Multiplication table

In mathematics, a multiplication table is a mathematical table used to define a multiplication operation for an algebraic system. The decimal multiplication table was traditionally taught as an essential part of elementary arithmetic around the world, as it lays the foundation for arithmetic operations with base-ten numbers. Many educators believe it is necessary to memorize the table up to 9 × 9; the oldest known multiplication tables were used by the Babylonians about 4000 years ago. However, they used a base of 60; the oldest known tables using a base of 10 are the Chinese decimal multiplication table on bamboo strips dating to about 305 BC, during China's Warring States period. The multiplication table is sometimes attributed to the ancient Greek mathematician Pythagoras, it is called the Table of Pythagoras in many languages, sometimes in English. The Greco-Roman mathematician Nichomachus, a follower of Neopythagoreanism, included a multiplication table in his Introduction to Arithmetic, whereas the oldest surviving Greek multiplication table is on a wax tablet dated to the 1st century AD and housed in the British Museum.

In 493 AD, Victorius of Aquitaine wrote a 98-column multiplication table which gave the product of every number from 2 to 50 times and the rows were "a list of numbers starting with one thousand, descending by hundreds to one hundred descending by tens to ten by ones to one, the fractions down to 1/144."In his 1820 book The Philosophy of Arithmetic, mathematician John Leslie published a multiplication table up to 99 × 99, which allows numbers to be multiplied in pairs of digits at a time. Leslie recommended that young pupils memorize the multiplication table up to 50 × 50; the illustration below shows a table up to 12 × 12, a size used in schools. The traditional rote learning of multiplication was based on memorization of columns in the table, in a form like This form of writing the multiplication table in columns with complete number sentences is still used in some countries, such as Bosnia and Herzegovina, instead of the modern grid above. There is a pattern in the multiplication table that can help people to memorize the table more easily.

It uses the figures below: Figure 1 is used for multiples of 1, 3, 7, 9. Figure 2 is used for the multiples of 2, 4, 6, 8; these patterns can be used to memorize the multiples of any number from 0 to 10, except 5. As you would start on the number you are multiplying, when you multiply by 0, you stay on 0; the pattern works with multiples of 10, by starting at 1 and adding 0, giving you 10 just apply every number in the pattern to the "tens" unit as you would do as usual to the "ones" unit. For example, to recall all the multiples of 7: Look at the 7 in the first picture and follow the arrow; the next number in the direction of the arrow is 4. So think of the next number after 7 that ends with 4, 14; the next number in the direction of the arrow is 1. So think of the next number after 14 that ends with 1, 21. After coming to the top of this column, start with the bottom of the next column, travel in the same direction; the number is 8. So think of the next number after 21 that ends with 8, 28. Proceed in the same way until the last number, 3, corresponding to 63.

Next, use the 0 at the bottom. It corresponds to 70. Start again with the 7; this time it will correspond to 77. Continue like this. Tables can define binary operations on groups, fields and other algebraic systems. In such contexts they can be called Cayley tables. Here are the addition and multiplication tables for the finite field Z5. For every natural number n, there are addition and multiplication tables for the ring Zn. For other examples, see group, octonion; the Chinese multiplication table consists of eighty-one sentences with four or five Chinese characters per sentence, making it easy for children to learn by heart. A shorter version of the table consists of only forty-five sentences, as terms such as "nine eights beget seventy-two" are identical to "eight nines beget seventy-two" so there is no need to learn them twice. A bundle of 21 bamboo slips dated 305 BC in the Warring States period in the Tsinghua Bamboo Slips collection is the world's earliest known example of a decimal multiplication table.

In 1989, the National Council of Teachers of Mathematics developed new standards which were based on the belief that all students should learn higher-order thinking skills, which recommended reduced emphasis on the teaching of traditional methods that relied on rote memorization, such as multiplication tables. Adopted texts such as Investigations in Numbers and Space omitted aids such as multiplication tables in early editions. NCTM made it clear in their 2006 Focal Points that basic mathematics facts must be learned, though there is no consensus on whether rote memorization is the best method. Chinese multiplication table Vedic square IBM 1620, an early computer that used tables stored in memory to perform addition and multiplication

Pythagorean theorem

In mathematics, the Pythagorean theorem known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides; the theorem can be written as an equation relating the lengths of the sides a, b and c called the "Pythagorean equation": a 2 + b 2 = c 2, where c represents the length of the hypotenuse and a and b the lengths of the triangle's other two sides. Although it is argued that knowledge of the theorem predates him, the theorem is named after the ancient Greek mathematician Pythagoras as it is he who, by tradition, is credited with its first proof, although no evidence of it exists. There is some evidence that Babylonian mathematicians understood the formula, although little of it indicates an application within a mathematical framework. Mesopotamian and Chinese mathematicians all discovered the theorem independently and, in some cases, provided proofs for special cases.

The theorem has been given numerous proofs – the most for any mathematical theorem. They are diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years; the theorem can be generalized in various ways, including higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, indeed, to objects that are not triangles at all, but n-dimensional solids. The Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical abstruseness, mystique, or intellectual power; the Pythagorean theorem was known long before Pythagoras, but he may well have been the first to prove it. In any event, the proof attributed to him is simple, is called a proof by rearrangement; the two large squares shown in the figure each contain four identical triangles, the only difference between the two large squares is that the triangles are arranged differently. Therefore, the white space within each of the two large squares must have equal area.

Equating the area of the white space yields the Pythagorean theorem, Q. E. D; that Pythagoras originated this simple proof is sometimes inferred from the writings of the Greek philosopher and mathematician Proclus. Several other proofs of this theorem are described below. If c denotes the length of the hypotenuse and a and b denote the lengths of the other two sides, the Pythagorean theorem can be expressed as the Pythagorean equation: a 2 + b 2 = c 2. If the length of both a and b are known c can be calculated as c = a 2 + b 2. If the length of the hypotenuse c and of one side are known the length of the other side can be calculated as a = c 2 − b 2 or b = c 2 − a 2; the Pythagorean equation relates the sides of a right triangle in a simple way, so that if the lengths of any two sides are known the length of the third side can be found. Another corollary of the theorem is that in any right triangle, the hypotenuse is greater than any one of the other sides, but less than their sum. A generalization of this theorem is the law of cosines, which allows the computation of the length of any side of any triangle, given the lengths of the other two sides and the angle between them.

If the angle between the other sides is a right angle, the law of cosines reduces to the Pythagorean equation. This theorem may have more known proofs than any other; this proof is based on the proportionality of the sides of two similar triangles, that is, upon the fact that the ratio of any two corresponding sides of similar triangles is the same regardless of the size of the triangles. Let ABC represent a right triangle, with the right angle located at C. Draw the altitude from point C, call H its intersection with the side AB. Point H divides the length of the hypotenuse c into parts d and e; the new triangle ACH is similar to triangle ABC, because they both have a right angle, they share the angle at A, meaning that the third angle will be the same in both triangles as well, marked as θ in the figure. By a similar reasoning, the triangle CBH is similar to ABC; the proof of similarity of the triangles requires the triangle postulate: the sum of the angles in a triangle is two right angles, is equivalent to the parallel postulate.

Similarity of the triangles leads to the equality of ratios of corresponding sides: B C A B = B H B C and A C A B = A H A C. The first result equates

Pythagoreanism

Pythagoreanism originated in the 6th century BC, based on the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in Italy. Early-Pythagorean communities lived throughout Magna Graecia. Espousing a rigorous life of the intellect and strict rules on diet and behavior comprised a cult of following Pythagorean's Code. For example, the Code's diet prohibits the consumption or touching any sort of bean or legume. Pythagoras’ death and disputes about his teachings led to the development of two philosophical traditions within Pythagoreanism; the practitioners of akousmatikoi were superseded in the 4th century BC as a significant mendicant school of philosophy by the Cynics. The Pythagorean mathēmatikoi philosophers were in the 4th century BC absorbed into the Platonic school. Following the political instability in the Magna Graecia, some Pythagorean philosophers fled to mainland Greece while others regrouped in Rhegium. By about 400 BC the majority of Pythagorean philosophers had left Italy.

Pythagorean ideas exercised a marked influence on Plato and through him, on all of Western philosophy. Many of the surviving sources on Pythagoras originate with Aristotle and the philosophers of the Peripatetic school; as a philosophic tradition, Pythagoreanism was revived in the 1st century BC, giving rise to Neopythagoreanism. The worship of Pythagoras continued in Italy and as a religious community Pythagoreans appear to have survived as part of, or influenced, the Bacchic cults and Orphism. Pythagoras was in ancient times well known for the mathematical achievement of the Pythagorean theorem. Pythagoras had discovered that "in a right-angled triangle the square of the hypotenuse is equal to the squares of the other two sides". In ancient times Pythagoras was noted for his discovery that music had mathematical foundations. Antique sources that credit Pythagoras as the philosopher who first discovered music intervals credit him as the inventor of the monochord, a straight rod on which a string and a movable bridge could be used to demonstrate the relationship of musical intervals.

Much of the surviving sources on Pythagoras originate with Aristotle and the philosophers of the Peripatetic school, which founded histographical academic traditions such as biography and the history of science. The surviving 5th century BC sources on Pythagoras and early Pythagoreanism are void of supernatural elements. While surviving 4th century BC sources on Pythagoreas' teachings introduced legend and fable. Philosophers who discussed Pythagoreanism, such as Anaximander, Andron of Ephesus and Neanthes had access to historical written sources as well as the oral tradition about Pythagoreanism, which by the 4th century BC was in decline. Neopythagorean philosophers, who authored many of the surviving sources on Pythagoreanism, continued the tradition of legend and fantasy; the earliest surviving ancient source on Pythagoras and his followers is a satire by Xenophanes, on the Pythagorean beliefs on the transmigration of souls. Xenophanes wrote of Pythagoras that: Once they say that he was passing by when a puppy was being whipped, And he took pity and said: "Stop!

Do not beat it! For it is the soul of a friend That I recognized when I heard it giving tongue." In a surviving fragment from Heraclitus and his followers are described as follows: Pythagoras, the son of Mnesarchus, practised inquiry beyond all other men and selecting of these writings made for himself a wisdom or made a wisdom of his own: a polymathy, an imposture. Two other surviving fragments of ancient sources on Pythagoras are by Ion of Empedocles. Both were born after Pythagoras' death. By that time he was known as a sage and his fame had spread throughout Greece. According to Ion, Pythagoras was:... distinguished for his many virtue and modesty in death has a life, pleasing to his soul, if Pythagoras the wise achieved knowledge and understanding beyond that of all men. Empedocles described Pythagoras as "a man of surpassing knowledge, master of all kinds of wise works, who had acquired the upmost wealth of understanding." In the 4th century BC the Sophist Alcidamas wrote that Pythagoras was honored by Italians.

Today scholars distinguish two periods of Pythagoreanism: early-Pythagoreanism, from the 6th till the 5th century BC, late-Pythagoreanism, from the 4th till the 3rd century BC. The Spartan colony of Taranto in Italy became the home for many practitioners of Pythagoreanism and for Neopythagorean philosophers. Pythagoras had lived in Crotone and Metaponto, both were Achaean colonies. Early-Pythagorean sects lived throughout Magna Graecia, they espoused to a rigorous life of the intellect and strict rules on diet and behavior. Their burial rites were tied to their belief in the immortality of the soul. Early-Pythagorean sects were closed societies and new Pythagoreans were chosen based on merit and discipline. Ancient sources record that early-Pythagoreans underwent a five year initiation period of listening to the teachings in silence. Initiates could through a test become members of the inner circle. However, Pythagoreans could leave the community if they wished. Iamblichus listed 235 Pythagoreans by name, among them 17 women who he described as the "most famous" women practitioners of Pythagoreanism.

It was customary that family members became Pythagoreans, as Pythagoreanism developed into a philosophic traditions that entailed rules for everyday life and Pythagoreans were bound by secrets. The home of a Pythagorean was known as the site of mysteries. Pythagoras had been born on the island of Samos at around 570 BC and left his homeland at around 530 BC in opposition

Pythagoras

Pythagoras of Samos was an ancient Ionian Greek philosopher and the eponymous founder of Pythagoreanism. His political and religious teachings were well known in Magna Graecia and influenced the philosophies of Plato, and, through them, Western philosophy. Knowledge of his life is clouded by legend, but he appears to have been the son of Mnesarchus, a seal 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, 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 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 devised the doctrine of musica universalis, which holds that the planets move according to mathematical equations 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 followers 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 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, 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 and that he was the first to divide the globe into five climatic zones.

Classical historians debate whether Pythagoras made these discoveries, many of the accomplishments credited to him 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 contributed to mathematics or natural philosophy. Pythagoras influenced Plato, whose dialogues his Timaeus, exhibit Pythagorean teachings. Pythagorean ideas on mathematical perfection 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, 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.

No authentic writings of Pythagoras have survived, nothing is known for certain about his life. The earliest sources on Pythagoras's life are brief and satirical; the earliest source on Pythagoras's teachings is a satirical poem written after his death by Xenophanes of Colophon, one of his contemporaries. In the poem, Xenophanes describes Pythagoras interceding on behalf of a dog, being beaten, professing to recognize in its cries the voice of a departed friend. Alcmaeon of Croton, a doctor who lived in Croton at around the same time Pythagoras lived there, incorporates many Pythagorean teachings into his writings and alludes to having known Pythagoras personally; the poet Heraclitus of Ephesus, born across a few miles of sea away from Samos and may have lived within Pythagoras's lifetime, mocked Pythagoras as a clever charlatan, remarking that "Pythagoras, son of Mnesarchus, practiced inquiry more than any other man, selecting from these writings he manufactured a wisdom for himself—much learning, artful knavery."The Greek poets Ion of Chios and Empedocles of Acragas both express admiration for Pythagoras in their poems.

The first concise description of Pythagoras comes from the historian Herodotus of Halicarnassus, who describes him as "not the most insignificant" of Greek sages and states that Pythagoras taught his followers how to attain immortality. The writings attributed to the Pythagorean philosopher Philolaus of Croton, who lived in the late fifth century BC, are the earliest texts to describe the numerological and musical theories that were ascribed to Pythagoras; the Athenian rhetorician Isocrates was the first to describe Pythagoras as having visited Egypt. Aristotle wrote a treatise On the Pythagoreans, no longer extant; some of it may be preserved in the Protrepticus. Aristotle's disciples Dicaearchus and Heraclides Ponticus wrote on the same subject. Most of the major sources on Pythagoras's life are from the Roman period, by which point, according to the German classicist Walter Burkert, "the history of Pythagoreanism was already... the laborious reconstruction of something lost and gone." Three lives of Pythagoras have survived from late antiquity, all of which are filled with myths and legends.

The earliest and most respectable of these is the one from Diogenes Laërtius's Lives and Opinions of Eminent Philosophers. The two lives were written by the Neoplatonist philosophers Porphyry and Iamblichus and were intended as po

Pythagorean hammers

According to legend, Pythagoras discovered the foundations of musical tuning by listening to the sounds of four blacksmith's hammers, which produced consonance and dissonance when they were struck simultaneously. According to Nicomachus in his 2nd century CE Enchiridion harmonices Pythagoras noticed that hammer A produced consonance with hammer B when they were struck together, hammer C produced consonance with hammer A, but hammers B and C produced dissonance with each other. Hammer D produced such perfect consonance with hammer A that they seemed to be "singing" the same note. Pythagoras rushed into the blacksmith shop to discover why, found that the explanation was in the weight ratios; the hammers weighed 12, 9, 8, 6 pounds respectively. Hammers A and D were in a ratio of 2:1, the ratio of the octave. Hammers B and C weighed 8 pounds, their ratios with hammer A were and. The space between B and C is a ratio of 9:8, equal to the musical whole tone, or whole step interval; the legend is, at least with respect to the hammers, demonstrably false.

It is a Middle Eastern folk tale. These proportions are indeed relevant to string length — using these founding intervals, it is possible to construct the chromatic scale and the basic seven-tone diatonic scale used in modern music, Pythagoras might well have been influential in the discovery of these proportions — but the proportions do not have the same relationship to hammer weight and the tones produced by them. However, hammer-driven chisels with equal cross-section, show an exact proportion between length or weight and Eigenfrequency. Earlier sources mention Pythagoras' interest in ratio. Xenocrates, while not as far as we know mentioning the blacksmith story, described Pythagoras' interest in general terms: "Pythagoras discovered that the intervals in music do not come into being apart from number. So he set out to investigate under what conditions concordant intervals come about, discordant ones, everything well-attuned and ill-tuned." Whatever the details of the discovery of the relationship between music and ratio, it is regarded as the first empirically secure mathematical description of a physical fact.

As such, it is symbolic of, leads to, the Pythagorean conception of mathematics as nature's modus operandi. As Aristotle was to write, "the Pythagoreans construct the whole universe out of numbers". Equal temperament Just intonation Pythagorean tuning

Pythagorean cup

A Pythagorean cup is a practical joke device in a form of a drinking cup, credited to Pythagoras of Samos. When it is filled beyond a certain point, a siphoning effect causes the cup to drain its entire contents through the base. A Pythagorean cup looks like a normal drinking cup, except that the bowl has a central column in it, giving it a shape like a Bundt pan; the central column of the bowl is positioned directly over the stem of the cup and over the hole at the bottom of the stem. A small open pipe runs from this hole to the top of the central column, where there is an open chamber; the chamber is connected by a second pipe to the bottom of the central column, where a hole in the column exposes the pipe to the bowl of the cup. When the cup is filled, liquid rises through the second pipe up to the chamber at the top of the central column, following Pascal's principle of communicating vessels; as long as the level of the liquid does not rise beyond the level of the chamber, the cup functions as normal.

If the level rises further, the liquid spills through the chamber into the first pipe and out the bottom. Gravity creates a siphon through the central column, causing the entire contents of the cup to be emptied through the hole at the bottom of the stem; some modern toilets operate on the same principle: when the water level in the bowl rises high enough, a siphon is created, flushing the toilet. Dribble glass Fuddling cup Heron's fountain List of practical joke topics Puzzle jug Soxhlet extractor, which uses the same mechanism. James Stanley — Towards a Better Pythagorean Cup A 2014 design for 3D printing your own Pythagoras Cup

Neopythagoreanism

Neopythagoreanism was a school of Hellenistic philosophy which revived Pythagorean doctrines. Neopythagoreanism was influenced in turn influenced Neoplatonism, it originated in the 1st century BCE and flourished during the 1st and 2nd centuries CE. The 1911 Britannica describes Neopythagoreanism as "a link in the chain between the old and the new" within Hellenistic philosophy; as such, it contributed to the doctrine of monotheism. Central to Neopythagorean thought was the concept of a soul and its inherent desire for a unio mystica with the divine; the word "Neopythagoreanism" is a modern term, coined as a parallel of "Neoplatonism". In the 1st century BCE Cicero's friend Nigidius Figulus made an attempt to revive Pythagorean doctrines, but the most important members of the school were Apollonius of Tyana and Moderatus of Gades in the 1st century CE. Other important Neopythagoreans include the mathematician Nicomachus of Gerasa, who wrote about the mystical properties of numbers. In the 2nd century, Numenius of Apamea sought to fuse additional elements of Platonism into Neopythagoreanism, prefiguring the rise of Neoplatonism..

Neopythagoreanism was an attempt to re-introduce a mystical religious element into Hellenistic philosophy in place of what had come to be regarded as an arid formalism. The founders of the school sought to invest their doctrines with the halo of tradition by ascribing them to Pythagoras and Plato, they went back to the period of Plato's thought, the period when Plato endeavoured to combine his doctrine of Ideas with Pythagorean number theory, identified the Good with the Monad, the source of the duality of the Infinite and the Measured with the resultant scale of realities from the One down to the objects of the material world. They emphasized the fundamental distinction between the body. God must be worshipped spiritually by the will to be good, not in outward action; the soul must be freed from its material surrounding, the "muddy vesture of decay," by an ascetic habit of life. Bodily pleasures and all sensuous impulses must be abandoned as detrimental to the spiritual purity of the soul. God is the principle of Matter the groundwork of Evil.

In this system can be distinguished not only the asceticism of Pythagoras and the mysticism of Plato, but the influence of the Orphic mysteries and of Oriental philosophy. The Ideas of Plato are no longer self-subsistent entities but are the elements which constitute the content of spiritual activity; the non-material universe is regarded as the sphere of spirit. The Porta Maggiore Basilica where Neopythagoreans held their meetings in the 1st century, believed to have been constructed by the Statilus family was found near Porta Maggiore on Via Praenestina in Rome. School of the Sextii Allegorical interpretations of Plato Charles H. Kahn and the Pythagoreans: A Brief History, Indianapolis: Hackett 2001 ISBN 0-87220-575-4 ISBN 978-0872205758 This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed.. "Neopythagoreanism". Encyclopædia Britannica. Cambridge University Press