The Egyptians and Babylonians used all the elementary arithmetic operations as early as 2000 BC. Later Roman numerals, descended from tally marks used for counting. The continuous development of modern arithmetic starts with ancient Greece, although it originated much later than the Babylonian and Egyptian examples. Euclid is often credited as the first mathematician to separate study of arithmetic from philosophical and mystical beliefs. Greek numerals were used by Archimedes, Diophantus and others in a positional notation not very different from ours. The ancient Chinese had advanced arithmetic studies dating from the Shang Dynasty and continuing through the Tang Dynasty, from basic numbers to advanced algebra. The ancient Chinese used a positional notation similar to that of the Greeks. The gradual development of the Hindu–Arabic numeral system independently devised the place-value concept and positional notation, which combined the simpler methods for computations with a decimal base and the use of a digit representing zero (0). This allowed the system to consistently represent both large and small integers. This approach eventually replaced all other systems. In the Middle Ages, arithmetic was one of the seven liberal arts taught in universities. The flourishing of algebra in the medievalIslamic world and in RenaissanceEurope was an outgrowth of the enormous simplification of computation through decimal notation.
Each of the six rows is a different permutation of three distinct balls
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set. (Full article...)
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Scientific notation is a way of expressing numbers that are too large or too small (usually would result in a long string of digits) to be conveniently written in decimal form. It may be referred to as scientific form or standard index form, or standard form in the United Kingdom. This base ten notation is commonly used by scientists, mathematicians, and engineers, in part because it can simplify certain arithmetic operations. On scientific calculators it is usually known as "SCI" display mode. (Full article...)
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In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers x, y, the greatest common divisor of x and y is denoted $\gcd(x,y)$. For example, the GCD of 8 and 12 is 4, that is, $\gcd(8,12)=4$. (Full article...)
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Finding the median in sets of data with an odd and even number of values
In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic feature of the median in describing data compared to the mean (often simply described as the "average") is that it is not skewed by a small proportion of extremely large or small values, and therefore provides a better representation of a "typical" value. Median income, for example, may be a better way to suggest what a "typical" income is, because income distribution can be very skewed. The median is of central importance in robust statistics, as it is the most resistant statistic, having a breakdown point of 50%: so long as no more than half the data are contaminated, the median is not an arbitrarily large or small result. (Full article...)
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In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple and an orange; or a pear and an orange. More formally, a k-combination of a setS is a subset of k distinct elements of S. So, two combinations are identical if and only if each combination has the same members. (The arrangement of the members in each set does not matter.) If the set has n elements, the number of k-combinations, denoted as $C_{k}^{n}$, is equal to the binomial coefficient (Full article...)
In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or it may be considered both positive and negative (having both signs).^{[citation needed]} Whenever not specifically mentioned, this article adheres to the first convention. (Full article...)
In mathematics, the natural numbers are those numbers used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country"). (Full article...)
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A cake with one quarter (one fourth) removed. The remaining three fourths are shown by dotted lines and labeled by the fraction 1/4.
A fraction (from Latinfractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A common, vulgar, or simple fraction (examples: ${\tfrac {1}{2}}$ and ${\tfrac {17}{3}}$) consists of a numerator, displayed above a line (or before a slash like 1⁄2), and a non-zero denominator, displayed below (or after) that line. Numerators and denominators are also used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals. (Full article...)
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The rational numbers ($\mathbb {Q}$) are included in the real numbers ($\mathbb {R}$), while themselves including the integers ($\mathbb {Z}$), which in turn include the natural numbers ($\mathbb {N}$)
In mathematics, a rational number is a number that can be expressed as the quotient or fractionp/q of two integers, a numeratorp and a non-zero denominatorq. For example, −3/7 is a rational number, as is every integer (e.g. {{{1}}}). The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by boldface Q, or blackboard bold$\mathbb {Q} .$ (Full article...)
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A Venn diagram showing the least common multiples of combinations of 2, 3, 4, 5 and 7 (6 is skipped as it is 2 × 3, both of which are already represented). For example, a card game which requires its cards to be divided equally among up to 5 players requires at least 60 cards, the number at the intersection of the 2, 3, 4, and 5 sets, but not the 7 set.
In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integersa and b, usually denoted by lcm(a, b), is the smallest positive integer that is divisible by both a and b. Since division of integers by zero is undefined, this definition has meaning only if a and b are both different from zero. However, some authors define lcm(a,0) as 0 for all a, since 0 is the only common multiple of a and 0. (Full article...)
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The unitary method is a technique for solving a problem by first finding the value of a single unit, and then finding the necessary value by multiplying the single unit value. (Full article...)
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A cake with one quarter (one fourth) removed. The remaining three fourths are shown by dotted lines and labeled by the fraction 1/4.
A fraction (from Latinfractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A common, vulgar, or simple fraction (examples: ${\tfrac {1}{2}}$ and ${\tfrac {17}{3}}$) consists of a numerator, displayed above a line (or before a slash like 1⁄2), and a non-zero denominator, displayed below (or after) that line. Numerators and denominators are also used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals. (Full article...)
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Notation for the (principal) square root of x.
In mathematics, a square root of a number x is a number y such that y^{2} = x; in other words, a number y whose square (the result of multiplying the number by itself, or y ⋅ y) is x. For example, 4 and −4 are square roots of 16, because 4^{2} = (−4)^{2} = 16. (Full article...)
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Visualization of distributive law for positive numbers
Cuisenaire rods: 5 (yellow) cannot be evenly divided in 2 (red) by any 2 rods of the same color/length, while 6 (dark green) can be evenly divided in 2 by 3 (lime green).
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it isn't. For example, −4, 0, 82 are even because
In arithmetic, a quotient (from Latin: quotiens 'how many times', pronounced /ˈkwoʊʃənt/) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a division (in the case of Euclidean division), or as a fraction or a ratio (in the case of proper division). For example, when dividing 20 (the dividend) by 3 (the divisor), the quotient is "6 with a remainder of 2" in the Euclidean division sense, and $6{\tfrac {2}{3}}$ in the proper division sense. In the second sense, a quotient is simply the ratio of a dividend to its divisor. (Full article...)
Image 4Cycles of the unit digit of multiples of integers ending in 1, 3, 7 and 9 (upper row), and 2, 4, 6 and 8 (lower row) on a telephone keypad (from Multiplication table)
Image 6If ${\tfrac {1}{2}}$ of a cake is to be added to ${\tfrac {1}{4}}$ of a cake, the pieces need to be converted into comparable quantities, such as cake-eighths or cake-quarters. (from Fraction)
Image 7Leibniz's Stepped Reckoner was the first calculator that could perform all four arithmetic operations. (from Arithmetic)
Image 9Arithmetic tables for children, Lausanne, 1835 (from Arithmetic)
Image 10A cake with one quarter (one fourth) removed. The remaining three fourths are shown by dotted lines and labeled by the fraction 1/4. (from Fraction)
Image 11The Ishango bone, found near Lake Edward, possibly displaying a numbering system from more than 20,000 years ago. (from History of arithmetic)
Image 12Arithmetics. Pinturikkyo's list. Borgia's apartments. 1492 — 1495. Rome, Vatican palaces (from History of arithmetic)
He is best known for being the first person known to calculate the circumference of the Earth, which he did by using the extensive survey results he could access in his role at the Library; his calculation was remarkably accurate. He was also the first to calculate Earth's axial tilt, which also proved to have remarkable accuracy. He created the first global projection of the world, incorporating parallels and meridians based on the available geographic knowledge of his era.
Eratosthenes was the founder of scientific chronology; he endeavoured to revise the dates of the main events of the semi-mythological Trojan War, dating the Sack of Troy to 1183 BC. In number theory, he introduced the sieve of Eratosthenes, an efficient method of identifying prime numbers.
He was a figure of influence in many fields who yearned to understand the complexities of the entire world. His devotees nicknamed him Pentathlos after the Olympians who were well rounded competitors, for he had proven himself to be knowledgeable in every area of learning. Yet, according to an entry in the Suda (a 10th-century encyclopedia), some critics scorned him, calling him Beta (the second letter of the Greek alphabet) because he always came in second in all his endeavours. (Full article...)
Since the discovery and publication of his work by European scholars in the nineteenth century, Pāṇini has been considered the "first descriptive linguist", and even labelled as “the father of linguistics”.
Thales is recognized for breaking from the use of mythology to explain the world and the universe, instead explaining natural objects and phenomena by offering naturalistictheories and hypotheses. Almost all the other pre-Socratic philosophers followed him in explaining nature as deriving from a unity of everything based on the existence of a single ultimate substance instead of using mythological explanations. Aristotle regarded him as the founder of the Ionian School of philosophy, and reported Thales' hypothesis that the originating principle of nature and the nature of matter was a single material substance: water.
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 biography to be substantially revised. It is generally agreed that he spent his career under Ptolemy I in Alexandria and lived around 300 BCE, after Plato and before Archimedes. There is some speculation that Euclid was a student of the Platonic Academy. Euclid is often regarded as bridging between the earlier Platonic tradition in Athens with the later tradition of Alexandria.
In the Elements, Euclid deduced the theorems from a small set of axioms. He also wrote works on perspective, conic sections, spherical geometry, number theory, and mathematical rigour. In addition to the Elements, Euclid wrote a central early text in the optics field, Optics, and lesser-known works including Data and Phaenomena. Euclid's authorship of two other texts—On Divisions of Figures, Catoptrics—has been questioned. He is thought to have written many now lost works. (Full article...)
Ibn al-Haytham was an early proponent of the concept that a hypothesis must be supported by experiments based on confirmable procedures or mathematical evidence—an early pioneer in the scientific method five centuries before Renaissance scientists. On account of this, he is sometimes described as the world's "first true scientist". He was also a polymath, writing on philosophy, theology and medicine.
Born in Basra, he spent most of his productive period in the Fatimid capital of Cairo and earned his living authoring various treatises and tutoring members of the nobilities. Ibn al-Haytham is sometimes given the bynameal-Baṣrī after his birthplace, or al-Miṣrī ("the Egyptian"). Al-Haytham was dubbed the "Second Ptolemy" by Abu'l-Hasan Bayhaqi and "The Physicist" by John Peckham. Ibn al-Haytham paved the way for the modern science of physical optics. (Full article...)
Pythagoras of Samos (Ancient Greek: Πυθαγόρας ὁ Σάμιος, romanized: Pythagóras ho Sámios, lit. 'Pythagoras the Samian', or simply Πυθαγόρας; Πυθαγόρης in Ionian Greek; c. 570 – c. 495 BC) was an ancient IonianGreek philosopher and the eponymous founder of Pythagoreanism. His political and religious teachings were well known in Magna Graecia and influenced the philosophies of Plato, Aristotle, and, through them, the West in general. Knowledge of his life is clouded by legend, but he appears to have been the son of Mnesarchus, a gem-engraver on the island of Samos. Modern scholars disagree regarding Pythagoras's education and influences, but they do agree that, around 530 BC, he travelled to Croton in southern Italy, where he founded a school in which initiates were sworn to secrecy and lived a communal, ascetic lifestyle. This lifestyle entailed a number of dietary prohibitions, traditionally said to have included vegetarianism, although modern scholars doubt that he ever advocated complete vegetarianism.
The teaching most securely identified with Pythagoras is metempsychosis, or the "transmigration of souls", which holds that every soul is immortal and, upon death, enters into a new body. He may have also devised the doctrine of musica universalis, which holds that the planets move according to mathematicalequations and thus resonate to produce an inaudible symphony of music. Scholars debate whether Pythagoras developed the numerological and musical teachings attributed to him, or if those teachings were developed by his later followers, particularly Philolaus of Croton. Following Croton's decisive victory over Sybaris in around 510 BC, Pythagoras's followers came into conflict with supporters of democracy and Pythagorean meeting houses were burned. Pythagoras may have been killed during this persecution, or escaped to Metapontum, where he eventually died.
In antiquity, Pythagoras was credited with many mathematical and scientific discoveries, including the Pythagorean theorem, Pythagorean tuning, the five regular solids, the Theory of Proportions, the sphericity of the Earth, and the identity of the morning and evening stars as the planet Venus. It was said that he was the first man to call himself a philosopher ("lover of wisdom") and that he was the first to divide the globe into five climatic zones. Classical historians debate whether Pythagoras made these discoveries, and many of the accomplishments credited to him likely originated earlier or were made by his colleagues or successors. Some accounts mention that the philosophy associated with Pythagoras was related to mathematics and that numbers were important, but it is debated to what extent, if at all, he actually contributed to mathematics or natural philosophy.
Pythagoras influenced Plato, whose dialogues, especially his Timaeus, exhibit Pythagorean teachings. Pythagorean ideas on mathematical perfection also impacted ancient Greek art. His teachings underwent a major revival in the first century BC among Middle Platonists, coinciding with the rise of Neopythagoreanism. Pythagoras continued to be regarded as a great philosopher throughout the Middle Ages and his philosophy had a major impact on scientists such as Nicolaus Copernicus, Johannes Kepler, and Isaac Newton. Pythagorean symbolism was used throughout early modern European esotericism, and his teachings as portrayed in Ovid's Metamorphoses influenced the modern vegetarian movement. (Full article...)
Hero published a well-recognized description of a steam-powered device called an aeolipile (sometimes called a "Hero engine"). Among his most famous inventions was a windwheel, constituting the earliest instance of wind harnessing on land. He is said to have been a follower of the atomists. In his work Mechanics, he described pantographs. Some of his ideas were derived from the works of Ctesibius.
In mathematics he is mostly remembered for Heron's formula, a way to calculate the area of a triangle using only the lengths of its sides.
Much of Hero's original writings and designs have been lost, but some of his works were preserved including in manuscripts from the Eastern Roman Empire and to a lesser extent, in Latin or Arabic translations. (Full article...)
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An imaginary rendition of Al Biruni on a 1973 Soviet postage stamp
Al-Biruni was well versed in physics, mathematics, astronomy, and natural sciences, and also distinguished himself as a historian, chronologist, and linguist. He studied almost all the sciences of his day and was rewarded abundantly for his tireless research in many fields of knowledge. Royalty and other powerful elements in society funded Al-Biruni's research and sought him out with specific projects in mind. Influential in his own right, Al-Biruni was himself influenced by the scholars of other nations, such as the Greeks, from whom he took inspiration when he turned to the study of philosophy. A gifted linguist, he was conversant in Khwarezmian, Persian, Arabic, Sanskrit, and also knew Greek, Hebrew, and Syriac. He spent much of his life in Ghazni, then capital of the Ghaznavids, in modern-day central-eastern Afghanistan. In 1017 he travelled to the Indian subcontinent and wrote a treatise on Indian culture entitled Tārīkh al-Hind (History of India), after exploring the Hindu faith practiced in India. He was, for his time, an admirably impartial writer on the customs and creeds of various nations, his scholarly objectivity earning him the title al-Ustadh ("The Master") in recognition of his remarkable description of early 11th-century India. (Full article...)
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A stamp of Zhang Heng issued by China Post in 1955
Zhang Heng began his career as a minor civil servant in Nanyang. Eventually, he became Chief Astronomer, Prefect of the Majors for Official Carriages, and then Palace Attendant at the imperial court. His uncompromising stance on historical and calendrical issues led to his becoming a controversial figure, preventing him from rising to the status of Grand Historian. His political rivalry with the palace eunuchs during the reign of Emperor Shun (r. 125–144) led to his decision to retire from the central court to serve as an administrator of Hejian Kingdom in present-day Hebei. Zhang returned home to Nanyang for a short time, before being recalled to serve in the capital once more in 138. He died there a year later, in 139.
Zhang applied his extensive knowledge of mechanics and gears in several of his inventions. He invented the world's first water-poweredarmillary sphere to assist astronomical observation; improved the inflow water clock by adding another tank; and invented the world's first seismoscope, which discerned the cardinal direction of an earthquake 500 km (310 mi) away. He improved previous Chinese calculations for pi. In addition to documenting about 2,500 stars in his extensive star catalog, Zhang also posited theories about the Moon and its relationship to the Sun: specifically, he discussed the Moon's sphericity, its illumination by reflected sunlight on one side and the hidden nature of the other, and the nature of solar and lunareclipses. His fu (rhapsody) and shi poetry were renowned in his time and studied and analyzed by later Chinese writers. Zhang received many posthumous honors for his scholarship and ingenuity; some modern scholars have compared his work in astronomy to that of the Greco-Roman Ptolemy (AD 86–161). (Full article...)
Cardano partially invented and described several mechanical devices including the combination lock, the gimbal consisting of three concentric rings allowing a supported compass or gyroscope to rotate freely, and the Cardan shaft with universal joints, which allows the transmission of rotary motion at various angles and is used in vehicles to this day. He made significant contributions to hypocycloids, published in De proportionibus, in 1570. The generating circles of these hypocycloids were later named Cardano circles or cardanic circles and were used for the construction of the first high-speed printing presses.
Brahmagupta was the first to give rules to compute with zero. The texts composed by Brahmagupta were in elliptic verse in Sanskrit, as was common practice in Indian mathematics. As no proofs are given, it is not known how Brahmagupta's results were derived.
In 628 CE, Brahmagupta first described gravity as an attractive force, and used the term "gurutvākarṣaṇam (गुरुत्वाकर्षणम्)]" in Sanskrit to describe it. (Full article...)
Al-Khwarizmi's popularizing treatise on algebra (The Compendious Book on Calculation by Completion and Balancing, c. 813–833 CE) presented the first systematic solution of linear and quadratic equations. One of his principal achievements in algebra was his demonstration of how to solve quadratic equations by completing the square, for which he provided geometric justifications. Because he was the first to treat algebra as an independent discipline and introduced the methods of "reduction" and "balancing" (the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation), he has been described as the father or founder of algebra. The term algebra itself comes from the title of his book (the word al-jabr meaning "completion" or "rejoining"). His name gave rise to the terms algorism and algorithm, as well as Spanish, Italian and Portuguese terms algoritmo, and Spanish guarismo and Portuguesealgarismo meaning "digit".
In addition to his best-known works, he revised Ptolemy's Geography, listing the longitudes and latitudes of various cities and localities. He further produced a set of astronomical tables and wrote about calendaric works, as well as the astrolabe and the sundial. He also made important contributions to trigonometry, producing accurate sine and cosine tables, and the first table of tangents. (Full article...)