The Mathematical Intelligencer
The Mathematical Intelligencer is a mathematical journal published by Springer Verlag that aims at a conversational and scholarly tone, rather than the technical and specialist tone more common among academic journals. It was started by mathematicians Bruce Chandler and Harold Edwards Jr. and first appeared in 1979. Marjorie Senechal is the editor-in-chief. Branislav Kisacanin Review of Mathematical Conversations. Mathematical Association of America. Home page for Mathematical Intelligencer at Springer Verlag
History of mathematics
The area of study known as the history of mathematics is an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BC the Mesopotamian states of Sumer and Assyria, together with Ancient Egypt and Ebla began using arithmetic and geometry for purposes of taxation, trade and in the field of astronomy and to formulate calendars and record time; the most ancient mathematical texts available are from Mesopotamia and Egypt - Plimpton 322, the Rhind Mathematical Papyrus and the Moscow Mathematical Papyrus. All of these texts mention the so-called Pythagorean triples and so, by inference, the Pythagorean theorem, seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry; the study of mathematics as a "demonstrative discipline" begins in the 6th century BC with the Pythagoreans, who coined the term "mathematics" from the ancient Greek μάθημα, meaning "subject of instruction".
Greek mathematics refined the methods and expanded the subject matter of mathematics. Although they made no contributions to theoretical mathematics, the ancient Romans used applied mathematics in surveying, structural engineering, mechanical engineering, creation of lunar and solar calendars, arts and crafts. Chinese mathematics made early contributions, including a place value system and the first use of negative numbers; the Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics through the work of Muḥammad ibn Mūsā al-Khwārizmī. Islamic mathematics, in turn and expanded the mathematics known to these civilizations. Contemporaneous with but independent of these traditions were the mathematics developed by the Maya civilization of Mexico and Central America, where the concept of zero was given a standard symbol in Maya numerals.
Many Greek and Arabic texts on mathematics were translated into Latin from the 12th century onward, leading to further development of mathematics in Medieval Europe. From ancient times through the Middle Ages, periods of mathematical discovery were followed by centuries of stagnation. Beginning in Renaissance Italy in the 15th century, new mathematical developments, interacting with new scientific discoveries, were made at an increasing pace that continues through the present day; this includes the groundbreaking work of both Isaac Newton and Gottfried Wilhelm Leibniz in the development of infinitesimal calculus during the course of the 17th century. At the end of the 19th century the International Congress of Mathematicians was founded and continues to spearhead advances in the field; the origins of mathematical thought lie in the concepts of number and form. Modern studies of animal cognition have shown; such concepts would have been part of everyday life in hunter-gatherer societies. The idea of the "number" concept evolving over time is supported by the existence of languages which preserve the distinction between "one", "two", "many", but not of numbers larger than two.
Prehistoric artifacts discovered in Africa, dated 20,000 years old or more suggest early attempts to quantify time. The Ishango bone, found near the headwaters of the Nile river, may be more than 20,000 years old and consists of a series of marks carved in three columns running the length of the bone. Common interpretations are that the Ishango bone shows either a tally of the earliest known demonstration of sequences of prime numbers or a six-month lunar calendar. Peter Rudman argues that the development of the concept of prime numbers could only have come about after the concept of division, which he dates to after 10,000 BC, with prime numbers not being understood until about 500 BC, he writes that "no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, some numbers that are multiples of 10." The Ishango bone, according to scholar Alexander Marshack, may have influenced the development of mathematics in Egypt as, like some entries on the Ishango bone, Egyptian arithmetic made use of multiplication by 2.
Predynastic Egyptians of the 5th millennium BC pictorially represented geometric designs. It has been claimed that megalithic monuments in England and Scotland, dating from the 3rd millennium BC, incorporate geometric ideas such as circles and Pythagorean triples in their design. All of the above are disputed however, the oldest undisputed mathematical documents are from Babylonian and dynastic Egyptian sources. Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia from the days of the early Sumerians through the Hellenistic period to the dawn of Christianity; the majority of Babylonian mathematical work comes from two separated periods: The first few hundred years of the second millennium BC, the last few centuries of the first millennium BC. It is named Babylonian mathematics due to the central role of Babylon as a place of study. Under the Arab Empire, Mesopotamia Baghdad, once again became an important center of
The lunar phase or phase of the Moon is the shape of the directly sunlit portion of the Moon as viewed from Earth. The lunar phases and cyclically change over the period of a synodic month, as the orbital positions of the Moon around Earth and of Earth around the Sun shift; the Moon's rotation is tidally locked by Earth's gravity. This near side is variously sunlit, depending on the position of the Moon in its orbit. Thus, the sunlit portion of this face can vary from 0% to 100%; the lunar terminator is the boundary between the darkened hemispheres. Each of the four "intermediate" lunar phases is around 7.4 days, but this varies due to the elliptical shape of the Moon's orbit. Aside from some craters near the lunar poles, such as Shoemaker, all parts of the Moon see around 14.77 days of daylight, followed by 14.77 days of "night". In western culture, the four principal phases of the Moon are new moon, first quarter, full moon, third quarter; these are the instances when the Moon's ecliptic longitude and the Sun's ecliptic longitude differ by 0°, 90°, 180°, 270°, respectively.
Each of these phases occur at different times when viewed from different points on Earth. During the intervals between principal phases, the Moon's apparent shape is either crescent or gibbous; these shapes, the periods when the Moon shows them, are called the intermediate phases and last one-quarter of a synodic month, or 7.38 days, on average. However, their durations vary because the Moon's orbit is rather elliptical, so the satellite's orbital speed is not constant; the descriptor waxing is used for an intermediate phase when the Moon's apparent shape is thickening, from new to full moon, waning when the shape is thinning. The eight principal and intermediate phases are given the following names, in sequential order: Non-Western cultures may use a different number of lunar phases; when the Sun and Moon are aligned on the same side of the Earth, the Moon is "new", the side of the Moon facing Earth is not illuminated by the Sun. As the Moon waxes, the lunar phases progress through new moon, crescent moon, first-quarter moon, gibbous moon, full moon.
The Moon is said to wane as it passes through the gibbous moon, third-quarter moon, crescent moon, back to new moon. The terms old moon and new moon are not interchangeable; the "old moon" is a waning sliver until the moment it aligns with the Sun and begins to wax, at which point it becomes new again. Half moon is used to mean the first- and third-quarter moons, while the term quarter refers to the extent of the Moon's cycle around the Earth, not its shape; when an illuminated hemisphere is viewed from a certain angle, the portion of the illuminated area, visible will have a two-dimensional shape as defined by the intersection of an ellipse and circle. If the half-ellipse is convex with respect to the half-circle the shape will be gibbous, whereas if the half-ellipse is concave with respect to the half-circle the shape will be a crescent; when a crescent moon occurs, the phenomenon of earthshine may be apparent, where the night side of the Moon dimly reflects indirect sunlight reflected from Earth.
In the Northern Hemisphere, if the left side of the Moon is dark the bright part is thickening, the Moon is described as waxing. If the right side of the Moon is dark the bright part is thinning, the Moon is described as waning. Assuming that the viewer is in the Northern Hemisphere, the right side of the Moon is the part, always waxing. In the Southern Hemisphere, the Moon is observed from a perspective inverted, or rotated 180°, to that of the Northern and to all of the images in this article, so that the opposite sides appear to wax or wane. Closer to the Equator, the lunar terminator will appear horizontal during the evening. Since the above descriptions of the lunar phases only apply at middle or high latitudes, observers moving towards the tropics from northern or southern latitudes will see the Moon rotated anti-clockwise or clockwise with respect to the images in this article; the lunar crescent can open upward or downward, with the "horns" of the crescent pointing up or down, respectively.
When the Sun appears above the Moon in the sky, the crescent opens downward. The crescent Moon is most and brightly visible when the Sun is below the horizon, which implies that the Moon must be above the Sun, the crescent must open upward; this is therefore the orientation in which the crescent Moon is most seen from the tropics. The waxing and waning crescents look similar; the waxing crescent appears in the western sky in the evening, the waning crescent in the eastern sky in the morning. When the Moon as seen from Earth is a narrow crescent, Earth as viewed from the Moon is fully lit by the Sun; the dark side of the Moon is dimly illuminated by indirect sunlight reflected from Earth, but is bright enough to be visible from Earth. This phenomenon is called earthshine and sometimes picturesquely described as "the old moon in the new
Asymmetry is the absence of, or a violation of, symmetry. Symmetry is an important property of both physical and abstract systems and it may be displayed in precise terms or in more aesthetic terms; the absence of or violation of symmetry that are either expected or desired can have important consequences for a system. Due to how cells divide in organisms, asymmetry in organisms is usual in at least one dimension, with biological symmetry being common in at least one dimension. Louis Pasteur proposed that biological molecules are asymmetric because the cosmic forces that preside over their formation are themselves asymmetric. While at his time, now, the symmetry of physical processes are highlighted, it is known that there are fundamental physical asymmetries, starting with time. Asymmetry is an important and widespread trait, having evolved numerous times in many organisms and at many levels of organisation. Benefits of asymmetry sometimes have to do with improved spatial arrangements, such as the left human lung being smaller, having one fewer lobes than the right lung to make room for the asymmetrical heart.
In other examples, division of function between the right and left half may have been beneficial and has driven the asymmetry to become stronger. Such an explanation is given for mammal hand or paw preference, an asymmetry in skill development in mammals. Training the neural pathways in a skill with one hand may take less effort than doing the same with both hands. Nature provides several examples of handedness in traits that are symmetric; the following are examples of animals with obvious left-right asymmetries: Most snails, because of torsion during development, show remarkable asymmetry in the shell and in the internal organs. Fiddler crabs have one small claw; the narwhal's tusk is a left incisor which can grow up to 10 feet in length and forms a left-handed helix. Flatfish have evolved to swim with one side upward, as a result have both eyes on one side of their heads. Several species of owls exhibit asymmetries in the size and positioning of their ears, thought to help locate prey. Many animals have asymmetric male genitalia.
The evolutionary cause behind this is, in most cases, still a mystery. Certain disturbances during the development of the organism, resulting in birth defects. Injuries after cell division that cannot be biologically repaired, such as a lost limb from an accident. Since birth defects and injuries are to indicate poor health of the organism, defects resulting in asymmetry put an animal at a disadvantage when it comes to finding a mate. In particular, a degree of facial symmetry is associated with physical attractiveness, but complete symmetry is both impossible and unattractive. Pre-modern architectural styles tended to place an emphasis on symmetry, except where extreme site conditions or historical developments lead away from this classical ideal. To the contrary and postmodern architects became much more free to use asymmetry as a design element. While most bridges employ a symmetrical form due to intrinsic simplicities of design and fabrication and economical use of materials, a number of modern bridges have deliberately departed from this, either in response to site-specific considerations or to create a dramatic design statement.
Some asymmetrical structures In fire-resistance rated wall assemblies, used in passive fire protection, but not limited to, high-voltage transformer fire barriers, asymmetry is a crucial aspect of design. When designing a facility, it is not always certain, that in the event of fire, which side a fire may come from. Therefore, many building codes and fire test standards outline, that a symmetrical assembly, need only be tested from one side, because both sides are the same. However, as soon as an assembly is asymmetrical, both sides must be tested and the test report is required to state the results for each side. In practical use, the lowest result achieved is the one. Neither the test sponsor, nor the laboratory can go by an opinion or deduction as to which side was in more peril as a result of contemplated testing and test only one side. Both must be tested in order to be compliant with test standards and building codes There are no a and b such that a < b and b < a. This form of asymmetry is an asymmetrical relation.
Certain molecules are chiral. Chemically identical molecules with different chirality are called enantiomers. Asymmetry arises in physics in a number of different realms; the original non-statistical formulation of thermodynamics was asymmetrical in time: it claimed that the entropy in a closed system can only increase with time. This was using the Clausius' Theorem; the theory of statistical mechanics, however, is symmetric in time. Although it states that a system below maximum entropy is likely to evolve towards higher entropy, it states that such a system is likely to have evolved from higher entropy. Symmetry is one of the most powerful tools in particle physics, because it has become evident that all laws of nature originate in symmetries. Violations of symmetry therefore present theoretical and experimental puzzles that lead to
Tally marks called hash marks, are a unary numeral system. They are a form of numeral used for counting, they are most useful in counting or tallying ongoing results, such as the score in a game or sport, as no intermediate results need to be erased or discarded. However, because of the length of large numbers, tallies are not used for static text. Notched sticks, known as tally sticks, were historically used for this purpose. Counting aids other than body parts appear in the Upper Paleolithic; the oldest tally sticks date to between 35,000 and 25,000 years ago, in the form of notched bones found in the context of the European Aurignacian to Gravettian and in Africa's Late Stone Age. The so-called Wolf bone is a prehistoric artifact discovered in 1937 in Czechoslovakia during excavations at Vestonice, led by Karl Absolon. Dated to the Aurignacian 30,000 years ago, the bone is marked with 55 marks which may be tally marks; the head of an ivory Venus figurine was excavated close to the bone. The Ishango bone, found in the Ishango region of the present-day Democratic Republic of Congo, is dated to over 20,000 years old.
Upon discovery, it was thought to portray a series of prime numbers. In the book How Mathematics Happened: The First 50,000 Years, Peter Rudman argues that the development of the concept of prime numbers could only have come about after the concept of division, which he dates to after 10,000 BC, with prime numbers not being understood until about 500 BC, he writes that "no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, some numbers that are multiples of 10." Alexander Marshack examined the Ishango bone microscopically, concluded that it may represent a six-month lunar calendar. Tally marks are clustered in groups of five for legibility; the cluster size 5 has the advantages of easy conversion into decimal for higher arithmetic operations and avoiding error, as humans can far more correctly identify a cluster of 5 than one of 10. Roman numerals, the Chinese numerals for one through three, rod numerals were derived from tally marks, as was the ogham script.
Base 1 arithmetic notation system is an unary positional system similar to tally marks. It is used as a practical base for counting due to its difficult readability, it is made by the concatenation of zero. The numbers 1, 2, 3, 4, 5... would be represented in this system as 0, 00, 000, 0000, 00000... Base 1 notation is used in type numbers of flour.
The Belgian Congo was a Belgian colony in Central Africa from 1908 until independence in 1960. The former colony adopted its present-day name, the Democratic Republic of the Congo, in 1964. Colonial rule in the Congo began in the late 19th century. King Leopold II of Belgium attempted to persuade the Belgian government to support colonial expansion around the then-largely unexplored Congo Basin, their ambivalence resulted in Leopold's establishing a colony himself. With support from a number of Western countries, Leopold achieved international recognition for a personal colony, the Congo Free State, in 1885. By the turn of the century, the violence used by Free State officials against indigenous Congolese and a ruthless system of economic exploitation led to intense diplomatic pressure on Belgium to take official control of the country, which it did by creating the Belgian Congo in 1908. Belgian rule in the Congo was based on the "colonial trinity" of state and private-company interests; the privileging of Belgian commercial interests meant that large amounts of capital flowed into the Congo and that individual regions became specialised.
On many occasions, the interests of the government and of private enterprise became linked, the state helped companies to break strikes and to remove other barriers raised by the indigenous population. The colony was divided into hierarchically organised administrative subdivisions, run uniformly according to a set "native policy"; this contrasted the practice of British and French colonial policy, which favoured systems of indirect rule, retaining traditional leaders in positions of authority under colonial oversight. During the 1940s and 1950s the Belgian Congo experienced extensive urbanisation, the colonial administration began various development programmes aimed at making the territory into a "model colony". One result saw the development of a new middle-class of Europeanised African "évolués" in the cities. By the 1950s the Congo had a wage labour force twice as large as that in any other African colony. In 1960, as the result of a widespread and radical pro-independence movement, the Congo achieved independence, becoming the Republic of Congo-Léopoldville under Prime Minister Patrice Lumumba and President Joseph Kasa-Vubu.
Poor relations between political factions within the Congo, the continued involvement of Belgium in Congolese affairs, the intervention by major parties during the Cold War led to a five-year-long period of war and political instability, known as the Congo Crisis, from 1960 to 1965. This ended with the seizure of power by Joseph-Désiré Mobutu in November 1965; until the part of the 19th century, few Europeans had ventured into the Congo basin. The rainforest and accompanying malaria and other tropical diseases, such as sleeping sickness, made it a difficult environment for European exploration and exploitation. In 1876, King Leopold II of Belgium organized the International African Association with the cooperation of the leading African explorers and the support of several European governments for the promotion of African exploration and colonization. After Henry Morton Stanley had explored the region in a journey that ended in 1878, Leopold courted the explorer and hired him to help his interests in the region.
Leopold II had been keen to acquire a colony for Belgium before he ascended to the throne in 1865. The Belgian civil government showed little interest in its monarch's dreams of empire-building. Ambitious and stubborn, Leopold decided to pursue the matter on his own account. European rivalry in Central Africa led to diplomatic tensions, in particular with regard to the unclaimed Congo River basin. In November 1884 Otto von Bismarck convened a 14-nation conference to find a peaceful resolution to the Congo crisis. Though the Berlin Conference did not formally approve the territorial claims of the European powers in Central Africa, it did agree on a set of rules to ensure a conflict-free partitioning of the region; the rules recognised the Congo basin as a free-trade zone. But Leopold II emerged triumphant from the Berlin Conference and his single-shareholder "philanthropic" organization received a large share of territory to be organized as the Congo Free State; the Congo Free State operated as a corporate state controlled by Leopold II through a non-governmental organization, the Association Internationale Africaine.
The state included the entire area of the present Democratic Republic of the Congo and existed from 1885 to 1908, when the government of Belgium reluctantly annexed the area. Under Leopold II's administration, the Congo Free State became a humanitarian disaster; the lack of accurate records makes it difficult to quantify the number of deaths caused by the ruthless exploitation and the lack of immunity to new diseases introduced by contact with European colonists – like the 1889–90 flu pandemic, which caused millions of deaths on the European continent, including Prince Baudouin of Belgium, who succumbed to the deadly virus in 1891. William Rubinstein wrote: "More it appears certain that the population figures given by Hochschild are inaccurate. There is, of course, no way of ascertaining the population of the Congo before the twentieth century, estimates like 20 million are purely guesses. Most of the interior of the Congo was unexplored if not inaccessible." Leopold's Force Publique, a private army that terrorized natives to work as forced labour for resource extraction, disrupted their societies and killed and abused natives indiscrimina
Uganda the Republic of Uganda, is a landlocked country in East-Central Africa. It is bordered to the east by Kenya, to the north by South Sudan, to the west by the Democratic Republic of the Congo, to the south-west by Rwanda, to the south by Tanzania; the southern part of the country includes a substantial portion of Lake Victoria, shared with Kenya and Tanzania. Uganda is in the African Great Lakes region. Uganda lies within the Nile basin, has a varied but a modified equatorial climate. Uganda takes its name from the Buganda kingdom, which encompasses a large portion of the south of the country, including the capital Kampala; the people of Uganda were hunter-gatherers until 1,700 to 2,300 years ago, when Bantu-speaking populations migrated to the southern parts of the country. Beginning in 1894, the area was ruled as a protectorate by the UK, who established administrative law across the territory. Uganda gained independence from the UK on 9 October 1962; the period since has been marked by intermittent conflicts, including a lengthy civil war against the Lord's Resistance Army in the Northern Region led by Joseph Kony, which has caused hundreds of thousands of casualties.
The official languages are English and Swahili, although "any other language may be used as a medium of instruction in schools or other educational institutions or for legislative, administrative or judicial purposes as may be prescribed by law." Luganda, a central language, is spoken across the country, several other languages are spoken including Runyoro, Rukiga and Lusoga. The president of Uganda is Yoweri Kaguta Museveni, who came to power in January 1986 after a protracted six-year guerrilla war, he has since eliminated the presidential term limits and the presidential age limit, becoming president for life. The residents of Uganda were hunter-gatherers until 1,700–2,300 years ago. Bantu-speaking populations, who were from central Africa, migrated to the southern parts of the country. According to oral tradition, the Empire of Kitara covered an important part of the great lakes area, from the northern lakes Albert and Kyoga to the southern lakes Victoria and Tanganyika. Bunyoro-Kitara is claimed as the antecedent of the Buganda, Toro and Busoga kingdoms.
Some Luo invaded the area of Bunyoro and assimilated with the Bantu there, establishing the Babiito dynasty of the current Omukama of Bunyoro-Kitara. Arab traders moved inland from the Indian Ocean coast of East Africa in the 1830s, they were followed in the 1860s by British explorers searching for the source of the Nile. British Anglican missionaries arrived in the kingdom of Buganda in 1877 and were followed by French Catholic missionaries in 1879; the British government chartered the Imperial British East Africa Company to negotiate trade agreements in the region beginning in 1888. From 1886, there were a series of religious wars in Buganda between Muslims and Christians and from 1890, between ba-Ingleza Protestants and ba-Fransa Catholics; because of civil unrest and financial burdens, IBEAC claimed that it was unable to "maintain their occupation" in the region. British commercial interests were ardent to protect the trade route of the Nile, which prompted the British government to annex Buganda and adjoining territories to create the Uganda Protectorate in 1894.
In the 1890s, 32,000 labourers from British India were recruited to East Africa under indentured labour contracts to construct the Uganda Railway. Most of the surviving Indians returned home, but 6,724 decided to remain in East Africa after the line's completion. Subsequently, some took control of cotton ginning and sartorial retail. From 1900 to 1920, a sleeping sickness epidemic in the southern part of Uganda, along the north shores of Lake Victoria, killed more than 250,000 people. Uganda gained independence from Britain on 9 October 1962 with Queen Elizabeth II as head of state and Queen of Uganda. In October 1963, Uganda became a republic but maintained its membership in the Commonwealth of Nations; the first post-independence election, held in 1962, was won by an alliance between the Uganda People's Congress and Kabaka Yekka. UPC and KY formed the first post-independence government with Milton Obote as executive prime minister, with the Buganda Kabaka Edward Muteesa II holding the ceremonial position of president.
Uganda's immediate post-independence years were dominated by the relationship between the central government and the largest regional kingdom – Buganda. From the moment the British created the Uganda protectorate, the issue of how to manage the largest monarchy within the framework of a unitary state had always been a problem. Colonial governors had failed to come up with a formula; this was further complicated by Buganda's nonchalant attitude to its relationship with the central government. Buganda never sought independence, but rather appeared to be comfortable with a loose arrangement that guaranteed them privileges above the other subjects within the protectorate or a special status when the British left; this was evidenced in part by hostilities between the British colonial authorities and Buganda prior to independence. Within Buganda there were divisions – between those who wanted the Kabaka to remain a dominant monarch, those who wanted to join with the rest of Uganda to create a modern secular state.
The split resulted in the creation of two dominant Buganda based parties – the Kabaka Yekka KY, the Democratic Party that had roots in the Catholic Church. The bitterness between these two parties was intense especiall