came up with the Robinson hypothesis. This helped Yuri Matijasevic to find the final solution to the problem in 1970.