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.
In
mathematics, the irrational numbers (from in-
prefix assimilated to ir- (negative prefix,
privative) + rational) are all the
real numbers which 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...)
Image 2
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 (i.e, {{{1}}}) at which the
probability mass function takes its maximum value. In other words, it is the value that is most likely to be sampled. (Full article...)
Image 3
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...)
Image 4
An operation $\circ$ is commutative if and only if$x\circ y=y\circ x$ for each $x$ and $y$. This image illustrates this property with the concept of an operation as a "calculation machine". It doesn't matter for the output $x\circ y$ or $y\circ x$ respectively which order the arguments $x$ and $y$ have – the final outcome is the same.
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. Most familiar as the name of the property that says something like "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 corresponding 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...)
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...)
Image 7
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, which is the result of taking the lcm to be the
least upper bound in the
lattice of divisibility. (Full article...)
Image 8
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. Every
nonnegativereal numberx has a unique nonnegative square root, called the principal square root, which is denoted by ${\sqrt {x}},$ where the symbol ${\sqrt {~^{~}}}$ is called the radical sign or radix. For example, the principal square root of 9 is 3, which is denoted by ${\sqrt {9}}=3,$ because 3^{2} = 3 ⋅ 3 = 9 and 3 is nonnegative. The term (or number) whose square root is being considered is known as the radicand. The radicand is the number or expression underneath the radical sign, in this case 9. (Full article...)
Image 9
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, which is the result of taking the lcm to be the
least upper bound in the
lattice of divisibility. (Full article...)
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...)
Image 12
20 / 4 = 5, illustrated here with apples. This is said verbally, "Twenty divided by four equals five."
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...)
Image 15
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. In essence, this method is used to find the value of a unit from the value of a multiple, and hence the value of a multiple. (Full article...)
Image 16
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 UK. 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...)
Image 17
Composite numbers can be arranged into
rectangles but prime numbers cannot
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...)
Image 18
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 number x, 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 function f(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, 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...)
Image 23
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...)
In
mathematics, a ratio indicates 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...)
Image 25
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...)
General images
The following are images from various arithmetic-related articles on Wikipedia.
Image 1A 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 4If ${\tfrac {1}{2}}$ 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 5Cycles 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 6Multiplication table from 1 to 10 drawn to scale with the upper-right half labeled with prime factorisations (from Multiplication table)
Ibn al-Haytham was the first to explain that vision occurs when light reflects from an object and then passes to one's eyes. He was also the first to demonstrate that vision occurs in the brain, rather than in the eyes. Building upon a naturalistic, empirical method pioneered by
Aristotle in ancient Greece, 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.
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 ("of Egypt"). 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 (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,
Western philosophy. 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 for 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...)
Image 3
17th-century German depiction of Hero
Hero of Alexandria (/ˈhɪəroʊ/;
Greek: Ἥρων ὁ Ἀλεξανδρεύς, Heron ho Alexandreus, also known as Heron of Alexandria/ˈhɛrən/; c. 10 AD – c. 70 AD), was a Greek
mathematician and
engineer who was active in his native city of
Alexandria,
Roman Egypt. He is often considered the greatest experimenter of antiquity and his work is representative of the
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...)
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. (Full article...)
Thales is recognized for breaking from the use of
mythology to explain the world and the universe, and instead explaining natural objects and phenomena by
naturalistictheories and
hypotheses, in a precursor to modern
science. 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 and reported Thales' hypothesis that the
originating principle of
nature and the nature of
matter was a single material
substance:
water.
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...)
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 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...)
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.
In Iran, Abu Rayhan Biruni's birthday is celebrated as the day of the surveying engineer. (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”.
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.
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. According to an entry in the
Suda (a 10th-century encyclopedia), his critics scorned him, calling him Beta (the second letter of the Greek alphabet) because he always came in second in all his endeavours. Nonetheless, 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. Eratosthenes yearned to understand the complexities of the entire world. (Full article...)
In an article for
Science, a team of researchers based at
Bournemouth University conclude that a series of human footprints preserved at
White Sands National Park in the
U.S. state of
New Mexico date back between 21,000 and 23,000 years ago during the
Last Glacial Maximum, suggesting that humans lived in the
Americas 5,000 years earlier than previously thought. The team also determined that most of the footprints came from children and teenagers.
(NPR)