HighFour History of Mathematics Category B: Grades 6 – 8 Round 5 Thursday, January 12, 2017 The use of calculator is not required. Answer #1: Explanation: Answer #2: Explanation: Answer #3: Explanation: Answer #4: Explanation: Answer #5: Explanation: Prime number Topics on prime numbers and their properties are extensively studied in number theory. Many famous mathematicians in the past studied prime numbers. Eratosthenes devised an algorithm that calculates primes called the Sieve of Eratosthenes. In Euclid’s Elements, it was proven that there are infinitely many prime numbers. Fundamental Theorem of Arithmetic The Fundamental Theorem of Arithmetic emphasizes the importance of prime numbers. Also, one important aspect of this theorem is the uniqueness of the prime factorization of every number, which has a challenging proof. Carl Friedrich Gauss Carl Friedrich Gauss is a German mathematics prodigy who had a great contribution in various fields of mathematics. He is also the famous mathematician in an anecdote who figured out how to add all integers from 1 up to 100 at the age of eight. Carl considers the book Disquisitiones Arithmeticae as his magnum opus (masterpiece). Fibonacci sequence The Fibonacci sequence was invented by Italian named Leonardo Pisano Bigollo. In mathematical history, he was also known as “Fibonacci” and “Leonardo of Pisa”. Composite number Composite numbers are simply positive integers greater than 1 which are not prime numbers. The first few composite numbers are 4,6,8,9 and 10. HighFour History of Mathematics Category B: Grades 6 – 8 Round 5 Thursday, January 12, 2017 The use of calculator is not required. Answer #6: Explanation: Answer #7: Explanation: Answer #8: Explanation: Answer #9: Explanation: Answer #10: Explanation: Grigori Perelman Grigori presented a proof of the Poincaré Conjecture in three papers in 2003. This breakthrough qualified him for a Fields Medal award and a million dollar prize. However, he declined both stating that he was not interested with money or fame. Five (5) In the beginning of Euclid’s book, Elements, these five postulates are stated together with some definitions. These laid the foundation of Euclidean geometry. Among the five postulates, the fifth postulate known as the Parallel Postulate is the most controversial for being so verbose as compared to the other four. Rational numbers The set of rational numbers is commonly denoted as ℚ. This was denoted by Giuseppe Peano in 1895 after the word “quotient”. The term rational is due to the fact that a rational number is a ratio of two integers. Dodecagon A dodecagon is a polygon with 12 sides. It comes from the Greek words duodeka, which means “twelve”, and gon, which means “angle”. A regular dodecagon, that is, a 12-­‐sided polygon whose sides and internal angles have equal measurement, is constructible by a compass and a straightedge. Twin primes Twin primes are pairs of prime numbers (𝑥, 𝑦) such that |𝑥 − 𝑦| = 2. One of the famous unsolved problems in number theory now is the Twin Prime Conjecture which claims that there are infinitely many twin primes. The first few twin primes are (3, 5), (5, 7), (11, 13) and (17, 19). HighFour History of Mathematics Category B: Grades 6 – 8 Round 5 Thursday, January 12, 2017 The use of calculator is not required. Answer #11: Explanation: Answer #12: Explanation: Answer #13: Explanation: Answer #14: Explanation: Answer #15: Explanation: Icosahedron There are many kinds of icosahedron, but the most familiar one is the regular icosahedron. An icosahedron has 30 edges and 20 equilateral triangle faces meeting at 12 vertices. Pierre de Fermat This theorem is famously known as Fermat’s Last Theorem. Fermat wrote this conjecture in the margin of a copy of Diophantus’ Arithmetica in which he claimed that he had proven this claim but was too long to fit in the margin. This theorem was successfully proven in 1994 by British mathematician named Andrew John Wiles. Abscissa A point in the Cartesian coordinate system can be represented by the ordered pair (𝑥, 𝑦) where 𝑥 is called the abscissa, and 𝑦 is called the ordinate. The word abscissa originated from the Latin word abscissa (linea) which means “(a line) cut off”. Chord (of a circle) The study on the chords of a circle and their properties contributed greatly in the development of Trigonometry. A chord that passes through the center point of a circle is called the diameter. In fact, in a circle, the diameter is the longest chord. Perfect number Perfect numbers are one of the most studied numbers in field of number theory. An example of a perfect number is the number 6 because given 1,2 and 3 as all the proper positive divisors of 6, the sum 1 + 2 + 3 = 6. A few other examples of a perfect number are 28,496 and 8182. HighFour History of Mathematics Category B: Grades 6 – 8 Round 5 Thursday, January 12, 2017 The use of calculator is not required. Answer #16: Explanation: Answer #17: Explanation: Answer #18: Explanation: Answer #19: Explanation: Answer #20: Explanation: Radian Radian is denoted by the symbol 𝑟𝑎𝑑. In a unit circle, the length of the arc is equal to the radian measurement of the angle it subtends. The concept of the radian measurement is credited to Roger Cotes. Parallel lines In the two-­‐dimensional Euclidean space, two lines are parallel if they do not intersect at a point. The concept of parallel lines is the main subject of the Parallel Postulate by Euclid. Heron’s formula This formula was credited to Heron of Alexandria, a Greek mathematician and engineer. Hypotenuse The word hypotenuse comes from Late Latin hypotenusa, from Greek hypoteinousa which means “stretching under”. The hypotenuse of a right triangle can also be defined as the side opposite to the right angle among the interior angles of the triangle. La Géométrie René Descartes published La Géométrie in 1637. This paved the way to the creation of a subject called analytic geometry which translated every geometric problem into an algebraic problem.