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THE FIRST MEMORY COMIC
Category - PRINT AND PUBLICATION
posted on 01-04-2010
World’s First Memory Comic is the first of its kind in the world where Neerja, the author has combined the ancient wisdom and modern science to create a comic, which can help a student to learn the memory technique, in a result they can memorize the entire dictionary

THE SMALLEST MANUSCRIPT
Category - PRINT AND PUBLICATION
posted on 01-04-2010
Ramchandra Karunakar (born on May 10, 1983) of Hardoi (Uttar Pradesh) has made smallest religious book, “Dhammapadam” of 20mm X 10mm X 10mm size. The total pages of the book are 104. He has also written this book in three types : opposite alphabet volume of 40mm X 15mm X 27mm size, whole alphabet volume of 29.7cm X 21cm

FIRST INDIAN TO WIN BOOKER PRIZE
Category - PRINT AND PUBLICATION
posted on 01-04-2010
Arundhati Roy (born on November 24, 1961) won the Booker prize for her brilliant and poetic novel “The God of Small Things” in the year 1997 making her the first Indian to receive the prestigious award.

EXPERT IN MIRROR WRITING
Category - PRINT AND PUBLICATION
posted on 01-04-2010
Piyush Dadriwala of Dadri (Uttar Pradesh) has written holy book “The Bhagwat Geeta” in mirror writing with 318,900 backward words

THE YOUNGEST AUTHOR ON ASTROLOGY
Category - PRINT AND PUBLICATION
posted on 01-04-2010
Mr. Praveen Kochar (born on December 28, 1983) of Faridabad ( Harayana) became the youngest author at the age of 22 yrs 8 months on the subject of Astrology by launching his first book in Hindi- “Jyotish Ek Samadhan” on August 28, 2006.

FIRST INDIAN TO WIN BEST OF BOOKERS
Category - PRINT AND PUBLICATION
posted on 01-04-2010
Midnight’s Children by Sir Salman Rushdie won the “Best of the Booker prize” based on an online poll on July 10, 2008. The Best of the Booker is a special prize awarded in commemoration of the Booker Prize’s 40th anniversary. In 1993, the Booker of Bookers Prize was awarded to Salman Rushdie for Midnight’s Children (the 1981 winner), as the best novel to win the award in the first 25 years of its existence

A VERSATILE CARTOONIST
Category - CREATIVITY
posted on 01-04-2010
B.V Panduranga Rao (born on September 20, 1944) of Bangalore (Karnataka) got his cartoons recognized 25 times at International level in cartoon contest, festivals and exhibitions. He first time participated in 1996 at the international cartoon contest held in Korea. He started his cartooning career in 1964 and also conducted 33 one man cartoon show and won 12 prizes at state as well as national level.

THE BIRTH OF FIRST CARTOON
Category - CREATIVITY
posted on 01-04-2010
Pran Kumar Sharma (born on August 15, 1938) of Pakistan, now moved to Gwalior (India) created a comic character. Chacha Chaudhary. It is the most popular Indian comic. This has more than 10 million readers in newspapers and comic books in ten languages. Chacha Chaudhary was created in 1969 and is the first Indian comic character to be made into a TV serial (in 2002).

FIRST ON LINE ANIMATED HUMOUR
Category - CREATIVITY
posted on 01-04-2010
Bijan Samaddar and Aditya Dubey, two animation professional from Ghazidabad (Uttar Pradesh) had started the site praaji.com in a spring of 2004, motivated by a western site joecartoon.com, they thought to develop a site which will be completely animated humour. In a small room in Surya Nagar, Ghaziabad they started their production. After two years with lots of research and development they launched the site for public viewing. Their aim is to bring smile on the face of the visitors.

THE LONGEST RUNNING
Category - CREATIVITY
posted on 01-04-2010
Vasantha Hosabettu (born on 21 May 1965) of Bangalore (Karnataka) wrote a longest running weekly column called Wa Re Wah…! which was written only about cartooning, cartoonist and cartoon. It was published in Kannada Prabha, Kannada Daily from the New Indian Express Group. This column appeared without a break between December 1, 2004 to February 11, 2009, on every Wednesday. These columns have also come out into two books, “Wa Re Wah…!” and “Munduvarida Wa Re Wah…!” (collection of 101 columns each) published by Sadana Prakashana- Bangalore.

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November 03, 2009
The Amazing Number 9 and the Mathematical Finger Print of God.

I am going to share what a Beautiful Number 9 is. This has been personified by the works of a mathematician Marko Rodin who calls his work Vortex Based Mathematics.Vortex-Based Mathematics (VBM) is completely different because it is a dynamic math that shows the relationships and thus the qualities of numbers rather than the quantities.



From his website :-





Marko studied all the world's great religions. He decided to take The Most Great Name of Bahaullah (prophet of the Bahai Faith) which is Abha and convert it into numbers. He did this in an effort to discover the true precise mystical intonation of The Most Great Name of God. Since the Bahai sacred scripture was originally written in Persian and Arabic, Marko used the Abjad numerical notation system for this letter to number translation. This was a sacred system of allocating a unique numerical value to each letter of the 27 letters of the alphabet so that secret quantum mechanic physics could be encoded into words. What Marko discovered was that (A=1, b=2, h=5, a=1) = 9. The fact that The Most Great Name of God equaled 9 seemed very important to him as everything he had read in both the Bahai scriptures and other religious text spoke of nine being the omni-potent number. So next he drew out a circle with nine on top and 1 through 8 going around the circle clockwise. Then he discovered a very intriguing number system within this circle. Marko knew he had stumbled upon something very profound. This circle with its hidden number sequence was the "Symbol of Enlightenment." This is the MATHEMATICAL FINGER PRINT OF GOD.





Follow along as the amazing properties of this symbol unveil themselves to you. Put your pencil on number 1 and without picking up your pencil, move your pencil in a straight line to number 2, then 4, then across the center to 8. Notice that you are doubling. So next should be 16 and it is, but 1+6=7. So move your pencil to 7. Then 16 doubled is 32, but 3+2=5. So move your pencil to 5. Then 32 doubled is 64 and 6+4=10 and 1+0=1. And you're back to 1. So move the pencil across the center and back up to 1. The significance of the Mayan calender is that 64 is one complete cycle of infinity. Then it begins again with 64 doubled is 128 and 1+2+8=11, then 1+1=2. And so on. You will never get off this track as you keep doubling. Notice the infinity symbol has formed underneath your pencil, creating an ever-repeating pattern of 1, 2, 4, 8, 7, 5. Amazingly, this number sequence stays intact as you half numbers as well. Start again at the 1 but this time go backwards on the infinity symbol. Half of 1 is .5, so move your pencil to the 5. Then half of .5 is .25, and 2+5=7. So move your pencil to the 7. And half of .25 is .125 and 1+2+5=8. So move to the 8. Next half of .125 is .0625 and 0+6+2+5=13 and 1+3=4. So go across to the 4. And half of .0625 is .03125 and 0+3+1+2+5=11 and 1+1=2. So move to the 2. Forever staying on the route of 1,2,4,8,7,5 even backwards.

At this point some of you might be thinking, "What in the world do these number patterns have to do with real world applications?" These number groupings piece together into a jig-saw-like puzzle pattern that perfectly demonstrates the way energy flows. Our base-ten decimal system is not man made, rather it is created by this flow of energy. Amazingly, after twenty years of working with this symbol and collaborating with engineers and scientists, Marko discovered that the 1,2,4,8,7,5 was a doubling circuit for a very efficient electrical coil. There was still one more very important number pattern to be realized. On the MATHEMATICAL FINGER PRINT OF GOD notice how the 3, 9, and 6 is in red and does not connect at the base. That is because it is a vector. The 1,2,4,8,7,5 is the third dimension while the oscillation between the 3 and 6 demonstrates the fourth dimension, which is the higher dimensional magnetic field of an electrical coil. The 3, 9, and 6 always occur together with the 9 as the control. In fact, the Yin/Yang is not a duality but rather a trinary. This is because the 3 and 6 represent each side of the Yin/Yang and the 9 is the "S" curve between them. Everything is based on thirds. We think that the universe is based on dualities because we see the effects not the cause.


Clearly Marko has used the principle of Digit Sums and the Vedic Square to create something beautiful, which later looks like this.








More on Marko's Website http://rodin.freelancepartnership.com/content/view/7/26/


After watching Bizza's One Eye and Now Marko's Vortex I think 9 has to be looked into seriously to bring out all the facets of it.


Another Contributor , Piyush Dadriwala from Noida, India sends us this on the number 9's amazing properties.

AMAZING NUMBER NINE


It is very interesting to note that when you take any number of digits like 25 AND 32,
NOW YOU CAN WRITE THEM IN FOUR WAYS LIKE THAT,
25*32=800
25*23=575
52*23=1196
52*32=1664

NOW VERY AMAZING,SUBTRACT BIGGER ONE TO ANY LOWER,ONE BY ONE

1664-1196=468=4+6+8=­18=1+8=9
1664-575=1089=1+0+8+­9=18=1+8=9
1164-800=864=8+6+4=1­8=1+8=9
1196-575=621=6+2+1=9­­
1196-800=396=3+6+9=1­8=1+8=9
800-575=225=2+2+5=9

Always nine,it is amazing.
for any no of digits.it is always true!

Thanking you
Gaurav Tekriwal
www.vedicmathsinda.org

Algebra


Algebra is a branch of mathematics concerning the study of structure, relation and quantity. The name is derived from the treatise written in Arabic by the Persian mathematician, astronomer, astrologer and geographer, Muhammad bin Mūsā al-Khwārizmī titled Kitab al-Jabr wa-l-Muqabala (meaning "The Compendious Book on Calculation by Completion and Balancing"), which provided symbolic operations for the systematic solution of linear and quadratic equations.

Together with geometry, analysis, combinatorics, and number theory, algebra is one of the main branches of mathematics. Elementary algebra is often part of the curriculum in secondary education and provides an introduction to the basic ideas of algebra, including effects of adding and multiplying numbers, the concept of variables, definition of polynomials, along with factorization and determining their roots.

Algebra is much broader than elementary algebra and can be generalized. In addition to working directly with numbers, algebra covers working with symbols, variables, and set elements. Addition and multiplication are viewed as general operations, and their precise definitions lead to structures such as groups, rings and fields.

Classification
Algebra may be divided roughly into the following categories:

Elementary algebra, in which the properties of operations on the real number system are recorded using symbols as "place holders" to denote constants and variables, and the rules governing mathematical expressions and equations involving these symbols are studied (note that this usually includes the subject matter of courses called intermediate algebra and college algebra), also called second year and third year algebra;
Abstract algebra, sometimes also called modern algebra, in which algebraic structures such as groups, rings and fields are axiomatically defined and investigated; this includes, among other fields,
Linear algebra, in which the specific properties of vector spaces are studied (including matrices);
Universal algebra, in which properties common to all algebraic structures are studied.
Algebraic number theory, in which the properties of numbers are studied through algebraic systems. Number theory inspired much of the original abstraction in algebra.
Algebraic geometry in its algebraic aspect.
Algebraic combinatorics, in which abstract algebraic methods are used to study combinatorial questions.
In some directions of advanced study, axiomatic algebraic systems such as groups, rings, fields, and algebras over a field are investigated in the presence of a geometric structure (a metric or a topology) which is compatible with the algebraic structure. The list includes a number of areas of functional analysis:


Normed linear spaces
Banach spaces
Hilbert spaces
Banach algebras
Normed algebras
Topological algebras
Topological groups

Elementary algebra
Elementary algebra is the most basic form of algebra. It is taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of arithmetic. In arithmetic, only numbers and their arithmetical operations (such as +, −, ×, ÷) occur. In algebra, numbers are often denoted by symbols (such as a, x, or y). This is useful because:


It allows the general formulation of arithmetical laws (such as a + b = b + a for all a and b), and thus is the first step to a systematic exploration of the properties of the real number system.
It allows the reference to "unknown" numbers, the formulation of equations and the study of how to solve these (for instance, "Find a number x such that 3x + 1 = 10").
It allows the formulation of functional relationships (such as "If you sell x tickets, then your profit will be 3x - 10 dollars, or f(x) = 3x - 10, where f is the function, and x is the number to which the function is applied.").

Polynomials
A polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, and multiplication (where repeated multiplication of the same variable is standardly denoted as exponentiation with a constant positive whole number exponent). For example, x^2 + 2x -3, is a polynomial in the single variable x.

An important class of problems in algebra is factorization of polynomials, that is, expressing a given polynomial as a product of other polynomials. The example polynomial above can be factored as (x-1)(x+3),!. A related class of problems is finding algebraic expressions for the roots of a polynomial in a single variable.

Abstract algebra
Abstract algebra extends the familiar concepts found in elementary algebra and arithmetic of numbers to more general concepts.

Sets: Rather than just considering the different types of numbers, abstract algebra deals with the more general concept of sets: a collection of all objects (called elements) selected by property, specific for the set. All collections of the familiar types of numbers are sets. Other examples of sets include the set of all two-by-two matrices, the set of all second-degree polynomials (ax2 + bx + c), the set of all two dimensional vectors in the plane, and the various finite groups such as the cyclic groups which are the group of integers modulo n. Set theory is a branch of logic and not technically a branch of algebra.

Binary operations: The notion of addition (+) is abstracted to give a binary operation, * say. The notion of binary operation is meaningless without the set on which the operation is defined. For two elements a and b in a set S a*b gives another element in the set; this condition is called closure. Addition (+), subtraction (-), multiplication (×), and division (÷) can be binary operations when defined on different sets, as is addition and multiplication of matrices, vectors, and polynomials.

Identity elements: The numbers zero and one are abstracted to give the notion of an identity element for an operation. Zero is the identity element for addition and one is the identity element for multiplication. For a general binary operator * the identity element e must satisfy a * e = a and e * a = a. This holds for addition as a + 0 = a and 0 + a = a and multiplication a × 1 = a and 1 × a = a. However, if we take the positive natural numbers and addition, there is no identity element.

Inverse elements: The negative numbers give rise to the concept of inverse elements. For addition, the inverse of a is -a, and for multiplication the inverse is 1/a. A general inverse element a-1 must satisfy the property that a * a-1 = e and a-1 * a = e.

Associativity: Addition of integers has a property called associativity. That is, the grouping of the numbers to be added does not affect the sum. For example: (2+3)+4=2+(3+4). In general, this becomes (a * b) * c = a * (b * c). This property is shared by most binary operations, but not subtraction or division or octonion multiplication.

Commutativity: Addition of integers also has a property called commutativity. That is, the order of the numbers to be added does not affect the sum. For example: 2+3=3+2. In general, this becomes a * b = b * a. Only some binary operations have this property. It holds for the integers with addition and multiplication, but it does not hold for matrix multiplication or quaternion multiplication .

Groups—structures of a set with a single binary operation
Combining the above concepts gives one of the most important structures in mathematics: a group. A group is a combination of a set S and a single binary operation '*', defined in any way you choose, but with the following properties:


An identity element e exists, such that for every member a of S, e * a and a * e are both identical to a.
Every element has an inverse: for every member a of S, there exists a member a-1 such that a * a-1 and a-1 * a are both identical to the identity element.
The operation is associative: if a, b and c are members of S, then (a * b) * c is identical to a * (b * c).
If a group is also commutative - that is, for any two members a and b of S, a * b is identical to b * a – then the group is said to be Abelian.

For example, the set of integers under the operation of addition is a group. In this group, the identity element is 0 and the inverse of any element a is its negation, -a. The associativity requirement is met, because for any integers a, b and c, (a + b) + c = a + (b + c)

The nonzero rational numbers form a group under multiplication. Here, the identity element is 1, since 1 × a = a × 1 = a for any rational number a. The inverse of a is 1/a, since a × 1/a = 1.

The integers under the multiplication operation, however, do not form a group. This is because, in general, the multiplicative inverse of an integer is not an integer. For example, 4 is an integer, but its multiplicative inverse is 1/4, which is not an integer.

The theory of groups is studied in group theory. A major result in this theory is the classification of finite simple groups, mostly published between about 1955 and 1983, which is thought to classify all of the finite simple groups into roughly 30 basic types.


Examples
Set: Natural numbers mathbb{N} Integers mathbb{Z} Rational numbers mathbb{Q} (also real mathbb{R} and complex mathbb{C} numbers) Integers mod 3: {0,1,2}
Operation + × (w/o zero) + × (w/o zero) + − × (w/o zero) ÷ (w/o zero) + × (w/o zero)
Closed Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes
Identity 0 1 0 1 0 NA 1 NA 0 1
Inverse NA NA -a NA -a NA begin{matrix} frac{1}{a} end{matrix} NA 0,2,1, respectively NA, 1, 2, respectively
Associative Yes Yes Yes Yes Yes No Yes No Yes Yes
Commutative Yes Yes Yes Yes Yes No Yes No Yes Yes
Structure monoid monoid Abelian group monoid Abelian group quasigroup Abelian group quasigroup Abelian group Abelian group (mathbb{Z}_2)


Semigroups, quasigroups, and monoids are structures similar to groups, but more general. They comprise a set and a closed binary operation, but do not necessarily satisfy the other conditions. A semigroup has an associative binary operation, but might not have an identity element. A monoid is a semigroup which does have an identity but might not have an inverse for every element. A quasigroup satisfies a requirement that any element can be turned into any other by a unique pre- or post-operation; however the binary operation might not be associative. All are instance of groupoids, structures with a binary operation upon which no further conditions are imposed.

All groups are monoids, and all monoids are semigroups.

Rings and fields—structures of a set with two particular binary operations, (+) and (×)
Groups just have one binary operation. To fully explain the behaviour of the different types of numbers, structures with two operators need to be studied. The most important of these are rings, and fields.

Distributivity generalised the distributive law for numbers, and specifies the order in which the operators should be applied, (called the precedence). For the integers (a + b) × c = a×c+ b×c and c × (a + b) = c×a + c×b, and × is said to be distributive over +.

A ring has two binary operations (+) and (×), with × distributive over +. Under the first operator (+) it forms an Abelian group. Under the second operator (×) it is associative, but it does not need to have identity, or inverse, so division is not allowed. The additive (+) identity element is written as 0 and the additive inverse of a is written as −a.

The integers are an example of a ring. The integers have additional properties which make it an integral domain.

A field is a ring with the additional property that all the elements excluding 0 form an Abelian group under ×. The multiplicative (×) identity is written as 1 and the multiplicative inverse of a is written as a-1.

The rational numbers, the real numbers and the complex numbers are all examples of fields.

Objects called algebras
The word algebra is also used for various algebraic structures:


Algebra over a field or more generally Algebra over a ring
Algebra over a set
Boolean algebra
F-algebra and F-coalgebra in category theory
Sigma-algebra

History

The origins of algebra can be traced to the ancient Babylonians, who developed an advanced arithmetical system with which they were able to do calculations in an algebraic fashion. With the use of this system they were able to apply formulas and calculate solutions for unknown values for a class of problems typically solved today by using linear equations, quadratic equations, and indeterminate linear equations. By contrast, most Egyptians of this era, and most Indian, Greek and Chinese mathematicians in the first millennium BC, usually solved such equations by geometric methods, such as those described in the Rhind Mathematical Papyrus, Sulba Sutras, Euclid's Elements, and The Nine Chapters on the Mathematical Art. The geometric work of the Greeks, typified in the Elements, provided the framework for generalizing formulae beyond the solution of particular problems into more general systems of stating and solving equations.

Later, the Indian mathematicians developed algebraic methods to a high degree of sophistication. Although Diophantus and the Babylonians used mostly special ad hoc methods to solve equations, Brahmagupta was the first to solve equations using general methods. He solved the linear indeterminate equations, quadratic equations, second order indeterminate equations and equations with multiple variable.

The word "algebra" is named after the Arabic word "al-jabr" from the title of the book , meaning The book of Summary Concerning Calculating by Transposition and Reduction, a book written by the Persian mathematician Muhammad ibn Mūsā al-khwārizmī in 820. The word Al-Jabr means "reunion". The Hellenistic mathematician Diophantus has traditionally been known as "the father of algebra" but debate now exists as to whether or not Al-Khwarizmi should take that title. Those who support Al-Khwarizmi point to the fact that much of his work on reduction is still in use today and that he gave an exhaustive explanation of solving quadratic equations. Those who support Diophantus point to the fact that the algebra found in Al-Jabr is more elementary than the algebra found in Arithmetica and that Arithmetica is syncopated while Al-Jabr is fully rhetorical. Another Persian mathematician, Omar Khayyam, developed algebraic geometry and found the general geometric solution of the cubic equation. The Indian mathematicians Mahavira and Bhaskara II, and the Chinese mathematician Zhu Shijie, solved various cases of cubic, quartic, quintic and higher-order polynomial equations.

Another key event in the further development of algebra was the general algebraic solution of the cubic and quartic equations, developed in the mid-16th century. The idea of a determinant was developed by Japanese mathematician Kowa Seki in the 17th century, followed by Gottfried Leibniz ten years later, for the purpose of solving systems of simultaneous linear equations using matrices. Gabriel Cramer also did some work on matrices and determinants in the 18th century. Abstract algebra was developed in the 19th century, initially focusing on what is now called Galois theory, and on constructibility issues.

The stages of the development of symbolic algebra are roughly as follows:


Rhetorical algebra, which was developed by the Babylonians and remained dominant up to the 16th century;
Geometric constructive algebra, which was emphasised by the Vedic Indian and classical Greek mathematicians;
Syncopated algebra, as developed by Diophantus, Brahmagupta and the Bakhshali Manuscript; and
Symbolic algebra, which was initiated by Abū al-Hasan ibn Alī al-Qalasādī and sees its culmination in the work of Gottfried Leibniz.
A timeline of key algebraic developments are as follows:


Circa 1800 BC: The Old Babylonian Strassburg tablet seeks the solution of a quadratic elliptic equation.
Circa 1600 BC: The Plimpton 322 tablet gives a table of Pythagorean triples in Babylonian Cuneiform script.
Circa 800 BC: Indian mathematician Baudhayana, in his Baudhayana Sulba Sutra, discovers Pythagorean triples algebraically, finds geometric solutions of linear equations and quadratic equations of the forms ax2 = c and ax2 + bx = c, and finds two sets of positive integral solutions to a set of simultaneous Diophantine equations.
Circa 600 BC: Indian mathematician Apastamba, in his Apastamba Sulba Sutra, solves the general linear equation and uses simultaneous Diophantine equations with up to five unknowns.
Circa 300 BC: In Book II of his Elements, Euclid gives a geometric construction with Euclidean tools for the solution of the quadratic equation for positive real roots. The construction is due to the Pythagorean School of geometry.
Circa 300 BC: A geometric construction for the solution of the cubic is sought (doubling the cube problem). It is now well known that the general cubic has no such solution using Euclidean tools.
Circa 100 BC: Algebraic equations are treated in the Chinese mathematics book Jiuzhang suanshu (The Nine Chapters on the Mathematical Art), which contains solutions of linear equations solved using the rule of double false position, geometric solutions of quadratic equations, and the solutions of matrices equivalent to the modern method, to solve systems of simultaneous linear equations.
Circa 100 BC: The Bakhshali Manuscript written in ancient India uses a form of algebraic notation using letters of the alphabet and other signs, and contains cubic and quartic equations, algebraic solutions of linear equations with up to five unknowns, the general algebraic formula for the quadratic equation, and solutions of indeterminate quadratic equations and simultaneous equations.
Circa 150 AD: Hero of Alexandria treats algebraic equations in three volumes of mathematics.
Circa 200: Diophantus, who lived in Egypt and is often considered the "father of algebra", writes his famous Arithmetica, a work featuring solutions of algebraic equations and on the theory of numbers.
499: Indian mathematician Aryabhata, in his treatise Aryabhatiya, obtains whole-number solutions to linear equations by a method equivalent to the modern one, describes the general integral solution of the indeterminate linear equation and gives integral solutions of simultaneous indeterminate linear equations.
Circa 625: Chinese mathematician Wang Xiaotong finds numerical solutions of cubic equations.
628: Indian mathematician Brahmagupta, in his treatise Brahma Sputa Siddhanta, invents the chakravala method of solving indeterminate quadratic equations, including Pell's equation, and gives rules for solving linear and quadratic equations.
820: The word algebra is derived from operations described in the treatise written by the Persian mathematician Muḥammad ibn Mūsā al-Ḵwārizmī titled Al-Kitab al-Jabr wa-l-Muqabala (meaning "The Compendious Book on Calculation by Completion and Balancing") on the systematic solution of linear and quadratic equations. Al-Khwarizmi is often considered as the "father of algebra", much of whose works on reduction was included in the book and added to many methods we have in algebra now.
Circa 850: Persian mathematician al-Mahani conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra.
Circa 850: Indian mathematician Mahavira solves various quadratic, cubic, quartic, quintic and higher-order equations, as well as indeterminate quadratic, cubic and higher-order equations.
Circa 990: Persian Abu Bakr al-Karaji, in his treatise al-Fakhri, further develops algebra by extending Al-Khwarizmi's methodology to incorporate integral powers and integral roots of unknown quantities. He replaces geometrical operations of algebra with modern arithmetical operations, and defines the monomials x, x2, x3, ... and 1/x, 1/x2, 1/x3, ... and gives rules for the products of any two of these.
Circa 1050: Chinese mathematician Jia Xian finds numerical solutions of polynomial equations.
1072: Persian mathematician Omar Khayyam develops algebraic geometry and, in the Treatise on Demonstration of Problems of Algebra, gives a complete classification of cubic equations with general geometric solutions found by means of intersecting conic sections.
1114: Indian mathematician Bhaskara, in his Bijaganita (Algebra), recognizes that a positive number has both a positive and negative square root, and solves various cubic, quartic and higher-order polynomial equations, as well as the general quadratic indeterminant equation.
1202: Algebra is introduced to Europe largely through the work of Leonardo Fibonacci of Pisa in his work Liber Abaci.
Circa 1300: Chinese mathematician Zhu Shijie deals with polynomial algebra, solves quadratic equations, simultaneous equations and equations with up to four unknowns, and numerically solves some quartic, quintic and higher-order polynomial equations.
Circa 1400: Indian mathematician Madhava of Sangamagramma finds iterative methods for approximate solution of non-linear equations.
Circa 1450: Arab mathematician Abū al-Hasan ibn Alī al-Qalasādī took "the first steps toward the introduction of algebraic symbolism." He represented mathematical symbols using characters from the Arabic alphabet.
1535: Nicolo Fontana Tartaglia and others mathematicians in Italy independently solved the general cubic equation.
1545: Girolamo Cardano publishes Ars magna -The great art which gives Fontana's solution to the general quartic equation.
1572: Rafael Bombelli recognizes the complex roots of the cubic and improves current notation.
1591: Francois Viete develops improved symbolic notation for various powers of an unknown and uses vowels for unknowns and consonants for constants in In artem analyticam isagoge.
1631: Thomas Harriot in a posthumous publication uses exponential notation and is the first to use symbols to indicate "less than" and "greater than".
1682: Gottfried Wilhelm Leibniz develops his notion of symbolic manipulation with formal rules which he calls characteristica generalis.
1680s: Japanese mathematician Kowa Seki, in his Method of solving the dissimulated problems, discovers the determinant, and Bernoulli numbers.
1750: Gabriel Cramer, in his treatise Introduction to the analysis of algebraic curves, states Cramer's rule and studies algebraic curves, matrices and determinants.
1824: Niels Henrik Abel proved that the general quintic equation is insoluble by radicals.
1832: Galois theory is developed by Évariste Galois in his work on abstract algebra.

Arithmetic

Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. In common usage, the word refers to a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numbers. Professional mathematicians sometimes use the term (higher) arithmetic when referring to number theory, but this should not be confused with elementary arithmetic.
History
The prehistory of arithmetic is limited to a very small number of small artifacts indicating a clear conception of addition and subtraction, the best-known being the Ishango bone from central Africa, dating from somewhere between 18,000 and 20,000 BC.

It is clear that the Babylonians had solid knowledge of almost all aspects of elementary arithmetic by 1800 BC, although historians can only guess at the methods utilized to generate the arithmetical results - as shown, for instance, in the clay tablet Plimpton 322, which appears to be a list of Pythagorean triples, but with no workings to show how the list was originally produced. Likewise, the Egyptian Rhind Mathematical Papyrus (dating from c. 1650 BC, though evidently a copy of an older text from c. 1850 BC) shows evidence of addition, subtraction, multiplication, and division being used within a unit fraction system.

Nicomachus (c. AD60 - c. AD120) summarised the philosophical Pythagorean approach to numbers, and their relationships to each other, in his Introduction to Arithmetic. At this time, basic arithmetical operations were highly complicated affairs; it was the method known as the "Method of the Indians" (Latin "Modus Indorum") that became the arithmetic that we know today. Indian arithmetic was much simpler than Greek arithmetic due to the simplicity of the Indian number system, which had a zero and place-value notation. The 7th century Syriac bishop Severus Sebhokt mentioned this method with admiration, stating however that the Method of the Indians was beyond description. The Arabs learned this new method and called it "Hesab" or "Hindu Science". Fibonacci (also known as Leonardo of Pisa) introduced the "Method of the Indians" to Europe in 1202. In his book "Liber Abaci", Fibonacci says that, compared with this new method, all other methods had been mistakes. In the Middle Ages, arithmetic was one of the seven liberal arts taught in universities.

Modern algorithms for arithmetic (both for hand and electronic computation) were made possible by the introduction of Hindu-Arabic numerals and decimal place notation for numbers. Hindu-Arabic numeral based arithmetic was developed by the great Indian mathematicians Aryabhatta, Brahmagupta and Bhāskara I. Aryabhatta tried different place value notations and Brahmagupta added zero to the Indian number system. Brahmagupta developed modern multiplication, division, addition and subtraction based on Hindu-Arabic numerals. Although it is now considered elementary, its simplicity is the culmination of thousands of years of mathematical development. By contrast, the ancient mathematician Archimedes devoted an entire work, The Sand Reckoner, to devising a notation for a certain large integer. The flourishing of algebra in the medieval Islamic world and in Renaissance Europe was an outgrowth of the enormous simplification of computation through decimal notation.

Decimal arithmetic
Decimal notation constructs all real numbers from the basic digits, the first ten non-negative integers 0,1,2,...,9. A decimal numeral consists of a sequence of these basic digits, with the "denomination" of each digit depending on its position with respect to the decimal point: for example, 507.36 denotes 5 hundreds (10²), plus 0 tens (101), plus 7 units (100), plus 3 tenths (10-1) plus 6 hundredths (10-2). An essential part of this notation (and a major stumbling block in achieving it) was conceiving of zero as a number comparable to the other basic digits.

Algorism comprises all of the rules of performing arithmetic computations using a decimal system for representing numbers in which numbers written using ten symbols having the values 0 through 9 are combined using a place-value system (positional notation), where each symbol has ten times the weight of the one to its right. This notation allows the addition of arbitrary numbers by adding the digits in each place, which is accomplished with a 10 x 10 addition table. (A sum of digits which exceeds 9 must have its 10-digit carried to the next place leftward.) One can make a similar algorithm for multiplying arbitrary numbers because the set of denominations {...,10²,10,1,10-1,...} is closed under multiplication. Subtraction and division are achieved by similar, though more complicated algorithms.

Arithmetic operations
The traditional arithmetic operations are addition, subtraction, multiplication and division, although more advanced operations (such as manipulations of percentages, square root, exponentiation, and logarithmic functions) are also sometimes included in this subject. Arithmetic is performed according to an order of operations. Any set of objects upon which all four operations of arithmetic can be performed (except division by zero), and wherein these four operations obey the usual laws, is called a field.
Addition (+)
Addition is the basic operation of arithmetic. In its simplest form, addition combines two numbers, the addends or terms, into a single number, the sum.
Adding more than two numbers can be viewed as repeated addition; this procedure is known as summation and includes ways to add infinitely many numbers in an infinite series; repeated addition of the number one is the most basic form of counting.

Addition is commutative and associative so the order in which the terms are added does not matter. The identity element of addition (the additive identity) is 0, that is, adding zero to any number will yield that same number. Also, the inverse element of addition (the additive inverse) is the opposite of any number, that is, adding the opposite of any number to the number itself will yield the additive identity, 0. For example, the opposite of 7 is (-7), so 7 + (-7) = 0. Addition can be given geometrically as follows.

If a and b are the lengths of two sticks, then if we place the sticks one after the other, the length of the stick thus formed will be a+b

Subtraction (−)
Subtraction is essentially the opposite of addition. Subtraction finds the difference between two numbers, the minuend minus the subtrahend. If the minuend is larger than the subtrahend, the difference will be positive; if the minuend is smaller than the subtrahend, the difference will be negative; and if they are equal, the difference will be zero.
Subtraction is neither commutative nor associative. For that reason, it is often helpful to look at subtraction as addition of the minuend and the opposite of the subtrahend, that is a − b = a + (−b). When written as a sum, all the properties of addition hold.

Multiplication (×, ·, or *)
Multiplication is in essence repeated addition, or the sum of a list of identical numbers. Multiplication finds the product of two numbers, the multiplier and the multiplicand, sometimes both simply called factors.
Multiplication, as it is really repeated addition, is commutative and associative; further it is distributive over addition and subtraction. The multiplicative identity is 1, that is, multiplying any number by 1 will yield that same number. Also, the multiplicative inverse is the reciprocal of any number, that is, multiplying the reciprocal of any number by the number itself will yield the multiplicative identity, 1.

Division (÷ or /)
Division is essentially the opposite of multiplication. Division finds the quotient of two numbers, the dividend divided by the divisor. Any dividend divided by zero is undefined. For positive numbers, if the dividend is larger than the divisor, the quotient will be greater than one, otherwise it will be less than one (a similar rule applies for negative numbers). The quotient multiplied by the divisor always yields the dividend.
Division is neither commutative nor associative. As it is helpful to look at subtraction as addition, it is helpful to look at division as multiplication of the dividend times the reciprocal of the divisor, that is a ÷ b = a × 1⁄b. When written as a product, it will obey all the properties of multiplication.

Examples
Multiplication table

× 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
2 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50
3 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75
4 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100
5 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125
6 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120 126 132 138 144 150
7 7 14 21 28 35 42 49 56 63 70 77 84 91 98 105 112 119 126 133 140 147 154 161 168 175
8 8 16 24 32 40 48 56 64 72 80 88 96 104 112 120 128 136 144 152 160 168 176 184 192 200
9 9 18 27 36 45 54 63 72 81 90 99 108 117 126 135 144 153 162 171 180 189 198 207 216 225
10 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250
11 11 22 33 44 55 66 77 88 99 110 121 132 143 154 165 176 187 198 209 220 231 242 253 264 275
12 12 24 36 48 60 72 84 96 108 120 132 144 156 168 180 192 204 216 228 240 252 264 276 288 300
13 13 26 39 52 65 78 91 104 117 130 143 156 169 182 195 208 221 234 247 260 273 286 299 312 325
14 14 28 42 56 70 84 98 112 126 140 154 168 182 196 210 224 238 252 266 280 294 308 322 336 350
15 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300 315 330 345 360 375
16 16 32 48 64 80 96 112 128 144 160 176 192 208 224 240 256 272 288 304 320 336 352 368 384 400
17 17 34 51 68 85 102 119 136 153 170 187 204 221 238 255 272 289 306 323 340 357 374 391 408 425
18 18 36 54 72 90 108 126 144 162 180 198 216 234 252 270 288 306 324 342 360 378 396 414 432 450
19 19 38 57 76 95 114 133 152 171 190 209 228 247 266 285 304 323 342 361 380 399 418 437 456 475
20 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440 460 480 500
21 21 42 63 84 105 126 147 168 189 210 231 252 273 294 315 336 357 378 399 420 441 462 483 504 525
22 22 44 66 88 110 132 154 176 198 220 242 264 286 308 330 352 374 396 418 440 462 484 506 528 550
23 23 46 69 92 115 138 161 184 207 230 253 276 299 322 345 368 391 414 437 460 483 506 529 552 575
24 24 48 72 96 120 144 168 192 216 240 264 288 312 336 360 384 408 432 456 480 504 528 552 576 600
25 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 525 550 575 600 625

Number theory
The term arithmetic is also used to refer to number theory. This includes the properties of integers related to primality, divisibility, and the solution of equations by integers, as well as modern research which is an outgrowth of this study. It is in this context that one runs across the fundamental theorem of arithmetic and arithmetic functions. A Course in Arithmetic by Serre reflects this usage, as do such phrases as first order arithmetic or arithmetical algebraic geometry. Number theory is also referred to as 'the higher arithmetic', as in the title of H. Davenport's book on the subject.
Arithmetic in education
Primary education in mathematics often places a strong focus on algorithms for the arithmetic of natural numbers, integers, rational numbers (vulgar fractions), and real numbers (using the decimal place-value system). This study is sometimes known as algorism.
The difficulty and unmotivated appearance of these algorithms has long led educators to question this curriculum, advocating the early teaching of more central and intuitive mathematical ideas. One notable movement in this direction was the New Math of the 1960s and '70s, which attempted to teach arithmetic in the spirit of axiomatic development from set theory, an echo of the prevailing trend in higher mathematics.

Since the introduction of the electronic calculator, which can perform the algorithms far more efficiently than humans, an influential school of educators has argued that mechanical mastery of the standard arithmetic algorithms is no longer necessary. In their view, the first years of school mathematics could be more profitably spent on understanding higher-level ideas about what numbers are used for and relationships among number, quantity, measurement, and so on. However, most research mathematicians still consider mastery of the manual algorithms to be a necessary foundation for the study of algebra and computer science. This controversy was central to the "Math Wars" over California's primary school curriculum in the 1990s, and continues today.

Many mathematics texts for K-12 instruction were developed, funded by grants from the United States National Science Foundation based on standards created by the NCTM and given high ratings by United States Department of Education, though condemned by many mathematicians. Some widely adopted texts such as TERC were based on the spirit of research papers which found that instruction of basic arithmetic was harmful to mathematical understanding. Rather than teaching any traditional method of arithmetic, teachers are instructed to instead guide students to invent their own (some critics claim inefficient) methods, instead using such techniques as skip counting, and the heavy use of manipulatives, scissors and paste, and even singing rather than multiplication tables or long division. Although such texts were designed to be a complete curricula, in the face of intense protest and criticism, many districts have chosen to circumvent the intent of such radical approaches by supplementing with traditional texts. Other districts have since adopted traditional mathematics texts and discarded such reform-based approaches as misguided failures.

Mathematics
mathematics, 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.
Branches of Mathematics
Foundations
The term foundations is used to refer to the formulation and analysis of the language, axioms, and logical methods on which all of mathematics rests (see logic; symbolic logic). The scope and complexity of modern mathematics requires a very fine analysis of the formal language in which meaningful mathematical statements may be formulated and perhaps be proved true or false. Most apparent mathematical contradictions have been shown to derive from an imprecise and inconsistent use of language. A basic task is to furnish a set of axioms effectively free of contradictions and at the same time rich enough to constitute a deductive source for all of modern mathematics. The modern axiom schemes proposed for this purpose are all couched within the theory of sets, originated by Georg Cantor, which now constitutes a universal mathematical language.

Algebra
Historically, algebra is the study of solutions of one or several algebraic equations, involving the polynomial functions of one or several variables. The case where all the polynomials have degree one (systems of linear equations) leads to linear algebra. The case of a single equation, in which one studies the roots of one polynomial, leads to field theory and to the so-called Galois theory. The general case of several equations of high degree leads to algebraic geometry, so named because the sets of solutions of such systems are often studied by geometric methods.

Modern algebraists have increasingly abstracted and axiomatized the structures and patterns of argument encountered not only in the theory of equations, but in mathematics generally. Examples of these structures include groups (first witnessed in relation to symmetry properties of the roots of a polynomial and now ubiquitous throughout mathematics), rings (of which the integers, or whole numbers, constitute a basic example), and fields (of which the rational, real, and complex numbers are examples). Some of the concepts of modern algebra have found their way into elementary mathematics education in the so-called new mathematics.

Some important abstractions recently introduced in algebra are the notions of category and functor, which grew out of so-called homological algebra. Arithmetic and number theory, which are concerned with special properties of the integers—e.g., unique factorization, primes, equations with integer coefficients (Diophantine equations), and congruences—are also a part of algebra. Analytic number theory, however, also applies the nonalgebraic methods of analysis to such problems.

Analysis
The essential ingredient of analysis is the use of infinite processes, involving passage to a limit. For example, the area of a circle may be computed as the limiting value of the areas of inscribed regular polygons as the number of sides of the polygons increases indefinitely. The basic branch of analysis is the calculus. The general problem of measuring lengths, areas, volumes, and other quantities as limits by means of approximating polygonal figures leads to the integral calculus. The differential calculus arises similarly from the problem of finding the tangent line to a curve at a point. Other branches of analysis result from the application of the concepts and methods of the calculus to various mathematical entities. For example, vector analysis is the calculus of functions whose variables are vectors. Here various types of derivatives and integrals may be introduced. They lead, among other things, to the theory of differential and integral equations, in which the unknowns are functions rather than numbers, as in algebraic equations. Differential equations are often the most natural way in which to express the laws governing the behavior of various physical systems. Calculus is one of the most powerful and supple tools of mathematics. Its applications, both in pure mathematics and in virtually every scientific domain, are manifold.

Geometry
The shape, size, and other properties of figures and the nature of space are in the province of geometry. Euclidean geometry is concerned with the axiomatic study of polygons, conic sections, spheres, polyhedra, and related geometric objects in two and three dimensions—in particular, with the relations of congruence and of similarity between such objects. The unsuccessful attempt to prove the "parallel postulate" from the other axioms of Euclid led in the 19th cent. to the discovery of two different types of non-Euclidean geometry.

The 20th cent. has seen an enormous development of topology, which is the study of very general geometric objects, called topological spaces, with respect to relations that are much weaker than congruence and similarity. Other branches of geometry include algebraic geometry and differential geometry, in which the methods of analysis are brought to bear on geometric problems. These fields are now in a vigorous state of development.

Applied Mathematics
The term applied mathematics loosely designates a wide range of studies with significant current use in the empirical sciences. It includes numerical methods and computer science, which seeks concrete solutions, sometimes approximate, to explicit mathematical problems (e.g., differential equations, large systems of linear equations). It has a major use in technology for modeling and simulation. For example, the huge wind tunnels, formerly used to test expensive prototypes of airplanes, have all but disappeared. The entire design and testing process is now largely carried out by computer simulation, using mathematically tailored software. It also includes mathematical physics, which now strongly interacts with all of the central areas of mathematics. In addition, probability theory and mathematical statistics are often considered parts of applied mathematics. The distinction between pure and applied mathematics is now becoming less significant.

Development of Mathematics
The earliest records of mathematics show it arising in response to practical needs in agriculture, business, and industry. In Egypt and Mesopotamia, where evidence dates from the 2d and 3d millennia B.C., it was used for surveying and mensuration; estimates of the value of π (pi) are found in both locations. There is some evidence of similar developments in India and China during this same period, but few records have survived. This early mathematics is generally empirical, arrived at by trial and error as the best available means for obtaining results, with no proofs given. However, it is now known that the Babylonians were aware of the necessity of proofs prior to the Greeks, who had been presumed the originators of this important step.

Greek Contributions
A profound change occurred in the nature and approach to mathematics with the contributions of the Greeks. The earlier (Hellenic) period is represented by Thales (6th cent. B.C.), Pythagoras, Plato, and Aristotle, and by the schools associated with them. The Pythagorean theorem, known earlier in Mesopotamia, was discovered by the Greeks during this period.

During the Golden Age (5th cent. B.C.), Hippocrates of Chios made the beginnings of an axiomatic approach to geometry and Zeno of Elea proposed his famous paradoxes concerning the infinite and the infinitesimal, raising questions about the nature of and relationships among points, lines, and numbers. The discovery through geometry of irrational numbers, such as 2, also dates from this period. Eudoxus of Cnidus (4th cent. B.C.) resolved certain of the problems by proposing alternative methods to those involving infinitesimals; he is known for his work on geometric proportions and for his exhaustion theory for determining areas and volumes.

The later (Hellenistic) period of Greek science is associated with the school of Alexandria. The greatest work of Greek mathematics, Euclid's Elements (c.300 B.C.), appeared at the beginning of this period. Elementary geometry as taught in high school is still largely based on Euclid's presentation, which has served as a model for deductive systems in other parts of mathematics and in other sciences. In this method primitive terms, such as point and line, are first defined, then certain axioms and postulates relating to them and seeming to follow directly from them are stated without proof; a number of statements are then derived by deduction from the definitions, axioms, and postulates. Euclid also contributed to the development of arithmetic and presented a geometric theory of quadratic equations.

In the 3d cent. B.C., Archimedes, in addition to his work in mechanics, made an estimate of π and used the exhaustion theory of Eudoxus to obtain results that foreshadowed those much later of the integral calculus, and Apollonius of Perga named the conic sections and gave the first theory for them. A second Alexandrian school of the Roman period included contributions by Menelaus (c.A.D. 100, spherical triangles), Heron of Alexandria (geometry), Ptolemy (A.D. 150, astronomy, geometry, cartography), Pappus (3d cent., geometry), and Diophantus (3d cent., arithmetic).

Chinese and Middle Eastern Advances
Following the decline of learning in the West after the 3d cent., the development of mathematics continued in the East. In China, Tsu Ch'ung-Chih estimated π by inscribed and circumscribed polygons, as Archimedes had done, and in India the numerals now used throughout the civilized world were invented and contributions to geometry were made by Aryabhata and Brahmagupta (5th and 6th cent. A.D.). The Arabs were responsible for preserving the work of the Greeks, which they translated, commented upon, and augmented. In Baghdad, Al-Khowarizmi (9th cent.) wrote an important work on algebra and introduced the Hindu numerals for the first time to the West, and Al-Battani worked on trigonometry. In Egypt, Ibn al-Haytham was concerned with the solids of revolution and geometrical optics. The Persian poet Omar Khayyam wrote on algebra.

Western Developments from the Twelfth to Eighteenth Centuries
Word of the Chinese and Middle Eastern works began to reach the West in the 12th and 13th cent. One of the first important European mathematicians was Leonardo da Pisa (Leonardo Fibonacci), who wrote on arithmetic and algebra (Liber abaci, 1202) and on geometry (Practica geometriae, 1220). With the Renaissance came a great revival of interest in learning, and the invention of printing made many of the earlier books widely available. By the end of the 16th cent. advances had been made in algebra by Niccolò Tartaglia and Geronimo Cardano, in trigonometry by François Viète, and in such areas of applied mathematics as mapmaking by Mercator and others.

The 17th cent., however, saw the greatest revolution in mathematics, as the scientific revolution spread to all fields. Decimal fractions were invented by Simon Stevin and logarithms by John Napier and Henry Briggs; the beginnings of projective geometry were made by Gérard Desargues and Blaise Pascal; number theory was greatly extended by Pierre de Fermat; and the theory of probability was founded by Pascal, Fermat, and others. In the application of mathematics to mechanics and astronomy, Galileo and Johannes Kepler made fundamental contributions.

The greatest mathematical advances of the 17th cent., however, were the invention of analytic geometry by René Descartes and that of the calculus by Isaac Newton and, independently, by G. W. Leibniz. Descartes's invention (anticipated by Fermat, whose work was not published until later) made possible the expression of geometric problems in algebraic form and vice versa. It was indispensable in creating the calculus, which built upon and superseded earlier special methods for finding areas, volumes, and tangents to curves, developed by F. B. Cavalieri, Fermat, and others. The calculus is probably the greatest tool ever invented for the mathematical formulation and solution of physical problems.

The history of mathematics in the 18th cent. is dominated by the development of the methods of the calculus and their application to such problems, both terrestrial and celestial, with leading roles being played by the Bernoulli family (especially Jakob, Johann, and Daniel), Leonhard Euler, Guillaume de L'Hôpital, and J. L. Lagrange. Important advances in geometry began toward the end of the century with the work of Gaspard Monge in descriptive geometry and in differential geometry and continued through his influence on others, e.g., his pupil J. V. Poncelet, who founded projective geometry (1822).

In the Nineteenth Century
The modern period of mathematics dates from the beginning of the 19th cent., and its dominant figure is C. F. Gauss. In the area of geometry Gauss made fundamental contributions to differential geometry, did much to found what was first called analysis situs but is now called topology, and anticipated (although he did not publish his results) the great breakthrough of non-Euclidean geometry. This breakthrough was made by N. I. Lobachevsky (1826) and independently by János Bolyai (1832), the son of a close friend of Gauss, whom each proceeded by establishing the independence of Euclid's fifth (parallel) postulate and showing that a different, self-consistent geometry could be derived by substituting another postulate in its place. Still another non-Euclidean geometry was invented by Bernhard Riemann (1854), whose work also laid the foundations for the modern tensor calculus description of space, so important in the general theory of relativity.

In the area of arithmetic, number theory, and algebra, Gauss again led the way. He established the modern theory of numbers, gave the first clear exposition of complex numbers, and investigated the functions of complex variables. The concept of number was further extended by W. R. Hamilton, whose theory of quaternions (1843) provided the first example of a noncommutative algebra (i.e., one in which ab ≠ ba). This work was generalized the following year by H. G. Grassmann, who showed that several different consistent algebras may be derived by choosing different sets of axioms governing the operations on the elements of the algebra.

These developments continued with the group theory of M. S. Lie in the late 19th cent. and reached full expression in the wide scope of modern abstract algebra. Number theory received significant contributions in the latter half of the 19th cent. through the work of Georg Cantor, J. W. R. Dedekind, and K. W. Weierstrass. Still another influence of Gauss was his insistence on rigorous proof in all areas of mathematics. In analysis this close examination of the foundations of the calculus resulted in A. L. Cauchy's theory of limits (1821), which in turn yielded new and clearer definitions of continuity, the derivative, and the definite integral. A further important step toward rigor was taken by Weierstrass, who raised new questions about these concepts and showed that ultimately the foundations of analysis rest on the properties of the real number system.

In the Twentieth Century
In the 20th cent. the trend has been toward increasing generalization and abstraction, with the elements and operations of systems being defined so broadly that their interpretations connect such areas as algebra, geometry, and topology. The key to this approach has been the use of formal axiomatics, in which the notion of axioms as "self-evident truths" has been discarded. Instead the emphasis is on such logical concepts as consistency and completeness. The roots of formal axiomatics lie in the discoveries of alternative systems of geometry and algebra in the 19th cent.; the approach was first systematically undertaken by David Hilbert in his work on the foundations of geometry (1899).

The emphasis on deductive logic inherent in this view of mathematics and the discovery of the interconnections between the various branches of mathematics and their ultimate basis in number theory led to intense activity in the field of mathematical logic after the turn of the century. Rival schools of thought grew up under the leadership of Hilbert, Bertrand Russell and A. N. Whitehead, and L. E. J. Brouwer. Important contributions in the investigation of the logical foundations of mathematics were made by Kurt Gödel and A. Church.