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Geometric Progression
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1/2. Examples of a geometric sequence are powers ''r''''k'' of a fixed non-zero number ''r'', such as 2''k'' and 3''k''. The general form of a geometric sequence is :a,\ ar,\ ar^2,\ ar^3,\ ar^4,\ \ldots where ''r'' ≠ 0 is the common ratio and ''a'' ≠ 0 is a scale factor, equal to the sequence's start value. The sum of a geometric progression terms is called a ''geometric series''. Elementary properties The ''n''-th term of a geometric sequence with initial value ''a'' = ''a''1 and common ratio ''r'' is given by :a_n = a\,r^, and in general :a_n = a_m\,r^. Such a geometric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Sequences
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1/2. Examples of a geometric sequence are powers ''r''''k'' of a fixed non-zero number ''r'', such as 2''k'' and 3''k''. The general form of a geometric sequence is :a,\ ar,\ ar^2,\ ar^3,\ ar^4,\ \ldots where ''r'' ≠ 0 is the common ratio and ''a'' ≠ 0 is a scale factor, equal to the sequence's start value. The sum of a geometric progression terms is called a ''geometric series''. Elementary properties The ''n''-th term of a geometric sequence with initial value ''a'' = ''a''1 and common ratio ''r'' is given by :a_n = a\,r^, and in general :a_n = a_m\,r^. Such a geometric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of is , or . The logarithm of to ''base'' is denoted as , or without parentheses, , or even without the explicit base, , when no confusion is possible, or when the base does not matter such as in big O notation. The logarithm base is called the decimal or common logarithm and is commonly used in science and engineering. The natural logarithm has the number e (mathematical constant), as its base; its use is widespread in mathematics and physics, because of its very simple derivative. The binary logarithm uses base and is frequently used in computer science. Logarithms were introduced by John Napier in 1614 as a means of simplifying calculations. They were rapidly adopted by navigators, scientists, engineers, surveyors and oth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sequences And Series
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an ''arbitrary'' index set. For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be ''finite'', as in these examples, or ''infinit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Powers Of Two
A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer as the exponent. In a context where only integers are considered, is restricted to non-negative values, so there are 1, 2, and 2 multiplied by itself a certain number of times. The first ten powers of 2 for non-negative values of are: : 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ... Because two is the base of the binary numeral system, powers of two are common in computer science. Written in binary, a power of two always has the form 100...000 or 0.00...001, just like a power of 10 in the decimal system. Computer science Two to the exponent of , written as , is the number of ways the bits in a binary word of length can be arranged. A word, interpreted as an unsigned integer, can represent values from 0 () to () inclusively. Corresponding signed integer values can be positive, negative and zero; see signed num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclid's Elements
The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 300 BC. It is a collection of definitions, postulates, propositions ( theorems and constructions), and mathematical proofs of the propositions. The books cover plane and solid Euclidean geometry, elementary number theory, and incommensurable lines. ''Elements'' is the oldest extant large-scale deductive treatment of mathematics. It has proven instrumental in the development of logic and modern science, and its logical rigor was not surpassed until the 19th century. Euclid's ''Elements'' has been referred to as the most successful and influential textbook ever written. It was one of the very earliest mathematical works to be printed after the invention of the printing press and has been estimated to be second only to the Bible in the number of editions published since the first pri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclid
Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Elements'' treatise, which established the foundations of geometry that largely dominated the field until the early 19th century. His system, now referred to as Euclidean geometry, involved new innovations in combination with a synthesis of theories from earlier Greek mathematicians, including Eudoxus of Cnidus, Hippocrates of Chios, Thales and Theaetetus. With Archimedes and Apollonius of Perga, Euclid is generally considered among the greatest mathematicians of antiquity, and one of the most influential in the history of mathematics. Very little is known of Euclid's life, and most information comes from the philosophers Proclus and Pappus of Alexandria many centuries later. Until the early Renaissance he was often mistaken for the earlier philosopher Euclid of Megara, causing his biograph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Babylonian Mathematics
Babylonian mathematics (also known as ''Assyro-Babylonian mathematics'') are the mathematics developed or practiced by the people of Mesopotamia, from the days of the early Sumerians to the centuries following the fall of Babylon in 539 BC. Babylonian mathematical texts are plentiful and well edited. With respect to time they fall in two distinct groups: one from the Old Babylonian period (1830–1531 BC), the other mainly Seleucid from the last three or four centuries BC. With respect to content, there is scarcely any difference between the two groups of texts. Babylonian mathematics remained constant, in character and content, for nearly two millennia. In contrast to the scarcity of sources in Egyptian mathematics, knowledge of Babylonian mathematics is derived from some 400 clay tablets unearthed since the 1850s. Written in Cuneiform script, tablets were inscribed while the clay was moist, and baked hard in an oven or by the heat of the sun. The majority of recovered clay ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shuruppak
Shuruppak ( sux, , "the healing place"), modern Tell Fara, was an ancient Sumerian city situated about 55 kilometres (35 mi) south of Nippur on the banks of the Euphrates in Iraq's Al-Qādisiyyah Governorate. Shuruppak was dedicated to Ninlil, also called Sud, the goddess of grain and the air. Shuruppak and its environment Shuruppak is located in Al-Qādisiyyah Governorate, approximately 55 kilometres (35 mi) south of Nippur. The site of extends about a kilometer from north to south. The total area is about 120 hectares, with about 35 hectares of the mound being more than 3 meters above the surrounding plain, with a maximum of 9 meters. Archaeology After a brief survey by Hermann Volrath Hilprecht in 1900, it was first excavated in 1902 by Robert Koldewey and Friedrich Delitzsch of the German Oriental Society for eight months. Among other finds, hundreds of Early Dynastic tablets were collected, which ended up in the Berlin Museum and the Istanbul Muse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sumer
Sumer () is the earliest known civilization in the historical region of southern Mesopotamia (south-central Iraq), emerging during the Chalcolithic and early Bronze Ages between the sixth and fifth millennium BC. It is one of the cradles of civilization in the world, along with ancient Egypt, Elam, the Caral-Supe civilization, Mesoamerica, the Indus Valley civilisation, and ancient China. Living along the valleys of the Tigris and Euphrates rivers, Sumerian farmers grew an abundance of grain and other crops, the surplus from which enabled them to form urban settlements. Proto-writing dates back before 3000 BC. The earliest texts come from the cities of Uruk and Jemdet Nasr, and date to between c. 3500 and c. 3000 BC. Name The term "Sumer" (Sumerian: or , Akkadian: ) is the name given to the language spoken by the "Sumerians", the ancient non-Semitic-speaking inhabitants of southern Mesopotamia, by their successors the East Semitic-speaking Akkadians. The Sumerians ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Early Dynastic Period (Mesopotamia)
The Early Dynastic period (abbreviated ED period or ED) is an archaeological culture in Mesopotamia (modern-day Iraq) that is generally dated to c. 2900–2350 BC and was preceded by the Uruk and Jemdet Nasr periods. It saw the development of writing and the formation of the first cities and states. The ED itself was characterized by the existence of multiple city-states: small states with a relatively simple structure that developed and solidified over time. This development ultimately led to the unification of much of Mesopotamia under the rule of Sargon, the first monarch of the Akkadian Empire. Despite this political fragmentation, the ED city-states shared a relatively homogeneous material culture. Sumerian cities such as Uruk, Ur, Lagash, Umma, and Nippur located in Lower Mesopotamia were very powerful and influential. To the north and west stretched states centered on cities such as Kish, Mari, Nagar, and Ebla. The study of Central and Lower Mesopotamia has l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Mean
In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the '' average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results of an experiment or an observational study, or frequently a set of results from a survey. The term "arithmetic mean" is preferred in some contexts in mathematics and statistics, because it helps distinguish it from other means, such as the geometric mean and the harmonic mean. In addition to mathematics and statistics, the arithmetic mean is used frequently in many diverse fields such as economics, anthropology and history, and it is used in almost every academic field to some extent. For example, per capita income is the arithmetic average income of a nation's population. While the arithmetic mean is often used to report central tendencies, it is not a robust statistic, meaning that it is greatly in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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