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
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...)
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 3
Visualization of distributive law for positive numbers
In
mathematics, the irrational numbers (from in-
prefix assimilated to ir- (negative prefix,
privative) + 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...)
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 is not. For example, −4, 0, 82 are even because
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 10
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: ${\tfrac {1}{2}}$ 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 11
The
double-struck symbol, often used to denote the set of all integers (see
ℤ)
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...)
These complex numbers, two of eight values of
^{8}√1, are mutually opposite
In mathematics, the additive inverse of a
numbera is the number that, when
added to a, yields
zero. This number is also known as the opposite (number), sign change, and negation. For a
real number, it reverses its
sign: the additive inverse (opposite number) of a
positive number is negative, and the additive inverse of a
negative number is positive.
Zero is the additive inverse of itself. (Full article...)
Image 16
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: ${\tfrac {1}{2}}$ 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 17
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...)
for $\operatorname {Re} (s)>1$ and its analytic continuation elsewhere. (Full article...)
Image 20
Demonstration, with
Cuisenaire rods, of the divisors of the composite number 10
A composite number is a
positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one
divisor other than 1 and itself. Every positive integer is composite,
prime, or the
unit 1, so the composite numbers are exactly the numbers that are not prime and not a unit. (Full article...)
Image 21
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 22
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
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 $C(n,k)$ or $C_{k}^{n}$, is equal to the
binomial coefficient (Full article...)
Image 2Multiplication table from 1 to 10 drawn to scale with the upper-right half labeled with prime factorisations (from Multiplication table)
Image 3A scale calibrated in imperial units, with an associated cost display (from Arithmetic)
Image 4The symbols for each basic elementary operations. Starting from the top-left going clockwise, is addition, division, multiplication, and subtraction. (from Elementary arithmetic)
Image 5If ${\tfrac {1}{2}}$ 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 9Arithmetics. Pinturikkyo's list. Borgia's apartments. 1492 — 1495. Rome, Vatican palaces (from History of arithmetic)
Image 10The
Ishango bone, found near Lake Edward, possibly displaying a numbering system from more than 20,000 years ago. (from History of arithmetic)
Image 11Cycles 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 12A 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)
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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.
Today, he is well known for his achievements in
algebra. In his 1545 book Ars Magna, he made the first systematic use of
negative numbers in Europe, published with attribution the solutions of other mathematicians for the
cubic and
quartic equations, and acknowledged the existence of
imaginary numbers. [suggested edit: French Nicholas Chiquet in his text, Triparty en la science des nombres, discussed negative numbers, and thus this credit to Cardano might not be applicable. Source: A History of Mathematics 3rd edition by Merzbach and Boyer pages 249 and 250.] (Full article...)
Image 2
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...)
Image 3
17th-century German depiction of Heron
Hero of Alexandria (/ˈhɪəroʊ/;
Greek: Ἥρων ὁ Ἀλεξανδρεύς, Hērōn hò Alexandreús, also known as Heron of Alexandria/ˈhɛrən/;
fl. 60 AD) was a
Greek mathematician and
engineer who was active in his native city of
Alexandria in Egypt during the Roman era. 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...)
Image 4
Posthumous portrait of Thales by Wilhelm Meyer, based on a bust from the 4th century
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.
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 he may have escaped to
Metapontum and died there.
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...)
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 BC, after
Plato and before Archimedes. There is some speculation that Euclid was a student of the
Platonic Academy and later taught at the
Musaeum. Euclid is often regarded as bridging 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...)
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”.
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...)
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, the Spanish, Italian, and Portuguese terms algoritmo, and the Spanish guarismo and
Portuguesealgarismo, both meaning "
digit".^{[
citation needed]} In the 12th century,
Latin translations of
his textbook on arithmetic (Algorithmo de Numero Indorum) which codified the various
Indian numerals, introduced the
decimalpositional number system to the Western world. The Compendious Book on Calculation by Completion and Balancing, translated into Latin by
Robert of Chester in 1145, was used until the sixteenth century as the principal mathematical text-book of
European universities.
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.^{[
citation needed]} (Full article...)
Image 10
Ḥasan Ibn al-Haytham,
Latinized as Alhazen (/ælˈhæzən/; full name Abū ʿAlī al-Ḥasan ibn al-Ḥasan ibn al-Haythamأبو علي، الحسن بن الحسن بن الهيثم; c. 965 – c. 1040), was a medieval
mathematician,
astronomer, and
physicist of the
Islamic Golden Age from present-day Iraq. Referred to as "the father of modern optics", he made significant contributions to the principles of
optics and
visual perception in particular. His most influential work is titled Kitāb al-Manāẓir (
Arabic: كتاب المناظر, "Book of Optics"), written during 1011–1021, which survived in a Latin edition.
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. Ibn al-Haytham was the first to explain that vision occurs when light reflects from an object and then passes to one's eyes, and to argue that vision occurs in the brain, pointing to observations that it is subjective and affected by personal experience.
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...)
Image 11
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...)
Brahmagupta was the first to give rules for computing 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...)
Image 13
Etching of an ancient seal identified as Eratosthenes.
Philipp Daniel Lippert, Dactyliothec, 1767.
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 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 mythical
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...)