Quadratic Equations

It is often claimed that the Babylonians (about 400 BC) were the first to solve quadratic equations. This is an over simplification, for the Babylonians had no notion of 'equation'. What they did develop was an algorithmic approach to solving problems which, in our terminology, would give rise to a quadratic equation. The method is essentially one of completing the square. However all Babylonian problems had answers which were positive (more accurately unsigned) quantities since the usual answer was a length.

In about 300 BC Euclid developed a geometrical approach which, although later mathematicians used it to solve quadratic equations, amounted to finding a length which in our notation was the root of a quadratic equation. Euclid had no notion of equation, coefficients etc. but worked with purely geometrical quantities.

Hindu mathematicians took the Babylonian methods further so that Brahmagupta (598-665 AD) gives an, almost modern, method which admits negative quantities. He also used abbreviations for the unknown, usually the initial letter of a color was used, and sometimes several different unknowns occur in a single problem.

The Arabs did not know about the advances of the Hindus so they had neither negative quantities nor abbreviations for their unknowns. However al-Khwarizmi (c 800) gave a classification of different types of quadratics (although only numerical examples of each). The different types arise since al-Khwarizmi had no zero or negatives. He has six chapters each devoted to a different type of equation, the equations being made up of three types of quantities namely: roots, squares of roots and numbers i.e. x, x² and numbers.

1. Squares equal to roots.
2. Squares equal to numbers.
3. Roots equal to numbers.
4. Squares and roots equal to numbers, e.g. x² + 10x = 39.
5. Squares and numbers equal to roots, e.g. x² + 21 = 10x.
6. Roots and numbers equal to squares, e.g. 3x + 4 = x².

Al-Khwarizmi gives the rule for solving each type of equation, essentially the familiar quadratic formula given for a numerical example in each case, and then a proof for each example which is a geometrical completing the square.

Abraham bar Hiyya Ha-Nasi, often known by the Latin name Savasorda, is famed for his book Liber embadorum published in 1145 which is the first book published in Europe to give the complete solution of the quadratic equation.

A new phase of mathematics began in Italy around 1500. In 1494 the first edition of Summa de arithmetica, geometrica, proportioni et proportionalita, now known as the Suma, appeared. It was written by Luca Pacioli although it is quite hard to find the author's name on the book, Fra Luca appearing in small print but not on the title page. In many ways the book is more a summary of knowledge at the time and makes no major advances. The notation and setting out of calculations is almost modern in style:

Quadratics are equations of the second degree, having the form ax^2+bx+c=0, for a,b,c constant. They, or their equivalents, have been around for at least 4,000 years. The solution, which we know today as the "quadratic formula," has been around for a few hundred years in the exact form we use today but took thousands of years to formulate. Impediments included formulations that were largely geometric instead of numerical and unwillingness to recognize a second root, irrational numbers, or even negatives.

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Islamic Mathematics

Al-Khawrizmi

The Islamic Empire established across Persia, the Middle East, Central Asia, North Africa, Iberia, and in parts of India in the 8th century made significant contributions towards mathematics. Although most Islamic texts on mathematics were written in Arabic, most of them were not written by Arabs, since much like the status of Greek in the Hellenistic world, Arabic was used as the written language of non-Arab scholars throughout the Islamic world at the time. Persians contributed to the world of Mathematics alongside Arabs.

In the 9th century, the Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī wrote several important books on the Hindu-Arabic numerals and on methods for solving equations. His book On the Calculation with Hindu Numerals, written about 825, along with the work of Al-Kindi, were instrumental in spreading Indian mathematics and Indian numerals to the West. The word algorithm is derived from the Latinization of his name, Algoritmi, and the word algebra from the title of one of his works, Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa’l-muqābala (The Compendious Book on Calculation by Completion and Balancing). He gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots,[107] and he was the first to teach algebra in an elementary form and for its own sake.[108] He also discussed the fundamental method of "reduction" and "balancing", referring to 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. This is the operation which al-Khwārizmī originally described as al-jabr.[109] His algebra was also no longer concerned "with a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study." He also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems."[110]
Further developments in algebra were made by Al-Karaji in his treatise al-Fakhri, where he extends the methodology to incorporate integer powers and integer roots of unknown quantities. Something close to a proof by mathematical induction appears in a book written by Al-Karaji around 1000 AD, who used it to prove the binomial theorem, Pascal's triangle, and the sum of integral cubes.[111] The historian of mathematics, F. Woepcke,[112] praised Al-Karaji for being "the first who introduced the theory of algebraic calculus." Also in the 10th century, Abul Wafa translated the works of Diophantus into Arabic. Ibn al-Haytham was the first mathematician to derive the formula for the sum of the fourth powers, using a method that is readily generalizable for determining the general formula for the sum of any integral powers. He performed an integration in order to find the volume of a paraboloid, and was able to generalize his result for the integrals of polynomials up to the fourth degree. He thus came close to finding a general formula for the integrals of polynomials, but he was not concerned with any polynomials higher than the fourth degree.[113]
In the late 11th century, Omar Khayyam wrote Discussions of the Difficulties in Euclid, a book about what he perceived as flaws in Euclid's Elements, especially the parallel postulate. He was also the first to find the general geometric solution to cubic equations. He was also very influential in calendar reform.[citation needed]
In the 13th century, Nasir al-Din Tusi (Nasireddin) made advances in spherical trigonometry. He also wrote influential work on Euclid's parallel postulate. In the 15th century, Ghiyath al-Kashi computed the value of π to the 16th decimal place. Kashi also had an algorithm for calculating nth roots, which was a special case of the methods given many centuries later by Ruffini and Horner.
Other achievements of Muslim mathematicians during this period include the addition of the decimal point notation to the Arabic numerals, the discovery of all the modern trigonometric functions besides the sine, al-Kindi's introduction of cryptanalysis and frequency analysis, the development of analytic geometry by Ibn al-Haytham, the beginning of algebraic geometry by Omar Khayyam and the development of an algebraic notation by al-Qalasādī.[114]
During the time of the Ottoman Empire and Safavid Empire from the 15th century, the development of Islamic mathematics became stagnant.

Abu Ja'far Muhammad ibn Musa Al-Khwarizmi

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Born: about 780 in possibly Baghdad (now in Iraq)
Died: about 850

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We know few details of Abu Ja'far Muhammad ibn Musa al-Khwarizmi's life. One unfortunate effect of this lack of knowledge seems to be the temptation to make guesses based on very little evidence. In [1] Toomer suggests that the name al-Khwarizmi may indicate that he came from Khwarizm south of the Aral Sea in central Asia. He then writes:-

But the historian al-Tabari gives him the additional epithet "al-Qutrubbulli", indicating that he came from Qutrubbull, a district between the Tigris and Euphrates not far from Baghdad, so perhaps his ancestors, rather than he himself, came from Khwarizm ... Another epithet given to him by al-Tabari, "al-Majusi", would seem to indicate that he was an adherent of the old Zoroastrian religion. ... the pious preface to al-Khwarizmi's "Algebra" shows that he was an orthodox Muslim, so Al-Tabari's epithet could mean no more than that his forebears, and perhaps he in his youth, had been Zoroastrians.

However, Rashed [7], put a rather different interpretation on the same words by Al-Tabari:-

... Al-Tabari's words should read: "Muhammad ibn Musa al-Khwarizmi and al-Majusi al-Qutrubbulli ...", (and that there are two people al-Khwarizmi and al-Majusi al-Qutrubbulli): the letter "wa" was omitted in the early copy. This would not be worth mentioning if a series of conclusions about al-Khwarizmi's personality, occasionally even the origins of his knowledge, had not been drawn. In his article ([1]) G J Toomer, with naive confidence, constructed an entire fantasy on the error which cannot be denied the merit of making amusing reading.

This is not the last disagreement that we shall meet in describing the life and work of al-Khwarizmi. However before we look at the few facts about his life that are known for certain, we should take a moment to set the scene for the cultural and scientific background in which al-Khwarizmi worked.

Harun al-Rashid became the fifth Caliph of the Abbasid dynasty on 14 September 786, about the time that al-Khwarizmi was born. Harun ruled, from his court in the capital city of Baghdad, over the Islam empire which stretched from the Mediterranean to India. He brought culture to his court and tried to establish the intellectual disciplines which at that time were not flourishing in the Arabic world. He had two sons, the eldest was al-Amin while the younger was al-Mamun. Harun died in 809 and there was an armed conflict between the brothers.

Al-Mamun won the armed struggle and al-Amin was defeated and killed in 813. Following this, al-Mamun became Caliph and ruled the empire from Baghdad. He continued the patronage of learning started by his father and founded an academy called the House of Wisdom where Greek philosophical and scientific works were translated. He also built up a library of manuscripts, the first major library to be set up since that at Alexandria, collecting important works from Byzantium. In addition to the House of Wisdom, al-Mamun set up observatories in which Muslim astronomers could build on the knowledge acquired by earlier peoples.

Al-Khwarizmi and his colleagues the Banu Musa were scholars at the House of Wisdom in Baghdad. Their tasks there involved the translation of Greek scientific manuscripts and they also studied, and wrote on, algebra, geometry and astronomy. Certainly al-Khwarizmi worked under the patronage of Al-Mamun and he dedicated two of his texts to the Caliph. These were his treatise on algebra and his treatise on astronomy. The algebra treatise Hisab al-jabr w'al-muqabala was the most famous and important of all of al-Khwarizmi's works. It is the title of this text that gives us the word "algebra" and, in a sense that we shall investigate more fully below, it is the first book to be written on algebra.

Rosen's translation of al-Khwarizmi's own words describing the purpose of the book tells us that al-Khwarizmi intended to teach [11] (see also [1]):-

... what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, lawsuits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computations, and other objects of various sorts and kinds are concerned.

This does not sound like the contents of an algebra text and indeed only the first part of the book is a discussion of what we would today recognise as algebra. However it is important to realise that the book was intended to be highly practical and that algebra was introduced to solve real life problems that were part of everyday life in the Islam empire at that time. Early in the book al-Khwarizmi describes the natural numbers in terms that are almost funny to us who are so familiar with the system, but it is important to understand the new depth of abstraction and understanding here [11]:-

When I consider what people generally want in calculating, I found that it always is a number. I also observed that every number is composed of units, and that any number may be divided into units. Moreover, I found that every number which may be expressed from one to ten, surpasses the preceding by one unit: afterwards the ten is doubled or tripled just as before the units were: thus arise twenty, thirty, etc. until a hundred: then the hundred is doubled and tripled in the same manner as the units and the tens, up to a thousand; ... so forth to the utmost limit of numeration.

Having introduced the natural numbers, al-Khwarizmi introduces the main topic of this first section of his book, namely the solution of equations. His equations are linear or quadratic and are composed of units, roots and squares. For example, to al-Khwarizmi a unit was a number, a root was x, and a square was x². However, although we shall use the now familiar algebraic notation in this article to help the reader understand the notions, Al-Khwarizmi's mathematics is done entirely in words with no symbols being used.

He first reduces an equation (linear or quadratic) to one of six standard forms:

1. Squares equal to roots.
2. Squares equal to numbers.
3. Roots equal to numbers.
4. Squares and roots equal to numbers; e.g. x² + 10 x = 39.
5. Squares and numbers equal to roots; e.g. x² + 21 = 10 x.
6. Roots and numbers equal to squares; e.g. 3 x + 4 = x².

The reduction is carried out using the two operations of al-jabr and al-muqabala. Here "al-jabr" means "completion" and is the process of removing negative terms from an equation. For example, using one of al-Khwarizmi's own examples, "al-jabr" transforms x² = 40 x - 4 x²into 5 x² = 40 x. The term "al-muqabala" means "balancing" and is the process of reducing positive terms of the same power when they occur on both sides of an equation. For example, two applications of "al-muqabala" reduces 50 + 3 x + x² = 29 + 10 x to 21 + x² = 7 x (one application to deal with the numbers and a second to deal with the roots).

Al-Khwarizmi then shows how to solve the six standard types of equations. He uses both algebraic methods of solution and geometric methods. For example to solve the equation x² + 10 x = 39 he writes [11]:-

... a square and 10 roots are equal to 39 units. The question therefore in this type of equation is about as follows: what is the square which combined with ten of its roots will give a sum total of 39? The manner of solving this type of equation is to take one-half of the roots just mentioned. Now the roots in the problem before us are 10. Therefore take 5, which multiplied by itself gives 25, an amount which you add to 39 giving 64. Having taken then the square root of this which is 8, subtract from it half the roots, 5 leaving 3. The number three therefore represents one root of this square, which itself, of course is 9. Nine therefore gives the square.

The geometric proof by completing the square follows. Al-Khwarizmi starts with a square of side x, which therefore represents x² (Figure 1). To the square we must add 10x and this is done by adding four rectangles each of breadth 10/4 and length x to the square (Figure 2). Figure 2 has area x² + 10 xwhich is equal to 39. We now complete the square by adding the four little squares each of area ⁵/₂ × ⁵/₂= ²⁵/₄. Hence the outside square in Fig 3 has area 4 × ²⁵/₄ + 39 = 25 + 39 = 64. The side of the square is therefore 8. But the side is of length ⁵/₂ + x + ⁵/₂so x + 5 = 8, giving x = 3.

These geometrical proofs are a matter of disagreement between experts. The question, which seems not to have an easy answer, is whether al-Khwarizmi was familiar with Euclid'sElements. We know that he could have been, perhaps it is even fair to say "should have been", familiar with Euclid's work. In al-Rashid's reign, while al-Khwarizmi was still young, al-Hajjaj had translated Euclid's Elements into Arabic and al-Hajjaj was one of al-Khwarizmi's colleagues in the House of Wisdom. This would support Toomer's comments in [1]:-

... in his introductory section al-Khwarizmi uses geometrical figures to explain equations, which surely argues for a familiarity with Book II of Euclid's "Elements".

Rashed [9] writes that al-Khwarizmi's:-

... treatment was very probably inspired by recent knowledge of the "Elements".

However, Gandz in [6] (see also [23]), argues for a very different view:-

Euclid's "Elements" in their spirit and letter are entirely unknown to [al-Khwarizmi]. Al-Khwarizmi has neither definitions, nor axioms, nor postulates, nor any demonstration of the Euclidean kind.

I [EFR] think that it is clear that whether or not al-Khwarizmi had studied Euclid's Elements,he was influenced by other geometrical works. As Parshall writes in [35]:-

... because his treatment of practical geometry so closely followed that of the Hebrew text, Mishnat ha Middot, which dated from around 150 AD, the evidence of Semitic ancestry exists.

Al-Khwarizmi continues his study of algebra in Hisab al-jabr w'al-muqabala by examining how the laws of arithmetic extend to an arithmetic for his algebraic objects. For example he shows how to multiply out expressions such as

(a + b x) (c + d x)

although again we should emphasise that al-Khwarizmi uses only words to describe his expressions, and no symbols are used. Rashed [9] sees a remarkable depth and novelty in these calculations by al-Khwarizmi which appear to us, when examined from a modern perspective, as relatively elementary. He writes [9]:-

Al-Khwarizmi's concept of algebra can now be grasped with greater precision: it concerns the theory of linear and quadratic equations with a single unknown, and the elementary arithmetic of relative binomials and trinomials. ... The solution had to be general and calculable at the same time and in a mathematical fashion, that is, geometrically founded. ... The restriction of degree, as well as that of the number of unsophisticated terms, is instantly explained. From its true emergence, algebra can be seen as a theory of equations solved by means of radicals, and of algebraic calculations on related expressions...

If this interpretation is correct, then al-Khwarizmi was as Sarton writes:-

... the greatest mathematician of the time, and if one takes all the circumstances into account, one of the greatest of all time....

Arabic/Islamic mathematicians in our archive in chronological order
790  al-Khwarizmi 800  Al-Jawhari 805  al-Kindi 808  Hunayn 810  Banu Musa, Ahmad 810  Banu Musa, al-Hasan 810  Banu Musa, Muhammad 820  Al-Mahani 826  Thabit 835  Ahmed 850  Abu Kamil 850  al-Battani 875  Al-Nayrizi 880  Sinan 900  Al-Khazin	908  Ibrahim 920  al-Uqlidisi 940  Abu'l-Wafa 940  Al-Khujandi 940  al-Quhi 945  al-Sijzi 950  Yunus 953  Al-Karaji 965  al-Haitam 970  Mansur 973  al-Biruni 980  al-Baghdadi 980  Avicenna 989  Al-Jayyani 1010  Al-Nasawi	1048  Khayyam 1100  Aflah 1130  al-Samawal 1135  al-Tusi, Sharaf 1201  al-Tusi, Nasir 1220  al-Maghribi 1250  al-Samarqandi 1256  al-Banna 1260  al-Farisi 1320  al-Khalili 1364  Qadi Zada 1390  al-Kashi 1393  Ulugh Beg 1400  al-Umawi 1412  al-Qalasadi

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الجبرا

الجبرا

آزاد دائرۃ المعارف، ویکیپیڈیا سے

ریاضیات کی ایک شاخ الجبرا جس میں مطالعہ کیا جاتا ہے ریاضیاتی عالجوں کا، اور وہ اشیاء جو ان سے بنائی جا سکتی ہیں، جس میں اصطلاحات، کثیر رقمی، مساوات، اور الجبرائی ساختیں شامل ہیں۔ ہندسہ، تحلیل، وضعیت، تراکیب، اور نظریہ عدد، کے ساتھ الجبرا خالص ریاضی کا بڑا حصہ ہے۔

ثانوی تعلیم میں ابتدائی الجبرا نصاب کا حصہ ہوتا ہے جس میں اعداد کی نمائندگی کرنے والے متغیر کا تعارف کرایا جاتا ہے۔ ان متغیر پر مبنی بیانات کو عالجی قواعد، جیسا کہ جمع، کے زریعہ برتا جاتا ہے۔ یہ متنوع وجوہات کے لیے کیا جاتا ہے، جیسا کہ مساوات کے حل کرنے کے لیے۔

الجبرا کا شعبہ ابتدائی الجبرا سے بہت وسیع ہے، اور اس میں مختلف عالجی قواعد کا مطالعہ کیا جاتا ہے، اور یہ دیکھا جاتا ہے کہ کیا ہوتا ہے جب عالج اختراع کیے جائیں اور ان کا اعداد کے علاوہ اشیاء پر اطلاق کیا جاتا ہے۔ جمع اور ضرب کے عالجوں کو جامعاتی شکل دی جاتی ہے اور ان کی ٹھیک تعاریف الجبرائی ساختوں کی طرف لے جاتی ہیں، جیسا کہ گروہ، حلقۂ اور میدان۔

تاریخ [ترمیم]

الخوارزمی کی کتاب الكتاب المختصر في حساب الجبر والمقابلة کا ایک صفحہ

لفظ الجبرا عربی الجبر سے اخذ ہے، اور یہ الخوارزمی کی کتاب "الكتاب المختصر في حساب الجبر والمقابلة" سے لیا گیا ہے۔ یہ کتاب عرب ریاضیدان نے 820ء میں لکھی۔ الخوارزمی کو پدرِ الجبرا کہا جاتا ہے۔[1] اس تناظر میں لفظ الجبر کا مطلب "اتحاد مکرر" یا "بجالی" ہے۔ الخوارزمی نے گھٹاؤ اور ترازو متعارف کرایا (تفریق کی گئی اصطلاحات کو مساوات کی دوسری طرف لے جانا، یعنی، ہم مشابہ اصطلاحات کا مخالف اطراف میں کاٹنا ) جس کی طرف لفظ الجبر اصل طور پر اشارہ کرتا تھا۔[2] الخوارزمی نے چکوری مساوات کے حل کرنے کے طریقہ تفصیلی بھی بیان کیا،[3] جس کے ساتھ ہندساتی ثبوت دیے، اور اس کے ساتھ ساتھ الجبرا کو آزاد شعبہ کے طور سلوک کرتے ہوئے بیان کیا۔[4] "اس کا الجبرا مسائل کا سلسلہ حل کرنے سے متعلق نہیں تھا، بلکہ ایک توضیح تھا جو اولیٰ اصطلاحات سے شروع کر کے، اور جب ان اصطلاحات کے تولیفات ضرور مساوات کے لیے تمام ممکنہ نموزج دیں، اور اسطرح شئے کا سچا مطالعہ تشکیل دیں۔" اس نے مساوات کو ان کے اپنے واسطے مطالعہ کیا، "جامع طور پر، اور یہ سادہ طریقہ سے کسی مسئلہ کے حل کے دوران پیدا نہیں ہوتیں، بلکہ مسائل کی ایک لامتناہی جماعت کو تعریف کرنے کے لیے۔"[5]

فارسی ریاضیدان عمر خیام نے الجبرائی ہندسہ کی بنیاد استوار کی اور مکعب مساوات کا ہندساتی حل پیش کیا۔ ایک اور فارسی ریاضی دان شرف الدین الطوسی نے مکعب مساوات کے عددی اور الجبرائی حل مختلف ماجروں میں ڈھونڈے۔ [6] اس نے دالہ کا تخیل بھی پیش کیا۔ [7]

جماعت بندی [ترمیم]

الجبرا کو ان زمرہ جات میں تقسیم کیا جا سکتا ہے:

ابتدائی الجبرا، جس میں حقیقی اعداد پر عالج کے خاصوں کو لکھا جاتا ہے، علامات کو دائم اور متغیر کے لیے "جگہ پکڑ" استعمال کرتے ہوئے، اور ریاضیاتی اظہار اور مساوات پر لاگو قواعد کا مطالعہ کیا جاتا ہے جن میں یہ علامات استعمال ہوتی ہیں۔ عام طور پر یہ مدرسہ میں "الجبرا" کے نام سے پڑھایا جاتا ہے۔

تجریدی الجبرا، جسے مُتَاَخِّر الجبرا بھی کہتے ہیں، میں الجبرائی ساختیں جیسا کہ گروہ، حلقہ اور میدان کو مسلماتی تعریف اور تشویش کیا جاتا ہے۔

لکیری الجبرا، میں لکیری فضاء اور میٹرکس کے خاصوں کا مطالعہ کیا جاتا ہے۔

کائناتی الجبرا، جس میں تمام الجبرائی ساختوں میں مشترک خاصوں کو پڑھا جاتا ہے۔

نظریہ الجبرائی عدد، جس میں اعداد کے خاصوں کو الجبرائی نظام کے زریعہ پڑھا جاتا ہے۔

الجبرائی ہندسہ، ہندسہ اپنے الجبرائی پہلو سے۔

الجبرائی تالیفیات، جس میں الجبرائی طریقوں سے تولیفیاتی سوالوں کا مطالعہ کیا جاتا ہے۔

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Maths History

People seem compelled to organize. They also have a practical need to count certain things: cattle, cornstalks, and so on. There is the need to deal with simple geometrical situations in providing shelter and dealing with land. Once some form of writing is added into the mix, mathematics cannot be far behind. It might even be said that the symbolic approach precedes and leads to the invention of writing.
Archaeologists, anthropologists, linguists and others studying early societies have found that number ideas evolve slowly. There will typically be a different word or symbol for two people, two birds, or two stones. Only slowly does the idea of 'two' become independent from the things that there are two of. Similarly, of course, for other numbers. In fact, specific numbers beyond three are unknown in some lesser developed languages. A bit of this usage hangs on in our modern English when we speak, for example, of a flock of geese, but a school of fish.
The Maya, the Chinese, the Civilization of the Indus Valley, the Egyptians, and the region of Mesopotamia between the Tigris and Euphrates rivers -- all had developed impressive bodies of mathematical knowledge by the dawn of their written histories. In each case, what we know of their mathematics comes from a combination of archaeology, the references of later writers, and their own written record.
Mathematical documents from Ancient Egypt date back to 1900 B.C. The practical need to redraw field boundaries after the annual flooding of the Nile, and the fact that there was a small leisure class with time to think, helped to create a problem oriented, practical mathematics. A base-ten numeration system was able to handle positive whole numbers and some fractions. Algebra was developed only far enough to solve linear equations and, of course, calculate the volume of a pyramid. It is thought that only special cases of The Pythagorean Theorem were known; ropes knotted in the ratio 3:4:5 may have been used to construct right angles.
What we know of the mathematics of Mesopotamia comes from cuneiform writing on clay tablets which date back as far as 2100 B.C. Sixty was the number system base -- a system that we have inherited and preserve to this day in our measurement of time and angles. Among the clay tablets are found multiplication tables, tables of reciprocals, squares and square roots. A general method for solving quadratic equations was available, and a few equations of higher degree could be handled. From what we can see today, both the Egyptians and the Mesopotamians (or Babylonians) stuck to specific practical problems; the idea of stating and proving general theorems did not seem to arise in either civilization.
Chinese mathematics -- a vast and powerful body of knowledge --, although mainly practical and problem oriented, did contain general statements and proofs. A method similar to Gaussian Reduction with back-substitution for solving systems of linear equations was known two thousand years earlier in China than in the West. The value ofp was known to seven decimal places by 500 A.D., far in advance of the West.
In India mathematics was also mainly practical. Methods of solving equations were largely centered around problems in astronomy. Negative and irrational numbers were used. Of course, India is noted for developing the concept of zero, that was passed into Western mathematics via the Arabic tradition, and is so important as a place holder in our modern decimal number system.
The Classic Maya civilization (250 BC to 900 AD) also developed the zero and used it as a place holder in a base-twenty numeration system. Again, astronomy played a central role in their religion and motivated them to develop mathematics. It is noteworthy that the Maya calendar was more accurate than the European at the time the Spanish landed in The Yukatan Peninsula.
Ancient Greece
The axiomatic method came into full force in Ancient Greek times; it has characterized mathematics ever since. Geometry was center stage in ancient times. Mathematical models, or idealizations of the real world, were built around points, lines, and planes. Numbers were represented as lengths of line segments. Modern mathematics still relies on the axiomatic method, but tends to be more algebraically based.
Key to the axiomatic method are abstraction and proof. For example, the idea of a point as a pure location with no extension is an abstraction since a point cannot physically exist. A dot differs from a point in that a dot has extension, and represents only an approximate location. Nevertheless, since they can be seen, we use dots to represent points which cannot be seen. Lines, planes and circles are also abstract ideas. That is, they represent idealizations, rather than concrete objects which actually exist. After all, a plane has no thickness, and cannot be anything except a boundary between two regions in space.
An interest in investigating the properties of abstract objects characterizes Greek mathematics. Precise definitions; a small number of commonly accepted assumptions called axioms or postulates are made; then general results (lemmas, theorems, and corollaries) are proved using logic.

The Middle Ages

In 476 A.D. The Roman Empire came to an end in the West; the last author of mathematical textbooks, Boethius, was executed in 524; the Eastern Roman Emperor, Justinian, closed the academies in Athens in 529 -- The Middle Ages had been born -- Mathematics, along with the rest of scholarly life, would fall into a decline which would last 1000 years.
Fortunately, during this period Chinese mathematics, the mathematics of India and The Arabic World would continue to flourish. Our modern base-ten number system featuring zero as a place holder was developed in the Eighth Century in India. The basis for algebra was developed in The Arabic World in the Eighth and Ninth Centuries. In fact, the word algebra comes from the Arabic al-jabr which refers to transposing a quantity from one side of an equation to the other.
One of the few bright spots in European mathematics during this period was the work of Fibonacci (1175-1250 A.D). He was the son of an Italian merchant who traveled widely and studied under a Muslim teacher. He helped to open Europe to the Arabic mathematical methods, including the use of 'Arabic Numerals,' which actually were invented in India, as we have seen. Many cegep students will have studied the Fibonacci sequence which has broad use in far-flung areas of mathematics.
By about 1500 A.D. the intellectual climate of Europe was changing. The Middle Ages were coming to an end and the Modern World was being born. Each century from that time until the present day would see the creation of powerful, new, mathematics.
The Sixteenth Century
The 1500's saw the emergence of what we recognize as mathematics in the modern world as opposed to the geometrical discussions of ancient times. Negative numbers were slowly gaining acceptance; the + and - signs made their debut; in accounting, Arabic numerals and double-entry book keeping came into use. Many people contributed to the more symbolic, algebraically based, mathematics that was coming into being.
Girolamo Cardano published his Ars Magne in 1545. In this work he presented for the first time a general solution of the cubic equation, and some special cases of the quartic, or fourth-degree polynomial, equation. This sparked a great deal of enthusiasm and the impetus lasted for centuries as mathematicians tried to solve fifth, and higher, degree equations in a general way. Only in the 1820's did Galois and Abel show that a general solution of fifth and higher degree equations was not possible. In the mean time a great deal of fresh mathematics was spun off as a by-product of the quest. Cardano, himself, lead an outrageous life full enough for several good biographies.
François Viète developed the first system of symbolic algebra. He introduced the use of braces and parentheses, and used the + and - signs along with a number of abbreviations for other operations. He was the first to make the crucial distinction between variable quantities and constant unknowns. In his work is found most of the methods of ordinary algebra as we know it today. He even foresaw the invention of logarithms in the next century by using the trig identity
sin a + sin b = 2 sin(a+b)/2 cos(a-b)/2
The Seventeenth Century
However, the two big new ideas are Descartes' founding of analytic geometry, or geometry based on algebra, and the simultaneous but independent invention of calculus by Newton and Leibniz. This is also the time when Fermat proposed his famous "Last Theorem" which has only just been proved by Andrew Wiles in the 1990's. As with the quest to find a general solution for the fifth degree (quintic) polynomial equation, the challenge presented by Fermat occupied great minds over a period of centuries and produced enormously rich benefits, but no solution until recently. Fermat's main contribution to mathematics was, however, the founding of number theory -- that branch of mathematics which deals with the arithmetic properties of the natural numbers.The 1600's were an especially high point in scientific and mathematical history. This is the century of Kepler, Galileo, Descartes, Newton, and Leibniz . But, one of the most exciting advancements of the time was the introduction of Logarithms in 1614 by John Napier. Logarithms greatly reduced the labour involved in calculations, and was welcomed by a wider public than most mathematical ideas.
Blaise Pascal worked closely with Fermat on number theory and also founded probability as we now know it. His name is commemorated in Pascal's Triangle as well as the Pascal programming language. In England, John Wallis, developed the analytic approach to the conic sections, an area dear to the hearts of many students even today. The Binomial Theorem, another favourite which dates to this period, was introduced by Newton himself. The Seventeenth Century, along with the works of Ancient Greece, establish the roots of the mathematical tradition which lives to the present day.
The Eighteenth Century

With calculus at its center, an ever widening body of knowledge began to take shape. The frontiers of mathematics in the Eighteenth Century included differential equations, infinite series, the study of planetary orbits, the theory of numbers, solutions to algebraic equations, probability theory, and complex numbers. It is possible here only to mention a few of the key figures involved.
Joseph Louis Lagrange (1736-1813) may have been France's greatest mathematician of the century. Like several of the top mathematicians of the era he was appointed to the Berlin Academy as Court Mathematician to Frederic the Great. A shy and quiet man, he extended greatly our understanding of solutions to algebraic equations, and of planetary orbits. Another great French mathematician was Pierre Simon de Laplace (1749-1827.) A more outgoing and practical person than Lagrange, his greatest contribution may have been in the area of probability theory. Laplace treated mathematics as a tool, as a means to an end; whereas Lagrange considered mathematics as a thing of great beauty -- like poetry -- and created mathematics as an end in itself. Another important figure was D'Alembert (1717-1783), who did significant work in differential equations, sequences and series, mechanics, and astronomy; he was a member of the French Academie des Sciences.
At some point mention should be made of the Bernoulli's. This Swiss family produced at least thirteen mathematicians over a period of two centuries. In the Eighteenth Century two brothers, Jacob and Johann, played a major role. Another Bernoulli, Jean, tutored Leonard Euler (1707-1783) who was to become, without doubt, the century's greatest mathematician. Euler divided his career between the court of Frederic the Great, at Berlin, and the Russian Academy, at Saint Petersberg. His personal life was quiet and apparently happy despite much hardship, including blindness in old age. Amazingly, he was able to continue doing original mathematics even after the loss of his eyesight by dictating his work to others. Euler's output was truly amazing; his collected works fill 74 volumes. And, he frequently held back his results so that others could claim some credit. Besides developing the use of complex numbers and founding what we now know as topology, he introduced many of today's familiar notations, including p, S, e, log x, sin x, cos x, f(x) for functions, and others.
The Nineteenth Century

By the 1820's Augustin-Louis Cauchy (1789-1857) could state a formal definition for the limit equivalent to the modern d, e-definition; this advancement soothed centuries of haggling over the true meaning of differentials. Modern abstract algebra was getting started with the invention of group theory by Evariste Galois (1811-1832) and Niels Abel (1802-1829). Galois and Abel gave us some of the most beautiful mathematics ever written, but both men lead tragic lives -- Galois was killed in a senseless duel at age 22, and Abel died of disease and starvation brought on by extreme poverty at age 27.
Karl Frederic Gauss (1777-1854) was the greatest mathematician of the nineteenth century, and one of the greatest of all times. Despite having been born into a working-class family with a father who did not value education, he got his doctorate proving in his thesis The Fundamental Theorem of Algebra which states that an nth-degree polynomial has n roots in complex numbers.
Apart from extending calculus to the complex numbers and developing more abstract algebras, mathematics branched out in various ways. Non-Euclidean geometry is the study of geometries which result from modifications of Euclid's axioms. Along with Reimann, the names of Lobachevsky and Bolyai are impotortant here. George Boole andGeorg Cantor were key in the foundations of set theory and mathematical logic. Karl Pearson (1857-1936) founded statistics as we know it today.
The Twentieth Century

By the beginning of The Twentieth Century mathematics had grown wide and deep, so vast that it is impossible to summarize the subject here. Let us just mention one thread.
The invention of mathematical logic lead to a deep analysis of the fundamentals underlying mathematics. It seemed that in the background there was a desire to mechanize intelligent thought itself. All of this was, of course, closely related to the impending introduction of computers. Then, in 1931, Kurt Gödel proved that statements can be formed that are neither provable nor disprovable in any complete and consistent axiom set. It follows that within a given mathematical system it is not possible to prove or disprove all of the statements that can be formed. Essentially, what Gödel did was to confirm that the human mind, and its spark of insight, can never be replaced by mechanical processes.
A few of the key figures in the subject this century are:

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What is Mathematics

Definitions:

Mathematics:

The study of the measurement, relationships, and properties of quantities and sets, using numbers and symbols. Arithmetic, algebra, geometry, and calculus are branches of mathematics.
The systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically.
A group of related sciences, including algebra, geometry, and calculus, concerned with the study of number, quantity, shape, and space and their interrelationships by using a specialized notation
mathematical operations and processes involved in the solution of a problem or study of some scientific field

deductive study of numbers, geometry, and various abstract constructs, or structures; the latter often "abstract" the features common to several models derived from the empirical, or applied, sciences, although many emerge from purely mathematical or logical considerations. Mathematics is very broadly divided into foundations, algebra, analysis, geometry, and applied mathematics, which includes theoretical computer science.

In the 17th century, the great scientist and mathematician Galileo Galilei noted that the book of nature "cannot be understood unless one first learns to comprehend the language and read the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures, without which it is not humanly possible to understand a single word of it." For at least 4,000 years of recorded history, humans have engaged in the study of mathematics. Our progress in this field is a gripping narrative, a never-ending search for hidden patterns in numbers, a philosopher's quest for the ultimate meaning of mathematical relationships, a chronicle of amazing progress in practical fields like engineering and economics, a tale of astonishing scientific discoveries, a fantastic voyage into realms of abstract beauty, and a series of fascinating personal profiles of individuals such as:
The "Queen of the Sciences"
The history of mathematics concerns one of the most magnificent, surprising, and powerful of all human achievements. In the early 19th century, the noted German mathematician Carl Friedrich Gauss called mathematics the "queen of the sciences" because it was so successful at uncovering the nature of physical reality. Gauss's observation is even more accurate in today's age of quantum physics, string theory, chaos theory, information technology, and other mathematics-intensive disciplines that have transformed the way we understand and deal with the world.

The Queen of the Sciences takes you from ancient Mesopotamia—where the Pythagorean theorem was already in use more than 1,000 years before the Greek thinker Pythagoras traditionally proved it—to the Human Genome Project, which uses sophisticated mathematical techniques to decipher the 3 billion letters of the human genetic code.

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