H.O.M. Final
A complete algorithm for the division of one polynomial by another was developed first by the Islamic mathematician A) al-Samaw'al; B) al-Khayyami; C) al-Tusi.
A
A proof that √ 2 is irrational (or "incommensurate," as the Greeks would have called it) was known: A) already at the time of Aristotle (≈ 384 − 322 BC); B) already to the architects of Ramses II, but they left no record about it; C) only after a proof was found by Hypatia in 400 AD.
A
A sphere of radius r contains 2/3 of the volume of the smallest cylinder into which the sphere can be placed. This theorem was proved by A) the Greek mathematician Archimedes of Syracuse; B) an unnamed court mathematician of King Hamurabi; C) the Greek mathematician Euclid of Alexandria
A
Already the earliest ancient Greek mathematicians established combinatorial identities, such as 1 + 2 + 3 +· · ·+N = N(N + 1)/ 2 and 1 + 3 + 5 +· · ·+ (2N −1) = N^ 2 , which are also listed in Euclid's Elements. Their method of proof was A) using geometrical arguments to effectively count (say) pebbles arranged in rectangular lattice; B) the method of complete induction; C) by building the first mechanical calculator, the famous astrolab.
A
Apollonius figured that one can draw an ellipse on a flat piece of sand by using A) three wooden sticks and one long rope; B) a stick, a straight edge and a compass; C) two wooden sticks and a compass.
A
Apollonius of Perga used his theory of conic sections to A) solve the problem of doubling the cube; B) describe the motion of Mars about the sun; C) construct mirrors that focus sun light onto a single point.
A
Aristotle is often credited as being the greatest logician of all time. Yet he got a number of things wrong. Among those conepts where he erred is A) his law of motion, saying that the natural state of motion is rest, and to deviate from rest, a force has to be applied; B) his argument for why √ 2 is not a rational number; C) his argument that "p implies q" means that "(not q) implies (not p)."
A
Around 1000 AD, the Islamic mathematician Abu 'Abdallah al-Hasan ibn al-Baghdadi was apparently the first one to prove that A) between 1 and 2 there are infinitely many irrational numbers; B) between 1 and 2 there are infinitely many rational numbers; C) between 1 and 2 there are infinitely many numbers.
A
Euclid's "Elements" (or rather about half of the books) were translated into Chinese A) in the early 17th century AD, by Xu Guangqi, who was a student of the Jesuit priest Matteo Ricci; B) in the 13th century AD, after Marco Polo brought a copy to the court of Kublai Khan; C) in the 4th century BC, after Alexander the Great had invaded northern India and had the "Elements" translated into Hindu, from where they were brought to China by Buddhist monks and translated into Mandarin.
A
One of the earliest Indian sources of combinatorial problem solving is A) a medical text that considers the question of how many different tastes can be created by choosing from six basic tastes; B) a mathematical text that considers the question of how many different tastes can be created by choosing from six basic tastes; C) a cook book that considers the question of how many different tastes can be created by choosing from six basic tastes.
A
Pascal's triangle was actually discovered earlier. Two mathematicians who definitely had it were A) the Islamic mathematician ibn Mun'im of the early 13th century AD and the Chinese mathematician Yang Hui of the late 13th century AD; B) the 13th century AD Islamic mathematician ibn Mun'im and the 6th century BC Indian mathematician Susruta. C) the Chinese mathematician Yang Hui of the late 13th century AD, and the 6th century BC Indian mathematician Susruta.
A
The "Chinese Remainder Theorem" is a theorem about A) modular arithmetic; B) trigonometry; C) the difference (a.k.a. the remainder term) between the area π of the unit circle and the area of a regular inscribed polygon.
A
The Greek mathematician Zeno who lived in the 5th century BC is famous for his paradoxes. He is credited for having inspired A) the concept of proof by "reductio ad absurdum;" B) the paradox of Archimedes and the tortoise; C) the paradox of the prisoner's dilemma
A
The Islamic mathematician Omar Khayyam found a geometric way to find the positive roots of the cubic equation x 3+bx2+cx = d whenever it has positive roots. He focused on positive roots because A) at the time the term x 3 stood for the (positive) volume of a cube of side length x; B) the negative numbers had not yet been discovered; C) he didn't know how to handle negative numbers.
A
The Islamic mathematician al-Samaw'al gave an argument of the inductive type for the validity of the identity summation(k^3)=(summation(k))^2. However, the result was already reported, though without proof, by A) the 6th century AD Indian mathematician Aryabhata; B) the 9th century AD Indian mathematician Mahavira; C) the 6th century BC Indian mathematician Susruta.
A
The ancient Greek mathematical problem of "doubling the cube" refers to a construction that is to be carried out with the help of only straightedge and compass. The problem is A) to construct a cube with twice the volume of a given cube; B) to construct a cube with twice the surface area of a given cube; C) to construct a cube with twice the side length of a given cube.
A
The book "Exhaustive treatise of shadows" contains a wealth of mathematical knowledge about trigonometric functions, yet put to use exclusively for astronomical problems. It was merely one of about 140 books written by the prolific Islamic mathematician A) Abu l-Rayhan Muhammad ibn Ahmad al-Biruni; B) Ahmad ibn 'Abdallah al-Marwazi Habas al-Hasib; C) Abu 'Abdallah Muhammad ibn Jabir al-Battani.
A
The early 13th century AD Islamic mathematician Ahmad ibn Mun'im ibn al-Abdari (d. 1228 AD) discovered a formula for the number C n k that counts the number of ways in which k objects can be picked from a set of n objects (with n ≥ k, of course). This formula shows that C n 2 equals A) the sum of the first n − 1 natural numbers; B) the sum of the first n natural numbers; C) the sum of the squares of the first n − 1 natural numbers.
A
The fact that there are infinitely many prime numbers was first proved by A) Euclid; B) Archimedes; C) Pythagoras.
A
The insight that the conic sections describe the motion of planets about the sun is due to A) Johannes Kepler; B) Tycho Brahe; C) Ptolemy.
A
The mathematical meaning behind the statement "it is already Summer again" can be explained with the help of A) modular arithmetic; B) combinatorics; C) the iso-perimetric inequality
A
The mathematician Brahmagupta stated that 0/0 equals A) 0; B) 1; C) ∞.
A
The meaning of the Indian word that got mis-translated into "sine" is A) half chord; B) double chord; C) chord.
A
The modern terminology for the mathematical subject "algebra" derives from the Arabic word "al-jabr" in Al-Khwarizmi's book on how to solve equations. "Al-jabr" is a move that A) rewrites the equation a + b = c into a = c − b; B) rewrites the equation a + b = c into a + b + d = c + d; C) divides both sides of an equation by the same divisor.
A
Using only a straight edge and a compass, one can construct A) a line with length √ 2; B) a line with length 3 √ 2; C) a square with area π.
A
"Don't disturb my circles" are (supposedly) the famous last words of A) Hypatia, before she was brutally slaughtered by a mob of religiously incensed commoners; B) Archimedes, before he was killed by a Roman soldier who was sent to fetch him and bring him to the Roman commander; C) Plato, when he learned that Appolonius wanted to change his model of planets that move in circles around the Earth to a model in which they move on epicycles around the Earth.
B
Archimedes designed a mechanism to irrigate fields. This is known as A) Archimedean water wheel; B) Archimedean screw; C) Archimedean spiral.
B
Archimedes of Syracuse is A) an honorary professor title at Syracuse University in New York State; B) one of the greatest Greek mathematicians of ancient times; C) the founder of the Greek city of Syracuse on the island of Sicily.
B
For large numbers n it is increasingly unpractical to compute n! by simply multiplying out n(n − 1)(n − 2)· · · 2 · 1. Stirling's formula gives a numerical approximation that is easier to compute. The Islamic mathematicians of the 13th century AD could not develop this formula because A) they were not interested in large numbers; B) they did not have the concept of a Riemann integral; C) they did not have the concept of numerical approximation.
B
Problem 7 Al-Samaw'al and ibn al-Banna used inductive-method-type reasoning to prove combinatorial identities. However, the mathematician who first announced the inductive principle as method of proof was a contemporary of ibn al-Banna, the mathematician A) Yang Hui; B) Levi ben Gerson; C) ibn Mun'im.
B
Pythagoras is reported to have argued, more on aesthetical grounds than any empirical reason, that the Earth was not a disk but a ball. By around 500 BC this had become common wisdom in ancient Greece. But it was not until Aristotle (4th century BC) that logical arguments applied to empirical observations were offered in favor of a spherical Earth. Inspired by Aristotle's arguments, his contemporary Eratosthenes of Cyrene came up with an idea to measure the circumference of the Earth, which he then carried out. His computations belong in the subject of A) Combinatorics; B) Trigonometry; C) Number theory.
B
THE isoperimetric problem is to find the shape of the closed curve which encloses the largest area, given its length (the perimeter of the area). It was solved in all generality only in modern times, but already the ancient Greek mathematician Zenodorus proved that among all regular polygons of given perimeter, the limiting figure of the circle encloses the largest area. Related to this is the insight of the Greek mathematician Pappus who noticed that a bee honey comb is made of many (ideally) identical regular polygonal cells patched together without gaps, and among the only three possible cross sections which accomplish such a feat (the regular triangle, the square, the regular hexagon) it is the hexagon which encloses the largest area, given the perimeter. He concluded that A) it is one of the greatest mysteries of science how bees could possibly have figured that out; B) bees in their wisdom of course chose this least materially expensive method to store their harvest; C) the laws of nature seem to favor optimal design through natural selection among random trials
B
The "method of insufficient reason" was used by Archimedes to argue that A) the proof of Euclid that there are infinitely many prime numbers is wrong; B) two equal masses on the end points of a lever supported in its midpoint must be in equilibrium; C) there is no reason to doubt his calculation of the surface area of a sphere of radius R.
B
The British mathematician Andrew Wiles is famous for A) having proved that there are infinitely many triplets (x, y, z), with x, y, z natural numbers, that solve the Diophantine equation x 2 +y 2 = z 2 . B) having proved that the Diophantine equation x n + y n = z n cannot be solved by triplets (x, y, z) with x, y, and z natural numbers, when n > 2; C) having found a mistake in Euclid's proof that there are infinitely many triplets (x, y, z), with x, y, z natural numbers, that solve x 2 + y 2 = z 2 .
B
The Greek philosopher and logician Aristotle was A) a student of Plato and the teacher of Archimedes; B) a student of Plato and the teacher of Alexander the Great; C) a student of Pythagoras and the teacher of Ptolemy of Alexandria.
B
The Islamic mathematician Nasir al-Din al-Tusi, who lived in the 13th century, made the important discovery that A) the cubic equation can have negative solutions; B) different types of cubic equations can be transformed into each other by suitable variable transformations; C) a cubic equation can have 4 solutions, counted in multiplicity.
B
The Islamic mathematician Omar Khayyam found a geometric way to find the positive roots of the cubic equation x 3+bx2+cx = d whenever it has positive roots. His method used A) the analogue of "completing the square," a.k.a. "completing the cube"; B) the intersection of two suitable conic sections; C) the tetrahedral closest packing of spheres.
B
The ancient Greek philosopher Aristotle is generally known for A) being the father of Alexander the Great; B) having stipulated the rules of mathematical logic and proof; C) being the son of the Greek philosopher Plato.
B
The computation with arbitrary real numbers, using the decimal place value system, was developed by A) the Indian mathematician Brahmagupta; B) the Islamic mathematician Al-Samaw'al; C) the Islamic mathematician ibn al-Banna.
B
The first book devoted entirely to trigonometry for the sake of trigonometry, and not as a tool to address problems in astronomy, was published by the Islamic mathematician al-Jayyani in the 11th century. He spent most of his life in A) Baghdad; B) Cordoba; C) Marrakesh.
B
The modern mathematical word "algorithm" traces its origin to a mis-translation of the Arabic statement "[name] says," where [name] is the Islamic mathematician A) Al-Samaw'al al-Maghribi; B) Muhammad ibn Musa al-Khwarizmi; C) Al Khayyami.
B
The name of the Islamic mathematician Al-Samaw'al al-Maghribi indicates that he worked in A) Baghdad; B) Morocco; C) Persia.
B
The problem of finding integer pairs (x, y) that solve the quadratic equation x 2 − ny2 = 1 with n an arbitrary natural number was first solved in generality by A) the Greek mathematician Pythagoras; B) the Indian mathematician Acarya Jayadeva around 1000 AD C) the Indian mathematician Bhaskara I in the 7th century AD.
B
The recursion formula C n k = summation(C n−l k−1) was discovered by A) the 9th century AD Islamic mathematician al-Khwarizmi; B) the 13th century AD Islamic mathematician ibn-Mun'im; C) the 12th century AD Islamic mathematician al-Samaw'al.
B
The study of spherical trigonometry by Islamic mathematicians was motivated by A) their quest for finding a non-flat geometry; B) their faith that required Muslims to face to the direction of Mecca during prayer; C) their desire to improve the nautical navigation skills of their Indian ocean fleet.
B
The theorem that 3 √ 2 cannot be constructed with straightedge and compass was discovered by A) Archimedes of Syracuse, around 250 BC; B) the French mathematician Pierre Wantzel, in 1837 AD; C) Apollonius of Perga, arround 200 BC.
B
The thirteen books that comprise Euclid's "Elements" are A) exclusively about Euclidean geometry; B) about Euclidean geometry and also number theory; C) about Euclidean and non-Euclidean geometry.
B
The trigonometric functions used by Islamic mathematicians did not yet have their modern definitions. Namely, A) they were defined only for angles between 0 and π; B) they were defined only for angles for which they yielded positive answers; C) they were defined for angles between 0 and 2π but not continued periodically.
B
The word sine A) has been coined by sinologists to honor the contributions of Chinese mathematicians to the field of trigonometry; B) is the English adaptation of the Latin word sinus, which in turn is the Latin translation of the Arabic word jaib, which in turn is written in Arabic simply as jb, which in turn originally stood for jiba, which in turn was simply a phonetic Arabic rendering of the Indian ji-va, the abbreviation for the Indian word of what we now call sine; C) was invented by German mathematicians in their study of the shape of an orange, which in German is called Apfelsine.
B
Using successive approximations of a circle by inscribed regular polygons, the number π can be computed, in principle, to arbitrary precision. This technique is known as "method of exhaustion." It was invented by A) Archimedes; B) Eudoxos; C) Euclid.
B
According to legend, the proof that √ 2 is irrational (or "incommensurate") was revealed by Hippasus, who subsequently A) was given the title "Court Mathematician" by Alexander the Great; B) became admired as a genius; C) was murdered for divulging the secret.
C
Al Khayyami, a.k.a. Omar Khayyam (≈ 1048 − 1131), made the revolutionary discovery that A) the roots of the cubic equation can be expressed with the help of square and cube roots; B) the orbits of the planets are conic sections with the sun in a focal point; C) irrational quantities like √ 2 and 3 √ 2 (etc.) obey the same arithmetical rules as rational numbers.
C
Already some 8th century AD Islamic mathematicians pondered the problem of how many (say) six-letter words can be formed from an alphabeth with (say) 28 letters. This problem can be solved with the help of A) the Chinese remainder theorem; B) what became later known as Pell's equation; C) combinatorics.
C
Although we seem to have only indirect evidence based on texts written centuries after the putative events, the theorem that the sum of consecutive odd natural numbers, starting with 1, is always a square number, was first shown by A) Archimedes; B) Euclid; C) Pythagoras.
C
Archimedes figured out the laws of buoyancy. As a consequence of these, you can figure the following. If you sit in a small boat on a quiet, isolated pond of water, and carry with you a heavy stone, then drop the stone slowly into the pond (no splash), wait until the stone has sunk to the bottom of the pond and the water is quiet again, then you compare the water level, you will find that the water level has A) stayed the same; B) gone up; C) gone down.
C
Archimedes found that the area of a circle equals 1 2 of the product of its radius with its circumference. His method of proof for this finding was A) to use conic sections; B) to square the circle; C) by showing that the assumption that it was false leads to contradictions.
C
Archimedes had given rigorous lower and upper bounds on the value of π, thus 3 + 10 /71 < π < 3 + 1 /7 . To obtain his results he used the "method of exhaustion." This method is a predecessor of the modern concept of A) differentiation; B) topology; C) Riemann integration.
C
Archimedes had given rigorous lower and upper bounds on the value of π, thus 3 + 10/71 < π < 3 + 1/7 . Later the Islamic mathematician Abu Sal al-Kuhi arrived at the much smaller value 3+ 1 9 for π. He knew that his value was smaller than Archimedes' lower bound, and he therefore concluded that A) he must have made a mistake; B) Archimedes must have made a mistake; C) a mistake must have been made during the transmission of Archimedes' results.
C
Babylonian mathematicians expressed numbers A) in base 2; B) in base 10; C) in base 60.
C
Euclid's 5th postulate in his "Elements" that deals with planar geometry had for a long time bothered mathematicians, who felt that it is redundant with the other postulates and axioms. The first person to realize that Euclid's 5th postulate is independent of the others was A) the Islamic mathematician Nasir al-Din al-Tusi in the 13th century; B) his son Sadr al-Din al-Tusi in the 13th century; C) none of the above.
C
Hypatia is perhaps the most renowned Greek female mathematician. She was killed by A) a Roman soldier who came to arrest her, but she told him "Don't disturb me while I am drawing circles;" B) a crocodile while taking a swim in the river Nile; C) an angry mob of religious fanatics who believed that she and her work served the Devil.
C
In the earliest known Chinese mathematical texts the number π was reported to be equal to 3. The 3rd century AD Chinese mathematician Liu Hui is apparently the first one on record for criticizing the statement "π equals 3" as false. He supported his claim by noting that π had to be bigger than 3 because he discovered that 3 is exactly the area of a regular N-gon whose vertices are on a circle with radius 1 when A) N = 3; B) N = 6; C) N = 12.
C
In two-dimensional Euclidean space there are infinitely many regular polygons. The analog concept in three dimensions is Plato's regular polyhedra, solids whose faces are bounded by polygons that are all congruent to each other. There are A) ∞ many different regular polyhedra in three dimensions; B) six different regular polyhedra in three dimensions; C) five different regular polyhedra in three dimensions.
C
Indian mathematicians also studied integer-valued solution pairs (x, y) of what later became known as "Pell's equation:" Dx2 ± 1 = y 2 , with D a natural number. It is known by this name because Leonhard Euler mistakenly attributed its solution to John Pell. However, a comprehensive algorithm for its solution, though without proof, was presented already in the book "Lilavati" written by A) Bhaskara I in the 7th century AD; B) Brahmagupta in the 7th century AD; C) Bhaskara II in the 12th century AD.
C
Mahavira's formula for the number C n k of possibilities to choose k items from a set of n ≥ k items was rediscovered by the Islamic mathematician A) of the 12th century AD, al-Samaw'al; B) of the early 13th century AD, ibn-Mun'im; C) of the late 13th century AD, ibn al-Banna.
C
Originally a library of books of knowledge that soon turned into a scientific, mathematical, medicinal, etc. research institute, The House of Wisdom was located in A) Mecca; B) Damaskus; C) Baghdad.
C
Partly influenced by the advances in India that were translated into Arabic already late in the eight century AD, Islamic mathematicians made major advances in trigonometry. In addition to the Greek chord and the Indian sine function, by the 9th century they also used cosine, secant, cosecant, tangent, and cotangent. Of these latter five trigonometric functions, only the tangent had made its appearance earlier already, namely in the 8th century in A) India; B) Cambodia; C) China.
C
Thales of Miletus (born before 620 BC, died after 550 BC) is generally credited with having introduced A) the base 60 system of numbers; B) the transcendental numbers; C) the concepts of rigorous proof and scientific inquiry.
C
The "Almagest," which was the standard work for astronomers in Europe until the revolutionary ideas of Copernicus, was authored by A) Aristotle; B) Thales of Miletus; C) Ptolemy of Alexandria.
C
The 13th century AD Chinese mathematician Qin Jiushao is well-known for A) independently solving Pell's equation; B) approximating π to 7 significant digits; C) publishing the first general strategy for solving N simultaneous linear congruences.
C
The Greek "method of exhaustion" was used A) by Alexander the Great as battle strategy to conquer his enemies; B) by Aristoteles to win his philosophical disputes; C) by mathematicians to compute better and better approximations to π.
C
The Platonic solids have their faces bounded by regular A) triangles, squares, or hexagons; B) triangles, squares, or heptagons; C) triangles, squares, or pentagons.
C
The ancient Greek mathematical problem of "squaring the circle" refers to a construction that is carried out with the help of only straightedge and compass. The problem is A) to construct a square whose corners lie on a given circle; B) to construct a square whose four sides touch a given circle; C) to construct a square that encloses the same area as a given circle.
C
The formula for the number C n k of possibilities to choose k items from a set of n ≥ k items, in modern notation given by C n k = n(n − 1)· · ·(n − k + 1)/ k(k − 1)· · · 2 · 1 , was first stated (though without proof ) by A) the 6th century BC Indian mathematician Susruta; B) the 6th century AD Indian mathematician Varahamihira; C) the 9th century AD Indian mathematician Mahavira.
C
The full name Abu-l'-Abbas Ahmad al-Marrakushi ibn al-Banna indicates that ibn al-Banna was from A) India; B) Persia; C) Morocco.
C
The mathematician who perfected the study of conic sections in Euclidean geometry was A) the Greek mathematician Pythagoras; B) the Greek mathematician Thales of Miletus; C) the Greek mathematician Appolonius of Perga.
C
The relative sizes of earth, moon, and sun were computed by A) Ptolemy, about 140 AD; B) Apollonius, about 200 BC; C) Aristarchus, in the 3rd century BC.
C
The theory of epicycles plays a major role in the Almagest, where the planetary motions as seen in the heavens from the perspective of someone on earth, are described. The epicycle theory was invented by A) Ptolemy; B) Aristarchus; C) Apollonius
C
Using approximations of a circle by inscribed regular polygons, the number π was computed to a precision of 7 significant digits by the Chinese mathematician Zu Chongzhi. According to later Chinese writers (no original text is extant), the polygon he used had 24576 sides. To accomplish this feat A) he went to the Gobi desert where he found a vast plateau of sand on which to draw such a large polygon; B) he computed the area S2n of polygons with 2n from the area Sn of polygons with n vertices, through a doubling algorithm that starts with S4; C) he computed the area S2n of polygons with 2n from the area Sn of polygons with n vertices, through a doubling algorithm that starts with S3
C
Zeno's paradox of Archilles and the tortoise was really paradoxical to him because A) he couldn't imagine that tortoises could really run that fast; B) he couldn't imagine why Archilles would agree to become so humiliated; C) he knew the conclusion was wrong, but he could not see what was wrong with his reasoning that produced the wrong conclusion.
C
