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.
The reciprocal function: y = 1/x. For every x except 0, y represents its multiplicative inverse. The graph forms a rectangular hyperbola.
In mathematics, a multiplicative inverse or reciprocal for a numberx, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fractiona/b is b/a. For the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one fifth (1/5 or 0.2), and the reciprocal of 0.25 is 1 divided by 0.25, or 4. The reciprocal function, the functionf(x) that maps x to 1/x, is one of the simplest examples of a function which is its own inverse (an involution). (Full article...)
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20 / 4 = 5, illustrated here with apples. This is said verbally, "Twenty divided by four equals five." Division is one of the four basic operations of arithmetic. The other operations are addition, subtraction, and multiplication. What is being divided is called the dividend, which is divided by the divisor, and the result is called the quotient. (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 Latin: fractus, "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: and ) consists of an integer numerator, displayed above a line (or before a slash like 1⁄2), and a non-zero integer denominator, displayed below (or after) that line. If these integers are positive, then the numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. For example, in the fraction 3/4, the numerator 3 indicates that the fraction represents 3 equal parts, and the denominator 4 indicates that 4 parts make up a whole. The picture to the right illustrates 3/4 of a cake. (Full article...)
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Natural numbers can be used for counting: one apple; two apples are one apple added to another apple, three apples are one apple added to two apples, ...
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, etc., possibly excluding 0. Some define the natural numbers as the non-negative integers0, 1, 2, 3, ..., while others define them as the positive integers1, 2, 3, ... . Some authors acknowledge both definitions whenever convenient. Some texts define the whole numbers as the natural numbers together with zero, excluding zero from the natural numbers, while in other writings, the whole numbers refer to all of the integers (including negative integers). The counting numbers refer to the natural numbers in common language, particularly in primary school education, and are similarly ambiguous although typically exclude zero. (Full article...)
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In arithmetic, quotition and partition are two ways of viewing fractions and division. In quotitive division one asks "how many parts are there?" while in partitive division one asks "what is the size of each part?" (Full article...)
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In mathematics, the greatest common divisor (GCD), also known as greatest common factor (GCF), 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 . For example, the GCD of 8 and 12 is 4, that is, gcd(8, 12) = 4. (Full article...)
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order. (Full article...)
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In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a property of arithmetic, e.g. "3 + 4 = 4 + 3" or "2 × 5 = 5 × 2", the property can also be used in more advanced settings. The name is needed because there are operations, such as division and subtraction, that do not have it (for example, "3 − 5 ≠ 5 − 3"); such operations are not commutative, and so are referred to as noncommutative operations. The idea that simple operations, such as the multiplication and addition of numbers, are commutative was for many years implicitly assumed. Thus, this property was not named until the 19th century, when mathematics started to become formalized. A similar property exists for binary relations; a binary relation is said to be symmetric if the relation applies regardless of the order of its operands; for example, equality is symmetric as two equal mathematical objects are equal regardless of their order. (Full article...)
In mathematics, the irrational numbers (in- + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable, meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself. (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 Latin: fractus, "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: and ) consists of an integer numerator, displayed above a line (or before a slash like 1⁄2), and a non-zero integer denominator, displayed below (or after) that line. If these integers are positive, then the numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. For example, in the fraction 3/4, the numerator 3 indicates that the fraction represents 3 equal parts, and the denominator 4 indicates that 4 parts make up a whole. The picture to the right illustrates 3/4 of a cake. (Full article...)
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The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The notion of weighted mean plays a role in descriptive statistics and also occurs in a more general form in several other areas of mathematics. (Full article...)
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Finding the median in sets of data with an odd and even number of values The median of a set of numbers 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 the center. Median income, for example, may be a better way to describe the center of the income distribution because increases in the largest incomes alone have no effect on the median. For this reason, the median is of central importance in robust statistics. (Full article...)
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In mathematics, the greatest common divisor (GCD), also known as greatest common factor (GCF), 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 . For example, the GCD of 8 and 12 is 4, that is, gcd(8, 12) = 4. (Full article...)
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A visual graph representing associative operations;
Notation for the (principal) square root of x. In mathematics, a square root of a number x is a number y such that ; in other words, a number y whose square (the result of multiplying the number by itself, or ) is x. For example, 4 and −4 are square roots of 16 because . (Full article...)
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order. (Full article...)
Plot of y = 3√x. The plot is symmetric with respect to origin, as it is an odd function. At x = 0 this graph has a vertical tangent. In mathematics, a cube root of a number x is a number y such that y3 = x. All nonzero real numbers have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. For example, the real cube root of 8, denoted , is 2, because 23 = 8, while the other cube roots of 8 are and . The three cube roots of −27i are: : (Full article...)
A mean is a numeric quantity representing the center of a collection of numbers and is intermediate to the extreme values of a set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statistics. Each attempts to summarize or typify a given group of data, illustrating the magnitude and sign of the data set. Which of these measures is most illuminating depends on what is being measured, and on context and purpose. (Full article...)
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This thermometer is indicating a negative Fahrenheit temperature (−4 °F).
In mathematics, a negative number represents an opposite. In the real number system, a negative number is a number that is less thanzero. Negative numbers are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset. If a quantity, such as the charge on an electron, may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. Negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common-sense idea of an opposite is reflected in arithmetic. For example, −(−3) = 3 because the opposite of an opposite is the original value. (Full article...)
The following are images from various arithmetic-related articles on Wikipedia.
Image 1The symbols for elementary-level math operations. From top-left going clockwise: addition, division, multiplication, and subtraction. (from Elementary arithmetic)
Image 2Abacuses are tools to perform arithmetic operations by moving beads. (from Arithmetic)
Image 3Irrational numbers are sometimes required to describe magnitudes in geometry. For example, the length of the hypotenuse of a right triangle is irrational if its legs have a length of 1. (from Arithmetic)
Image 4Multiplication table from 1 to 10 drawn to scale with the upper-right half labeled with prime factorisations (from Multiplication table)
Image 5Some historians interpret the Ishango bone as one of the earliest arithmetic artifacts. (from Arithmetic)
Image 6Example of modular arithmetic using a clock: after adding 4 hours to 9 o'clock, the hand starts at the beginning again and points at 1 o'clock. (from Arithmetic)
Image 8Example of long multiplication. The black numbers are the multiplier and the multiplicand. The green numbers are intermediary products gained by multiplying the multiplier with only one digit of the multiplicand. The blue number is the total product calculated by adding the intermediary products. (from Arithmetic)
Image 9If of a cake is to be added to of a cake, the pieces need to be converted into comparable quantities, such as cake-eighths or cake-quarters. (from Fraction)
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 main arithmetic operations are addition, subtraction, multiplication, and division. (from Arithmetic)
Image 13Using the number line method, calculating is performed by starting at the origin of the number line then moving five units to right for the first addend. The result is reached by moving another two units to the right for the second addend. (from Arithmetic)
Image 14Example of long multiplication. The black numbers are the multiplier and the multiplicand. The green numbers are intermediary products gained by multiplying the multiplier with only one digit of the multiplicand. The blue number is the total product calculated by adding the intermediary products. (from Arithmetic)
Image 15Irrational numbers are sometimes required to describe magnitudes in geometry. For example, the length of the hypotenuse of a right triangle is irrational if its legs have a length of 1. (from Arithmetic)
Image 16Calculations in mental arithmetic are done exclusively in the mind without relying on external aids. (from Arithmetic)
Image 17Some historians interpret the Ishango bone as one of the earliest arithmetic artifacts. (from Arithmetic)
Image 18The main arithmetic operations are addition, subtraction, multiplication, and division. (from Arithmetic)
Image 19Using the number line method, calculating is performed by starting at the origin of the number line then moving five units to right for the first addend. The result is reached by moving another two units to the right for the second addend. (from Arithmetic)
Image 20Different types of numbers on a number line. Integers are black, rational numbers are blue, and irrational numbers are green. (from Arithmetic)
Image 21Leibniz's stepped reckoner was the first calculator that could perform all four arithmetic operations. (from Arithmetic)
Image 22Example of addition with carry. The black numbers are the addends, the green number is the carry, and the blue number is the sum. (from Arithmetic)
Image 23Hieroglyphic numerals from 1 to 10,000 (from Arithmetic)
Image 24Different types of numbers on a number line. Integers are black, rational numbers are blue, and irrational numbers are green. (from Arithmetic)
Image 25Calculations in mental arithmetic are done exclusively in the mind without relying on external aids. (from Arithmetic)
Image 26Leibniz's stepped reckoner was the first calculator that could perform all four arithmetic operations. (from Arithmetic)
Image 27Example of addition with carry. The black numbers are the addends, the green number is the carry, and the blue number is the sum. In the rightmost digit, the addition of 9 and 7 is 16, carrying 1 into the next pair of the digit to the left, making its addition 1 + 5 + 2 = 8. Therefore, 59 + 27 = 86. (from Elementary arithmetic)
Image 28Abacuses are tools to perform arithmetic operations by moving beads. (from Arithmetic)
Image 29Cycles 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 30Example of addition with carry. The black numbers are the addends, the green number is the carry, and the blue number is the sum. (from Arithmetic)
Image 31Hieroglyphic numerals from 1 to 10,000 (from Arithmetic)
Image 32Example of modular arithmetic using a clock: after adding 4 hours to 9 o'clock, the hand starts at the beginning again and points at 1 o'clock. (from Arithmetic)
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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 (his error margin turned out to be less than 1%). He was also the first to calculate Earth's axial tilt, which has 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 used Egyptian and Persian records to estimate the dates of the main events of the 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 and composite 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 Number 2 because he always came in second in all his endeavours. (Full article...)
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Posthumous portrait of Thales by Wilhelm Meyer, based on a bust from the 4th century
The first philosophers followed him in explaining all of nature as based on the existence of a single ultimate substance. Thales theorized that this single substance was water. Thales thought the Earth floated on water.
In mathematics, Thales is the namesake of Thales's theorem, and the intercept theorem can also be known as Thales's theorem. Thales was said to have calculated the heights of the pyramids and the distance of ships from the shore. In science, Thales was an astronomer who reportedly predicted the weather and a solar eclipse. The discovery of the position of the constellation Ursa Major is also attributed to Thales, as well as the timings of the solstices and equinoxes. He was also an engineer, known for having diverted the Halys River. (Full article...)
Ibn al-Haytham was the first to correctly explain the theory of vision, and to argue that vision occurs in the brain, pointing to observations that it is subjective and affected by personal experience. He also stated the principle of least time for refraction which would later become Fermat's principle. He made major contributions to catoptrics and dioptrics by studying reflection, refraction and nature of images formed by light rays. Ibn al-Haytham was an early proponent of the concept that a hypothesis must be supported by experiments based on confirmable procedures or mathematical reasoning—an early pioneer in the scientific method five centuries before Renaissance scientists, 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...)
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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...)
In 628 CE, Brahmagupta first described gravity as an attractive force, and used the term "gurutvākarṣaṇam (गुरुत्वाकर्षणम्)" in Sanskrit to describe it. He is also credited with the first clear description of the quadratic formula (the solution of the quadratic equation) in his main work, the Brāhma-sphuṭa-siddhānta. (Full article...)
Pythagoras of Samos (Ancient Greek: Πυθαγόρας; c. 570 – c. 495 BC) was an ancient IonianGreek philosopher, polymath 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; 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.
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.
The teaching most securely identified with Pythagoras is the "transmigration of souls" or metempsychosis, 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 he may have escaped to Metapontum and died there.
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 also used throughout early modern European esotericism, and his teachings as portrayed in Ovid's Metamorphoses would later influence the modern vegetarian movement. (Full article...)
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17th-century German depiction of Heron
Hero of Alexandria (/ˈhɪəroʊ/; Greek: Ἥρωνὁ Ἀλεξανδρεύς, Hērōn hò Alexandreús, also known as Heron of Alexandria/ˈhɛrən/; probably 1st or 2nd century AD) was a Greek mathematician and engineer who was active in Alexandria in Egypt during the Roman era. He has been described as the greatest experimentalist of antiquity and a representative of the Hellenistic scientific tradition.
Hero published a well-recognized description of a steam-powered device called an aeolipile, also known as "Hero's engine". Among his most famous inventions was a windwheel, constituting the earliest instance of wind harnessing on land. In his work Mechanics, he described pantographs. Some of his ideas were derived from the works of Ctesibius.
In mathematics, he wrote a commentary on Euclid's Elements and a work on applied geometry known as the Metrica. 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 in manuscripts from the Byzantine Empire and, to a lesser extent, in Latin or Arabic translations. (Full article...)
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 ("The 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...)
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, later named "Cardano circles" or "cardanic circles", were used for the construction of the first high-speed printing presses.
His popularizing treatise on algebra, compiled between 813–33 as Al-Jabr (The Compendious Book on Calculation by Completion and Balancing), presented the first systematic solution of linear and quadratic equations. One of his achievements in algebra was his demonstration of how to solve quadratic equations by completing the square, for which he provided geometric justifications. Because al-Khwarizmi was the first person 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 English term algebra comes from the short-hand title of his aforementioned treatise (الجبرAl-Jabr, transl. "completion" or "rejoining"). His name gave rise to the English terms algorism and algorithm; the Spanish, Italian, and Portuguese terms algoritmo; and the Spanish term guarismo and Portuguese term algarismo, both meaning "digit".
Al-Khwarizmi revised Geography, the 2nd-century Greek-language treatise by the Roman polymath Claudius Ptolemy, listing the longitudes and latitudes of cities and localities. He further produced a set of astronomical tables and wrote about calendric works, as well as the astrolabe and the sundial. Al-Khwarizmi made important contributions to trigonometry, producing accurate sine and cosine tables and the first table of tangents. (Full article...)
Very little is known of Euclid's life, and most information comes from the scholars Proclus and Pappus of Alexandria many centuries later. Medieval Islamic mathematicians invented a fanciful biography, and medieval Byzantine and early Renaissance scholars mistook him for the earlier philosopher Euclid of Megara. It is now generally accepted that he spent his career in Alexandria and lived around 300 BC, after Plato's students and before Archimedes. There is some speculation that Euclid studied at the Platonic Academy and later taught at the Musaeum; he is regarded as bridging the earlier Platonic tradition in Athens with the later tradition of Alexandria.