In statistics, the mode is the value that appears most often in a set of data values. If X is a discrete random variable, the mode is the value x at which the probability mass function takes its maximum value (i.e., x=argmaxxi P(X = xi)). In other words, it is the value that is most likely to be sampled. (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 ; 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...)
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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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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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In
zero
, this is equivalent to saying that is an integer. (Full article...)
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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
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In
fractions. It simplifies adding, subtracting, and comparing fractions. (Full article...
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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...
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 by or , is equal to the
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fraction or ratio (in the case of a general division
). For example, when dividing 20 (the dividend) by 3 (the divisor), the quotient is 6 (with a remainder of 2) in the first sense and (a repeating decimal) in the second sense. (Full article...)
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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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
Full article...
)
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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
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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The ratio of width to height of standard-definition television In mathematics, a ratio (/ˈreɪʃ(i)oʊ/) shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ratio 4:3). Similarly, the ratio of lemons to oranges is 6:8 (or 3:4) and the ratio of oranges to the total amount of fruit is 8:14 (or 4:7). (Full article...)
The following are images from various arithmetic-related articles on Wikipedia.
Image 1Calculations in
mental arithmetic are done exclusively in the mind without relying on external aids. (from Arithmetic
)
Image 2Hieroglyphic numerals from 1 to 10,000 (from Arithmetic)
Image 3The
Warring States era decimal multiplication table of 305 BC (from Multiplication table
)
Image 4Example 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 5If 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 6A 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 7Leibniz's stepped reckoner was the first calculator that could perform all four arithmetic operations. (from Arithmetic)
Image 8Irrational 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 10Different types of numbers on a number line. Integers are black, rational numbers are blue, and irrational numbers are green. (from Arithmetic)
Image 11Example 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 12Using 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 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 14Some historians interpret the Ishango bone as one of the earliest arithmetic artifacts. (from Arithmetic)
Image 15Calculations in
mental arithmetic are done exclusively in the mind without relying on external aids. (from Arithmetic
)
Image 16Multiplication table from 1 to 10 drawn to scale with the upper-right half labeled with prime factorisations (from Multiplication table)
Image 17The main arithmetic operations are addition, subtraction, multiplication, and division. (from Arithmetic)
Image 18Irrational 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 19Example 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 20Different types of numbers on a number line. Integers are black, rational numbers are blue, and irrational numbers are green. (from Arithmetic)
Image 21Abacuses are tools to perform arithmetic operations by moving beads. (from Arithmetic)
Image 22The symbols for elementary-level math operations. From top-left going clockwise: addition, division, multiplication, and subtraction. (from Elementary arithmetic)
Image 23Leibniz's stepped reckoner was the first calculator that could perform all four arithmetic operations. (from Arithmetic)
Image 24Hieroglyphic numerals from 1 to 10,000 (from Arithmetic)
Image 25Example 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 26Cycles 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 27Example 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 28Example 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 29Abacuses are tools to perform arithmetic operations by moving beads. (from Arithmetic)
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. 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, the addition of 59 + 27 gives the result 86. (from Elementary arithmetic)
Image 31Some historians interpret the Ishango bone as one of the earliest arithmetic artifacts. (from Arithmetic)
Image 32The main arithmetic operations are addition, subtraction, multiplication, and division. (from Arithmetic)
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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...
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Khaṇḍakhādyaka ("edible bite", dated 665), a more practical text.
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...
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Hellenistic scientific tradition.
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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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...
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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...
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double-entry system of book-keeping on the continent. He was also called Luca di Borgo after his birthplace, Borgo Sansepolcro, Tuscany.
Several of his works were plagiarised from Piero della Francesca, in what has been called "probably the first full-blown case of plagiarism in the history of mathematics". (Full article...
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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...
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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