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Appendix 1- Calculations of Pi

Appendix 2 � Prime Numbers

James Joyce on eternity, infinity. Imagine a huge block of sand a million miles high, a million miles wide, a million miles long. Once every million years a bird comes along and pecks off one grain of sand. Eventually the entire block disappears, only to instantly reappear once more. By the time that cube of sand has reappeared for the millionth time, not one instant of Eternity has passed.

2010, Japanese systems engineer Shigeru Kondo computed the value of e to 1,000,000,000,000 decimal places.

14/10/2010, French-American mathematician who developed fractal geometry, Benoit Mandelbrot, died aged 85.

3/5/1995, Fermat�s Last Theorem was finally proved in generality, for all values of n. It was proposed by Pierre de Fermat (ca. 1607 � 1665) in 1637, that an + bn = cn had no solution for any value of n greater than 2.

6/10/1968, Phyllis Nicolson, English mathematician, died.

18/7/1960, William A. Dembski, USmathematician was born in Chicago.

2/8/1957, John von Neumann, Hungarian-US mathematician, died in Washington DC.

2/2/1950, Constantin Caratheodory, Greek mathematician, died, (born 1873).

27/4/1936, English mathematician Karl Pearson died in Coldharbour, Surrey.

6/1/1922, Jakob Rosanes, Ukrainian-German mathematician, died.

26/4/1920, Mathematician Srinivasa Ramanujan died.

12/2/1916, Julius Wilhelm Richard Dedekind, German mathematician, died in Braunschweig.

22/4/1910, Norman Steenrod, US mathematician (algebraic topology), was born in Dayton, Ohio (died 1971)

28/4/1906, Austrian-US mathematician Kurt Godel was born in Brno, Czechoslovakia.

106/1903, Luigi Cremona, Italian mathematician, died (born 7/12/1830).

28/4/1903, Josiah Gibbs, US mathematician, died (born 11/2/1839).

2/2/1903, Bartel Leendert van der Waerden, Dutch mathematician, was born in Amsterdam (died 1996)

24/1/1902, Oskar Morgenstern, German-US mathematician, was born in Silesia (Poland).

18/2/1899, Marius Lie, Norwegian mathematician, died.

27/1/1895, Sir James Cockle, English mathematician, died (born 14/1/1819).

26/1/1895, Arthur Cayley, British mathematician, died in Cambridge.

1894, The New York Mathematical Society was formed, later to become the American Mathematical Society.

10/2/1891, Sophie Kovalevsky, Russian mathematician, died (born 15/1/1850).

1888, Vector Space was defined by Guiseppe Peano (1858 � 1932) in his work, Geometrical Calculation According to the Ausdehnungslhere of H Grossman. Vectors such as (3,1) and (1,5) could be added to form a quadrilateral (0,0, 3,1,1,5, 6,4); vectors could also be scaled and added together. In physics they are useful for when different forces act on a particle.

14/4/1883, William Farr, English statistician, died (born 30/11/1807).

5/11/1879, James Clerk Maxwell, Scottish mathematician and physicist, died in Cambridge, England.

3/3/1879, William Clifford, English mathematician, died (born 4/5/1845).

31/3/1877, Antoine Cournot, French mathematician, died (born 28/8/1801).

1874, German mathematician Georg Cantor (1845 v- 1918) started the science of Set Theory with his paper, �On a Characteristic Property of All Real Agebriac Numbers�.

18/10/1871, Charles Babbage was born in Teignmouth, Devon died in London (born 26/12/1792 in Teignmouth, Devon).

26/9/1868, August Mobius, German mathematician, died (born 17/11/1790).

22/5/1868, Julius Plucker, German mathematician, died in Bonn, Germany.

15/10/1867, Jacques Inaudi, Italian child calculating prodigy, was born.

1865, The London Mathematical Society was founded and began to issue its journal, Proceedings.

1865, German mathematician August Ferdinand Mobius (born Schulpforte 17/11/1790) presented his discovery of a figure that had only one side and one edge, now known as the Mobius Strip.

2/9/1865, Sir William Hamilton, Scottish mathematician, died (born 4/8/1805).

8/11/1858, George Peacock, mathematician, died in Ely, England.

23/5/1857, Augustin Cauchy, mathematician, died (born 21/8/1789)

24/2/1856, Nicolas Lobachevsky, Russian mathematician, died (born 2/11/1793)

23/2/1855, Johann Karl Friedrich Gauss, mathematician, died in Gottingen, Germany.

29/4/1854, Jules Henri Poincare, French mathematician, was born in Nantes. In 1895 he effectively founded the science of topology, although some of its principles were already known.

1/11/1851, Thomas Galloway, Scottish mathematician, died (born 26/2/1796).

18/2/1851, Karl Jacobi, German mathematician, died (born 10/12/1804).

15/1/1850, Sophie Kovalevsky, Russian mathematician, was born (died 10/2/1891).

4/5/1845, William Clifford, English mathematician, was born (died 3/3/1879).

11/2/1839, Josiah Gibbs, US mathematician, was born (died 28/4/1903).

16/3/1838, Nathaniel Bowditch, mathematician, died in Boston, USA (born in Salem, Massachusetts, 1773)

16/1/1834, Jean Hachette, French mathematician, died (born 6/5/1769).

31/5/1832, Evariste Gallois, mathematician, died (born 25/10/1811)

7/12/1830, Luigi Cremona, Italian mathematician, was born (died 10/6/1903).

16/5/1830, Jean Fourier, French mathematician, died (born 23/3/1768).

6/4/1829, Neils Abel, Norwegian mathematician (born 1802) died inArendal.

23/2/1826, Nikolai Ivanovich Lobachevski (born near Nizhni Novgorod, Russia) gave a paper at Kazan University outlining the principles of non-Euclidean geometry.

7/4/1823, Jacques Charles, mathematician, died (born 12/11/1746)

16/8/1821, Arthur Cayley, English mathematician, was born (died 26/1/1895).

26/5/1821, Pafnutiy Chebichev, Russian mathematician, was born (died 8/12/1894).

14/1/1819, Sir James Cockle, English mathematician, was born (died 27/1/1895).

18/2/1816, Maurice Block, French statistician, was born in Berlin (died in Paris 9/1/1901).

2/11/1815, Mathematician George Boole was born in Lincoln, England. In 1847 he published his paper on symbolic logic.

10/4/1813, Joseph Lagrange, mathematician, died (born 25/1/1736)

25/10/1811, Evariste Gallois, mathematician, was born (died 31/5/1832).

30/11/1807, William Farr, English statistician, was born (died 14/4/1883).

13/7/1807, Jean Bernoulli, mathematician, died (born 4/11/1744)

4/8/1805, Sir William Hamilton, Scottish mathematician, was born (died 2/9/1865).

10/12/1804, Karl Jacobi, German mathematician, was born (died 18/2/1851).

25/8/1802, Neils Abel, Norwegian mathematician (died 1829) was born in Findoe.

28/8/1801, Antoine Cournot, French mathematician, was born (died 31/3/1877).

18/12/1799, Jean Montucla, French mathematician, died (born 5/9/1725).

20/2/1799, Jean Borda, French mathematician, died in Paris (born in Dax 4/5/1733).

26/2/1796, Thomas Galloway, Scottish mathematician, was born (died 1/11/1851).

2/11/1793, Nicolas Lobachevsky, Russian mathematician, was born (died 24/2/1856)

26/12/1792, Charles Babbage was born in Teignmouth, Devon (died 18/10/1871 in London).

17/11/1790, August Mobius, German mathematician, was born (died 26/9/1868).

15/1/1790, John Landen, English mathematician, died.

21/8/1789, Augustin Cauchy, mathematician, was born (died 23/5/1857).

22/11/1784, Paolo Frisi, Italian mathematician, died (born 13/4/1728).

29/10/1783, Jean Alembert, French mathematician, died in Paris.

18/9/1783, Leonhard Euler, Swiss mathematician, died in St Petersburg, Russia.

1782, Latin Squares were developed by Euler. These are n x n matrices with n different symbols,where each symbol occurs once in each row and once in each column. Sudoku puzzles are a type of Latin Square. These squares can be used in randomised drug experiments where n groups can be given different non-duplicated schedules of the drug.

20/5/1782, William Emerson, English mathematician, died (born 4/5/1701).

1777, Euler first used i to denote the square root of minus 1. He did not publish this until 1794, and it was Gauss who gained widespread acceptance for this notation with his work Disquisitiones arithmeticae in 1801.

25/9/1777, Johann Lambert, German mathematician, died (born 26/8/1728).

4/2/1774, Charles la Condamine, French mathematician, died (born 28/1/1701).

21/3/1768, Jean Fourier, French mathematician, was born (died 16/5/1830).

2/2/1768, Charles Camus, French mathematician, died (born 25/8/1699).

17/5/1765, Alexis Clairault, French mathematician, died (born 5/1713).

1761, Mathematician Thomas Bayes died (born 1701). He developed Bayes Theorem, an �intuitive�method of calculating probabilities based on prior observations; for example if the Sun rose yesterday morning, and the day before, it will likely rise this morning too. If every time the Sun rises (A) a cock is heard to crow (B), the probability that in the future the two events will occur together is Pr(A[B)= Pr(B[A) x PrPr(A) / Pr(B). In other words, the RHS terms are the numerical values for how often A occurs, B occurs and A and B occur together.

17/10/1759, Jacques Bernoulli, mathematician, was born (died 7/1789).

27/11/1754, Abraham Demoivre, English mathematician, died (born 26/5/1667).

15/4/1754, Jacopo Riccati, mathematician, died in Treviso, Italy.

12/11/1746, Jacques Charles, mathematician, was born (died 7/4/1823).

14/6/1746, Colin MacLaurin, Scottish mathematician, died.

10/5/1746, Gaspard Monge, inventor of descriptive geometry, was born (died 28/7/1818).

4/11/1744, Jean Bernoulli, mathematician, was born (died 13/7/1807).

1736, The symbol e for natural logarithms, devised by Leonhard Eulerin 1727, first appeared in print.

1736, Graph Theory was originated by Leonhard Euler. It had topological origins, dealing with networks of points connected by lines (the Konigsberg Bridge problem), and does not deal with graphs such as bar charts or pie charts.

25/1/1736, Joseph Lagrange, mathematician, was born (died 10/4/1813).

1734, Swiss mathematician Leonhardt Euler introduced the term f(x) to denote a mathematical function.

1733, The Bell Curve, showing that valuers for random processes couldcluster around a mean with values further off becoming less likely,was invented by Abraham de Moivre (1667 - 1754). This was developed in 1833 by German mathematician Friedrich Gauss (1777 � 1855).

4/5/1733, Jean Borda, French mathematician, was born in Dax (died in Paris 20/2/1799).

26/8/1728, Johann Lambert, German mathematician, was born (died 25/9/1777).

13/4/1728, Paolo Frisi, Italian mathematician, was born (died 22/11/1784).

27/6/1726, Nicolas Bernoulli, mathematician, died (born 27/1/1695).

5/9/1725, Jean Montucla, French mathematician, was born (died 18/12/1799).

16/5/1718, Maria Agnesi, Italian mathematician, was born (died 9/1/1799).

5/6/1716, Roger Cotes, English mathematician, died (born 10/7/1682).

15/10/1715, Humphry Ditton, English mathematician, died (born 29/5/1675).

15/4/1707, Leonhard Euler, Swiss mathematician, was born (died 18/9/1783).

20/3/1707, Jedediah Buxton, English arithmetician, was born (died 1772).

16/8/1705, Jacques Bernoulli, mathematician, died (born in Basel 27/12/1654).

4/5/1701, William Emerson, English mathematician, was born (died 20/5/1782).

28/1/1701, Charles la Condamine, French mathematician, was born (died 4/2/1774).

27/1/1695, Nicolas Bernoulli, mathematician, was born (died 26/7/1726).

1691, Leibniz first used the mathematical terms coordinate, ordinate and abscissa. He also introduced the term �function� to describe a mathematical equation where every unique input produces one unique output, for example if the function is x2, then input 4 produces output 16.

11/11/1688, Louis Castel, mathematician, was born (died 1757).

10/7/1682, Roger Cotes, English mathematician, was born (died 5/6/1716).

4/5/1677, Isaac Barrow, English mathematician, died (born 1630).

21/11/1675, Leibniz became the first mathematician to use the modern notation of f[x] dx for integration/differentiation.

29/5/1675, Humphry Ditton, English mathematician, was born (died 15/10/1715).

18/4/1674, John Graunt, English statistician, diedin London

26/5/1667, Abraham Demoivre, English mathematician, was born (died 27/11/1754).

1665, Isaac Newton worked out a system of �fluxions� � precursor of modern calculus. He also began work on a theory of gravity.

12/1/1665, French mathematician Pierre de Fermat (born 1601) died.

1664, Isaac Newton discovered the Binomial Theorem.

1659, First publication of the (division) sign, in Teutsche Algebra, written by Johann Heinrich and published long after his death in 1588.

27/12/1654, Jacques Bernoulli, mathematician, was born in Basel (died 16/8/1705).

1653, Mathematician Blaise Pascal (1623-1662) wrote his treatise, On the Arithmetical Traingle (published posthumously in 1665). His Triangle consisted of rows of numbers where each term is the sum of the two above, e.g. 1 � 1,1 � 1,2,1 � 1,3,3,1, �.This triangle is important in Probability Theory and for polynomial equations and fractals.

8/10/1652, John Greaves, English mathematician, died (born 1602).

1/9/1648, Marin Mersenne, French mathematician, died (born 8/9/1588).

30/9/1632, Thomas Allen, English mathematician, died at Gloucester Hall (born 21/12/1542 at Gloucester Hall).

1631, First recorded used of the multiplication sign, by William Oughtred. First use of the signs < and > to mean �less than� and �more than�.

26/1/1630, Henry Briggs, mathematician, died.

1629, Brackets were first used in mathematics by the Dutch mathematician, Albert Gerard.

10/12/1626, Edmund Gunter, English mathematician, died (born 1581).

2/7/1621, Thomas Harriot, English mathematician, died (born 1560).

1614, Logarithms were invented by the Scottish mathematician, John Napier (1550-1617), who in that year published a 97-page work entitled �Mirifici Logarithmorum Canonia Descriptio�. He coined the word �logarithm from two Greek words meaning �ratio� and �number�.

1603, Pietro A Cataldi discovered the 6th and 7th Perfect Numbers, which are 8,859,869,056 and 137,438,691,328.

17/8/1601, Pierre de Fermat, mathematician, was born (died 12/1/1665).

1591, Systematic use of letters in algebra, to denote both coefficients and unknown quantities, was introduced by French mathematician Francois Viete, aged 51, the �father of algebra�.

8/9/1588, Marin Mersenne, French mathematicianwas born (died 1/9/1648).

1586, Dutch mathematician Simon Stevin, aged 37, introduced decimal fractions, in his work La Disme.

5/3/1574, William Oughtred was born in Eton. Around 1621 he invented the slide rule.

1572, Complex / imaginary numbers were applied to solve equations in Rafael Bombelli�s work, Algebra. Imaginary numbers are those with a negative square root,and complex numbers are those with a real and imaginary element, such as 3 + 4i, which are expressed on a 2-d grid.

1556, The = sign was created by Robert Recorde, on the basis that nothing can be more equal than a pair of parallel lines.

21/12/1542, Thomas Allen, English mathematician, was born in Uttoxeter, Staffordshire (died 30/9/1632 at Gloucester Hall).

13/7/1527, John Dee, English mathematician, was born (died 1608).

1525, The first manual of geometry was compiled by the German engraver, Albrecht Durer.

1524, First recorded usage of the modern square root symbol, in Die Coss, by Christoff Rudolff.

1513, The fifth Perfect Number was discovered, 33,350,336. A Perfect Number is one whose factors add up to the number itself, like 6 is the sum of 1, 2 and 3.

1489, The first recorded use of the mathematical symbols + and -, in the book Mental Arithmetic by Johann Wildmann, published in Leipzig. They were used for �excess� and �deficiency�, and by the 1500s came to mean �add� and �subtract�.

1434, Leone Battista, born in Genoa, Italy, 14/2/1404, published a book on the geometrical laws of perspective in drawing.

1299, In Florence, Italy, the use of Arabic numerals was banned.

1240, Death of Italian mathematician Leonardo Pisano Bigollo (or, Fibonacci) (born ca. 1170). He developed the Fibonacci Series, 1,1,2,3,5,8,13,21,34�.where each term is the sum of the two previous ones. This series governs how biological systems reproduce.

1055, The Arabs introduced decimal notation to Spain.

15/5/1048, The Persian mathematician and poet Omar Khayyam was born at Nisipar. He was the first to solve cubic equations (those containing terms to the power of three).

830, The Arabic text Hisab al jabr w�al muqabalah (The Science of Reunion and Opposition) became the basis of algebra in the West. Renaissance scholars sometimes preferred the Latin term �analysis�.

665, Death of Indian mathematician Brahmagupta (ca. 598-665). He drew up formal rules for the use of zero in mathematics, as a number not just as a placeholder.

462, Birth of Aryabhata, Indian mathematician who wrote on the powers and roots of numbers.

17/4/485, Proclus, Greek mathematician, died in Athens.

250, Diophantes of Alexandria wrote the first known book on algebra.

200, Chinese mathematicians used powers of 10 to express numbers.

120, Death of Nichomachus,mathematician who first formulated the concept of Perfect Numbers. These are numbers whose factors, including 1, add to the number itself; 6 = 3+2+1, and 28 = 14+7+4+2+1. 496 and 98,128 are the next two Perfect Numbers. By 2007, just 44 such numbers were known.

95 BCE, First use of negative numbers, in China.

323 BCE, Euclid published his work �Elements�, the standard text on geometry. He proved that there must be infinitely many prime numbers.

395 BCE, Theodorus of Cyrene demonstrated that the square roots of 3,5,6,7,8,10, 12,13,14, 15 and 17 were irrational.

445 BCE, The earliest concept of irrational numbers (numbers like the square root of 2, or Pi, that have infinite decimal places). Hippacos of Metapontium discovered that some magnitudes are �not commensurable�, such as the diagonal and sode of a square; they have no common unit. Also at this time Zeno of Elea formulated paradoxes contrasting continuity with discreteness, such as the notion that a faster runner cannot ever catch a tortoise that has a headstart. These questions are still not fully answered today.

465 BCE, The dodecahedron, a solid with 12 faces, was described by Hippasus of Greece.

545 BCE, Death of Thales of Miletus. He derived the �Thales Proposition�; that triangles over the diameter of a circle are always right-angled.

628 CE, The Indian astronomer amd mathematician Brahmagupta first described the concept of �zero� as a true number. By 300 BCE use of the number zero was common in Babylonian mathematical texts. The name�zero�cpomes from the Sanskrit �sunya�, meaning �nothing�; it became �sifr� in Arabic, and was latinised by Leonardo Fibonacci into �zephirum�

876 BCE, First known use of a symbol for zero, in India. The actual concept of zero may have been known earlier than this.

1300 BCE, Decimal numerals in use in China.

1725 BCE, Egyptian geometrical uinderstanding was advanced, with formulae for the volume of a truncated pyramid.

1875 BCE, The Pythagorean Theorem was known in Mesopotamia.

1975 BCE, Quadratic equations, where symbols up to the power of two are used, were known and could be solved in Mesopotamia.

2000 BCE, Decimal notation first used in Mesopotamia.

3000 BCE, Numerology was being practised in Egypt. The art of numerology helped develop the science of mathematics, along with the need to keep accounts of trading and payments.

3400 BCE, In Sumeria, clay counting tokens and written mathematical symbols first used.

Appendix 1 � Calculations of Pi

1988, Japanese computer scientist Yasumasa Kanada calculated Pi to 201,326,000 decimal places.

12/4/1852, Ferdinand Lindemann was born in Hannover, Germany. In 1882 he proved that Pi is a transcendental number.

25/1/1812, Mathematician William Shanks was born in Corsenside, England. He attempted a calculation of Pi to 707 places in 1853. However in 1944 it was discovered he had made an error at place 528, causing all digits thereafter to be erroneous.

1717, Abraham Sharp calculated the value of Pi to 72 places.

605, Use of decimal notation in India. In China, Pi was calculated, by Tsu Chung Chi (430-501), to be between 3.1415926 and 3.1415927.

200, In China, a polygon of 3072 sides was used to calculate the value of Pi as 3.14159.

Appendix 2 � Prime Numbers

2018, The largest prime number so far known was calculated by Patrick Laroche. It had 24,862,048 digits.

1989, A prime number with 65,087 digits was calculated at the Amdahl Corporation, California, USA.

1985, The number composed of 1,031 ones in a row was found to be prime.

1985, The largest-then-known prime number,(2 to the power 216,065) minus 1, with 66,050 digits,was discovered.

1896, Jacques Hadamard proved that, for large values of a, the number of primes less than a approximates to a / log a.

1859, Bernhardt Reimann (1826-66) introduced the Reimann Hypothesis, which predicted the frequency of prime numbers amongst all integers.

1648, Death of French monk Marin Mersenne (born 1588), who first identified what are now know as Mersenne Prime Numbers; those whose formula is 2n � 1.

1588, Italian mathematician Pietro Cataldi discovedred the largest known prime number, 524,287. It remained the largest-known prime for almost two centuries.