History of algebra

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As a branch of mathematics, algebra emerged at the end of the 16th century in Europe, with the work of FranÃÆ'§ois ViÃÆ'¨te. Algebra can basically be regarded as performing calculations similar to arithmetic but with non-numerical mathematical objects. However, until the 19th century, algebra was essentially composed of the theory of equations. For example, the basic theorem of algebra belongs to the theory of equality and is not, currently, considered to be the property of algebra (It is impossible to provide pure algebraic evidence for this theorem, since this proposition involves the idea of ​​real numbers and therefore the notion of boundary, which does not belong to algebra ).

This article explains the history of the theory of equations, called here "algebra", from origin to the appearance of algebra as a separate area of ​​mathematics.

Video History of algebra

Etymology

The word "algebra" comes from the Arabic word ????? al-jabr , and this comes from a treatise written in 830 by medieval Persian mathematician Muhammad ibn M? al-Khw? rizm ?, whose Arabic title, Kit? b al-mu? ta? ar f? ? is? b al-? abr wa-l-muq? reinforcements , can be translated as Quick Start on Counting and Balancing Calculations . The treatises provided for the systematic solution of linear and quadratic equations. According to one narration, "[i] t is uncertain what the terms al-jabr and muqabalah are, but the usual interpretations are similar to those implied in the previous word 'al- jabr 'may mean something like' restoration 'or' settlement 'and seems to refer to the transposition of the term being reduced to the other side of the equation, the word' muqabalah 'is said to refer to' reduction 'or' balancing '- that is, the cancellation of similar terms in the opposite side of the equation The Arabic influence in Spain shortly after the time of al-Khwarizmi was found in Don Quixote , in which the word 'algebrista' is used for bone penyetel, the 'restorer'. "The term is used by al- Khwarizmi to describe the operations he introduced, "subtraction" and "balancing", refers to the transposition of diminished terms to the other side of an equation, that is, the cancellation of similar terms on the opposite side of the equation.

Maps History of algebra

Algebraic stages

Algebraic expression

Algebra does not always use the symbolism that now exists everywhere in mathematics; instead, it passes through three distinct phases. Stages in the development of symbolic algebra are approximately as follows:

• Rhetorical algebra , where the equations are written in full sentences. For example, the rhetorical form x 1 = 2 is "One plus one equals two" or maybe "Object plus 1 equals 2". Rhetorical algebra was first developed by the ancient Babylonians and remained dominant until the 16th century.
• Sync algebra , in which some symbolism is used, but does not contain all the characteristics of symbolic algebra. For example, there may be restrictions that reduction can only be used once on one side of the equation, which is not the case with symbolic algebra. The expression of syncopated algebra first appeared in Diophantus' Arithmetica (3rd century), followed by Brahmagupta Brahma Sphuta Siddhanta <7> (7th century).
• Symbolic algebra , where full symbolism is used. The first step towards this can be seen in the works of some Islamic mathematicians such as Ibn al-Banna (13-14 centuries) and al-Qalasadi (15th century), although symbolic algebra was fully developed by FranÃÆ'§ois ViÃÆ'¨te (16th ). century). Later, Renà © ¨ Descartes (17th century) introduces modern notation (eg, the use of x - see below) and shows that problems occurring in geometry can be expressed and solved in terms of algebra (Cartesian geometry ).

Sama pentingnya dengan penggunaan atau kurangnya simbolisme dalam aljabar adalah tingkat persamaan yang dibahas. Persamaan kuadrat memainkan peran penting dalam aljabar awal; dan sepanjang sebagian besar sejarah, hingga periode modern awal, semua persamaan kuadrat diklasifikasikan sebagai salah satu dari tiga kategori.

• ${{displaystyle x ^ {2} px = q}$
• ${{displaystyle x ^ {2} = px q}$
• ${{displaystyle x ^ {2} q = px}$

di mana p dan q positif. Trikotomi ini terjadi karena persamaan kuadrat dari bentuk ${\ displaystyle x2 px q = 0} Â Â$ , denote what is positif, tidak to memilize akar positif.

Di antara tahapan retoris dan sinkopasi aljabar simbolis, sebuah aljabar konstruktif geometrik dikembangkan oleh ahli matematika Yunani dan Veda Yunani klasik de mana persamaan aljabar diselesaikan melalui geometri. Misalnya, persamaan bentuk ${\ displaystyle x2} = A} Â$ diselesaikan dengan mencari sisi kotak persegi A .

Tahapan konseptual

In addition to the three stages of expressing algebraic ideas, some authors recognize four conceptual stages in algebraic development that occur in addition to changes in expression. These four stages are as follows:

• The geometric stage , where the concept of algebra is mostly geometric. This date returned to Babylonia and continued with the Greeks, and then revived by Omar KhayyÃÆ'¡m.
• Static equation completion stage , where the goal is to find numbers that satisfy a particular relationship. The shift from geometric algebra back to Diophantus and Brahmagupta, but algebra did not convincingly move to a static equation settlement stage until Al-Khwarizmi introduced a general algorithmic process to solve algebraic problems.
• Dynamic function stages , where motion is the underlying idea. The idea of ​​a function begins to emerge with Sharaf al-D? N al-T? S?, But algebra does not convincingly move to a dynamic function stage until Gottfried Leibniz.
• Abstract stage , in which the mathematical structure plays a central role. Abstract algebra is largely a product of the nineteenth and twentieth centuries.

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Babylonian algebra

The origins of algebra can be traced back to ancient Babylonia, which developed a number position system that greatly assisted them in solving their rhetorical algebraic equations. Babylonians are not interested in exact solutions but approximations, so they usually use linear interpolation to estimate intermediate values. One of the most famous tablets is the Plimpton 322 tablet, made around 1900-1600 BCE, which provides a triple table of Pythagoras and represents some of the most advanced mathematics before Greek mathematics.

Babylonian algebra was much more advanced than the Egyptian algebra of that time; while the Egyptians are especially concerned with the linear equations of Babylon more concerned with the equations of squares and cubic. The Babylonians have developed flexible algebraic operations by which they can add equally and multiply both sides of the equation by the same amount so as to eliminate fractions and factors. They are familiar with many simple forms of factoring, a three-term quadratic equation with positive roots, and many cubic equations although it is unknown if they are able to reduce the general cubic equation.

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Egyptian Algebra

Ancient Egyptian Algebra deals primarily with linear equations while the Babylonians find these equations too basic and develop mathematics to a higher level than the Egyptians.

The Rhind Papyrus, juga dikenal sebagai Ahmes Papyrus, adalah papirus Mesir kuno yang ditulis c. 1650 SM oleh Ahmes, yang mentranskripsinya dari karya sebelumnya yang ia kencani antara tahun 2000 dan 1800 SM. Ini adalah dokumen matematika Mesir kuno yang paling luas yang diketahui oleh para sejarawan. The Rhind Papyrus berisi masalah di mana persamaan linear dari bentuk ${\ displaystyle x ax = b}  ÂÂ$ dan ${\ displaystyle x ax bx = c}  ÂÂ$ diselesaikan, di mana a , b , dan c dikenal dan x , yang disebut sebagai "aha" atau heap, adalah yang tidak diketahui. Solusinya mungkin, tetapi tidak mungkin, tiba dengan menggunakan "metode posisi salah", atau regula falsi , di mana pertama nilai tertentu diganti ke sisi kiri persamaan, maka perhitungan aritmatika yang diperlukan dilakukan, ketiga hasilnya dibandingkan dengan sisi kanan persamaan, dan akhirnya jawaban yang benar ditemukan melalui penggunaan proporsi. Dalam beberapa masalah, penulis "memeriksa" solusinya, dengan demikian menulis salah satu bukti sederhana yang diketahui paling awal.

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Aljabar geometrik Yunani

It is sometimes said that Greeks do not have algebra, but this is not accurate. At the time of Plato, Greek mathematics had undergone drastic changes. The Greeks made geometric algebra where terms are represented by the sides of geometric objects, usually lines, which have corresponding letters, and with this new algebraic form they can find solutions to equations using a process they find , known as "territorial implications". "Implementation of territory" is only a part of geometric algebra and it is completely closed in Euclid Elements .

An example of geometric algebra will solve linear equations ax = bc. The ancient Greeks would solve this equation by viewing it as the equality of the area rather than as the equation between a: b and c: x. The Greeks will build a rectangle with the long sides of b and c, then extend the sides of the rectangle to the length, and finally they will complete the extended rectangle so finding the rectangular side of it is the solution.

Bloom of Thymaridas

Iamblichus in Introductio arithmatica tells us that Thymaridas (around 400 BC - 350 BC) work with simultaneous linear equations. In particular, he created the famous rule known as "Thymaridas bloom" or as "Thymaridas flower", which states that:

If the total amount of n is given, and also the number of each pair containing a certain quantity, then this particular quantity equals 1/(n - 2) of the difference between the sum of this sum. pair and the amount given first.

or using modern ideas, the solution of the following linear equation n system in n is unknown,

x x ₁ x ₂ ... x _n-1 = s
x x ₁ = m ₁
x x ₂ = m ₂
.
.
.
x x _n-1 = m _n-1

aku s,

${\ displaystyle x = {\ cfrac {(m_ 1 m2... m_ {n-1}) - s} {n- 2}} = {\ cfrac {(\ sum _ {i = 1} ^ {n-1} m {{i}) - s} {n-2}}} Â Â$

Iamblichus goes on to illustrate how some systems of linear equations that are not in this form can be placed into this form.

Euclid of Alexandria

Euclid (Greek: ????????? ) is a Greek mathematician who developed in Alexandria, Egypt, almost certainly during his time government of Ptolemy I (323-283Ã, BCE). Neither the year nor place of birth has been established, nor the circumstances of his death.

Euclid is considered the "father of geometry". The element is the most successful book in the history of mathematics. Although he is one of the most famous mathematicians in history, no new discoveries are associated with him, but he is remembered for his great explanatory skills. The Elements is not, as is sometimes thought, a collection of all Greek mathematical knowledge to date, but rather a basic introduction to it.

Elements

The geometrical work of the Greeks, typed in Euclid Elements , provides a framework for generalizing formulas beyond the solution of certain problems into the more general system of expressing and solving equations.

Book II of the Element contains fourteen propositions, which in Euclid's time was significant for performing geometric algebra. This proposition and its result are the geometric equivalents of our modern symbolic algebra and trigonometry. Today, using modern symbolic algebra, we let symbols represent known and unknown quantities (ie numbers) and then apply algebraic operations to them. While on the time scale Euclid is seen as a line segment and then the result is concluded using axiom or geometry theorem.

Many basic laws of addition and multiplication are embedded or geometrically proven within Elements . For example, Proposition 1 of Book II states:

If there are two straight lines, and one of them will be cut into any number of segments, the rectangle contained by two straight lines is equal to the rectangle contained by a straight line cut and each segment.

Tapi ini tidak lebih dari versi geometris dari (kiri) hukum distributif, ${\ displaystyle a (b c d) = ab ac iklan} Â Â$ ; Dan Balam V Dan VII gives Elemen hukum komutatif dan asosiatif untuk multiplikasi ditunjukkan.

Banyak persuasan dasar juga dibuktika secara geometris. Misalnya, proposisi 5 dalam Buku II membuktikan bahwa ${\ displaystyle a2 -b2 = = (a) (a-b)} Â Â$ , dan proposisi 4 dalam Buku II membuktikan bahwa ${\ displaystyle (a b) 2 = a2 2ab b2} Â Â$ .

In addition, there are also geometric solutions that are given to many equations. For example, proposition 6 of Book II provides a solution to the quadratic equation ax x ² = b ² , and the proposition 11 of Book II provides a solution for x ² = a ² .

Data

Data is a work written by Euclid for use in Alexandria schools and it is intended to be used as a companion volume for the first six books of the Element . This book contains about fifteen definitions and ninety-five statements, of which there are about two dozen statements that serve as rules or algebraic formulas. Some of these statements are geometric equivalents for the solution of quadratic equations. For example, Data contains solutions for dx ² - adx < i> b ² c = 0 and known Babylonian equations xy = a ² , x Ã, Â ± y = b .

The cone

The conic section is the curve generated from the cone intersection by plane. There are three main types of conic sections: ellipses (including circles), parabolas, and hyperboles. The famous cone parts have been discovered by Menaechmus (c) 380Ã, BCEÃ,Â,,,, 320Ã, BCE) and since dealing with conical parts is equivalent to dealing with their respective equations, they play a geometric role equivalent to cubic equations and high order equations others..

Menaechmus knows that in a parabola, the equation y ² = l x applies, where l is a constant called rectum latus, even though it is unaware the fact that each equation in two is unknown determines the curve. He apparently lowered the properties of cone and other leaves as well. Using this information, it is now possible to find a solution to the problem of cube duplication by solving the points at which two paraboles intersect, equivalent solutions to solving cubic equations.

We were told by Eutocius that the method he used to solve the cubic equation was due to Dionysodorus (250Ã, BCEÃ, - 190Ã, BCE). Dionysodorus solves the cubic by means of rectangular and parabolic hyperbola junctions. This is related to the problem in Archimedes' On Sphere and Cylinder . The cone section will be studied and used for thousands of years by the Greeks, and then Islamists and Europeans, mathematicians. Especially Apollonius of Perga's famous Conicia deals with conic sections, among other topics.

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Chinese Algebra

Chinese mathematics began at least 300 BC with Zhoubi Suanjing, generally regarded as one of China's oldest mathematical documents.

Nine Chapter on Mathematical Art

Chiu-chang suan-shu or The Nine Chapters on the Mathematical Art , written around 250 BC, is one of China's most influential mathematical books and consists of about 246 problems. Chapter eight discusses the solving of determinative and indefinite linear equations using positive and negative numbers, with one problem handling the settlement of four equations in five unknowns.

Sea-Mirror of Circle Measurement

Ts'e-yuan hai-ching , or Sea-Mirror of the Circle Measurements , is a collection of about 170 problems written by Li Zhi (or Li Ye) (1192 - 1279 CE). He uses the fan fan, or Horner's method, to solve equations of six degrees, although he does not explain his method of solving equations.

Mathematics Magazine in Nine Sections

Shu-shu chiu-chang , or The Mathematics of Nine Sections , written by rich governor and minister Ch'in Chiu-shao (c. 1202 - c 1261 CE) and with the discovery of a simultaneous congruent-solving method, now called Chinese residual theorem, it marks a high point in Chinese indeterminate analysis.

Magic box

The earliest known magic box appeared in China. In Nine Chapter the author solves a system of linear equations by placing coefficients and constants of linear equations into a magic square (ie a matrix) and performing a column reduction operation in the magic square. The most famous original magic boxes ordered over three are attributed to Yang Hui (fl.C 1261 - 1275), who work with magic boxes as tall as ten.

The Precious Mirror of the Four Elements

Ssy-yÃÆ'¼an yÃÆ'¼-chien ??????, or The Precious Mirror of the Four Elements , written by Chu Shih-chieh in 1303 and marks the peak in the development of Chinese algebra. The four elements, called heaven, earth, man and matter, represent four unknown quantities in the algebraic equation. The Ssy-yÃÆ'¼an yÃÆ'¼-chien deals with simultaneous equations and with equations of fourteen degrees. The author uses the fan fa method, today called the Horner method, to solve this equation.

The Precious Mirror opens with an arithmetic triangle diagram (Pascal triangle) using a zero round symbol, but Chu Shih-chieh denies the credit for it. A similar triangle appears in Yang Hui's work, but without the zero symbol.

Ada banyak persamaan seri penjumlahan yang diberikan tanpa bukti di Cermin Berharga . Beberapa seri penjumlahan adalah:

${\ displaystyle 1 2 2 2 2 3 2 \ nd2 nets = {n (n 1) (2n 1) \ over 3!}} Â Â$

${\ displaystyle 1 8 30 80 \ cdots {n ^ {2} (n 1) (n 2) \ over 3!} = {N (n 1 ) (n 2) (n 3) (4n 1) \ over 5!}} Â Â$

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Aljabar diophantine

Diophantus is a living Hellenistic mathematician c. 250 CE, but the uncertainty of this date is so great that it may be more than a century old. He is known for writing Arithmetica, a treatise originally thirteen books but only the first six survivors. Arithmetica has very little in common with traditional Greek mathematics because he is divorced from geometric methods, and it is different from Babylonian mathematics in Diophantus concerned primarily with the exact, decisive and indeterminate solution, not the simple one. estimates.

It is usually difficult to say whether a Diophantine equation can be solved. There is no evidence to suggest that Diophantus even realizes that there may be two solutions to quadratic equations. He also considered simultaneous quadratic equations. Also, there is no general method that can be abstracted from all Diophantus solutions.

In Arithmetica , Diophantus is the first to use symbols for unknown numbers as well as abbreviations for numerical strength, relationships and operations; so he uses what is now known as sync algebra. The main difference between the syncopation algebra of Diophantine and modern algebraic notation is that the former has no special symbols for operation, relationship, and exponential. So, for example, what will we write as

${\ displaystyle x ^ {3} -2x ^ {2} 10x-1 = 5}$

Diophantus will write this as

? ^? ???? ?? ? ^? Ã, ?? ? Ã, ?? ?? ? Ã, ??

where the symbol represents the following:

Perhatikan bahwa koefisien datang setelah variabel dan penambahan itu diwakili oleh penjajaran istilah. Sebuah Terjemahan Symbol Symbol Symbol harfiah dari persamaan sinkopasi Diophantus that Dalam persuasive symbolized modern adalah sebagai berikut:

${\ displaystyle {x ^ {3}} 1 {x} 10- {x2} 2 {x ^ 0}} 1 = { x ^}} 5} Â Â$

give, untuk memperjelas, jika tanda kurung modern dan plus digunakan maka persamaan di atas dapat ditulis ulang sebagai:

${\ displaystyle ({x ^ {3}} 1 x} 10) - ({x2}} 2 {x ^ 0} 1) = {x ^ {0}} 5} Â Â$

Arithmetica is a collection of about 150 problems that are solved with certain numbers and no postulateal development or general methods are explicitly described, although a general method may have been intended and no attempt to find all solutions to equations. Arithmetica does contain a completed problem involving some unknown quantity, which is solved, if possible, by disclosing an unknown amount only from one of them. Arithmetica also takes advantage of the identity:

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Indian Algebra

Indian mathematicians are active in studying number systems. The earliest Indian mathematical documents are known dated around the middle of the first millennium BC (circa 6th century BC).

The recurrent themes in Indian mathematics are, among other things, the determination and determination of linear and quadratic equations, simple measurements, and triple Pythagoras.

Aryabhata

Aryabhata (476-550 CE) adalah seorang matematikawan India yang menulis Aryabhatiya . Di dalamnya dia memberi aturan,

${\ displaystyle 1 2 2 2 \ cdots n2 = {n (n 1) (2n 1) \ over 6 }} Â Â$

dan

${\ displaystyle 13 2 3 \ cdots n3 = (1 2 \ cdots n )2} Â Â$

Brahma Sphuta Siddhanta

Brahmagupta (fl. 628) adalah seorang matematikawan India yang menulis Brahma Sphuta Siddhanta . Dalam karyanya Brahmagupta memecahkan persamaan kuadrat umum untuk akar positif dan negatif. Dalam analisis tak tentu Brahmagupta memberikan trias Pythagoras ${\ displaystyle m}  ÂÂ$ , ${\ displaystyle {1 \ over 2} ({m ^ {2} \ over n} -n)}  ÂÂ$ , ${\ displaystyle {1 \ over 2} ({m ^ {2} \ over n} n)}  ÂÂ$ , tetapi ini adalah bentuk modifikasi dari aturan Babylonia lama yang mungkin akrab dengan Brahmagupta. Dia adalah orang pertama yang memberikan solusi umum untuk kapak Diophantine linear = c, di mana a, b, dan c adalah bilangan bulat. Tidak seperti Diophantus yang hanya memberikan satu solusi untuk persamaan tak tentu, Brahmagupta memberi semua solusi integer; tetapi Brahmagupta menggunakan beberapa contoh yang sama seperti Diophantus telah membawa beberapa sejarawan untuk mempertimbangkan kemungkinan pengaruh Yunani pada karya Brahmagupta, atau setidaknya sumber Babilonia umum.

Like Diophantus algebra, Brahmagupta syncope algebra. Also shown by placing the numbers side by side, subtraction by placing the dot on the subtrahend, and dividing by placing the divisor below the dividend, similar to our notation but without the bar. Unknown multiplication, evolution, and quantity are represented by appropriate acronyms. The extent to which Greek influence on syncopation is, if any, is unknown and it is possible that Greek and Indian syncopation may originate from a common source of Babylon.

Bh? skara II

Bh? Skara II (1114 - c 1185) was a prominent mathematician of the twelfth century. In Algebra, he gives a general solution of the Pell equation. He is the author of Lilavati and Vija-Ganita, which contains problems relating to the determination and determination of linear and quadratic equations, and triple Pythagoras and he fails to distinguish between the right and the forecast. statement. Many problems in Lilavati and Vija-Ganita come from other Hindu sources, and so Bhaskara is the best in dealing with indeterminate analysis.

Bhaskara menggunakan simbol awal dari nama-nama untuk warna sebagai simbol dari variabel yang tidak diketahui. Jadi, misalnya, apa yang akan kita tulis hari ini sebagai

${\ displaystyle (-x-1) (2x-8) = x-9} Â Â$

Bhaskara akan menulis sebagai

. _ .

ya 1 ru 1

.

ya 2 ru 8

.

Jumlah ya 1 ru 9

where yes shows the first syllable for black , and ru is taken from the species . The points above the numbers show a reduction.

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Islamic Algebra

The first century of the Arabian Islamic Empire has hardly seen any scientific or mathematical achievements since the Arabs, with their newly conquered empire, have not gained intellectual impulse and research in other parts of the world have faded away. In the second half of the 8th century, Islam experienced a cultural revival, and research in mathematics and science increased. The Abbasid Caliph al-Mamun (809-833) is said to have dreamed where Aristotle appeared to him, and as a consequence al-Mamun ordered that Arabic translations be made of as many Greek works as possible, including Ptolemy> Almagest and Euclid's Elements . Greek works will be given to the Muslims by the Byzantine Empire to be exchanged for a treaty, because the two empires hold an uncomfortable peace. Many of these Greek works are translated by Thabit ibn Qurra (826-901), which translates books written by Euclid, Archimedes, Apollonius, Ptolemy, and Eutocius.

There are three theories about the origin of Arabic Algebra. The former emphasized the influence of Hinduism, the latter emphasizing the influence of Mesopotamia or Persia-Syriac and the third emphasizing Greek influence. Many scholars believe that this is the result of a combination of the three sources.

Throughout their time in power, before the fall of Islamic civilization, the Arabs used a full rhetorical algebra, where often even the numbers were spelled with words. The Arabs would eventually replace the spelled numbers (eg twenty-two) with Arabic numerals (eg 22), but the Arabs did not adopt or develop symbolic or synchronized algebra until Ibn al-Banna's work in the 13th century and Ab? al-Hasan bin Al? al-Qalas?

Source of the article : Wikipedia