David Hilbert

David Hilbert

German Mathematician David Hilbert was among the most influential mathematician of the 20^th century. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He reduced geometry to a series of axioms and contributed substantially to the establishment of the formalistic foundation of mathematics. He also formulated the theory of Hilbert spaces, one of the foundations of functional analysis.

He was born on January 23, 1862 in Wehlau, near Konigsberg, then the principal city of East Prussia. His father Otto Hilbert was a judge while his mother Maria Theresa was a lover of mathematics and Hilbert is said to have inherited the love of mathematics from the early childhood is from her. When Otto Hilbert became a senior judge the family moved into Koinsberg itself, where David was enrolled at the Royal Friedrichkolleg at the age of eight years. The school was not suitable for the child prodigy Hilbert so after completing one year in the school he attended the more progressive and science oriented Wilhelm Gymnasium.

He completed his graduation in 1880 from Wilhelm Gymnasium and later he enrolled at University of Koinsberg to complete his doctorate. He did doctorate under Lindemann. Lindemann was not among the first rank of mathematicians, but he had a wide range of interests, one of which was the theory of invariants. This was the subject on which Hilbert wrote his dissertation entitled Über invariante Eigenschaften specieller binärer Formen, insbesondere der Kugelfunctionen ("On the invariant properties of special binary forms, in particular the spherical harmonic functions"). One of Hilbert's friends there was Minkowski, who was also a doctoral student at Königsberg, and they were to strongly influence each others mathematical progress.

In 1884 Hurwitz was appointed to the University of Konigsberg and quickly became friends with Hilbert, a friendship which was another important factor in Hilbert's mathematical development.  In 1886, he was appointed as an assistant professor in the University of Konigsberg. He remained there as a professor from 1886 to 1895. The University of Gottingen had a flourishing tradition in mathematics, primarily as the result of the contributions of Carl Friedrich Gauss, Dirichlet, and Bernhard Riemann in the 19th century. In 1888 he demonstrated his first work on invariant function known as finiteness theorem. In 1892, Hilbert married Käthe Jerosch , the daughter of a Konigsberg merchant. They had a son named Franz who died young at the age of 21 and never lived a normal life.

 In 1893, Hilbert began a work on algebraic number theory. In 1895, as a result of intervention on his behalf by Felix Klein he obtained the position of Chairman of Mathematics at the University of Göttingen, at that time the best research center for mathematics in the world and where he remained for the rest of his life. Hilbert’s intense interest in mathematical physics also contributed to the university’s reputation in physics. Hilbert's work in geometry had the greatest influence in that area after Euclid. A systematic study of the axioms of Euclidean geometry led Hilbert to propose 21 such axioms and he analyzed their significance. He published Grundlagen der Geometrie in 1899 putting geometry in a formal axiomatic setting. 

Hilbert put forth a most influential list of 23 unsolved problems at the International Congress of Mathematicians in Paris in 1900. This is generally reckoned the most successful and deeply considered compilation of open problems ever to be produced by an individual mathematician. Hilbert's famous 23 Paris problems challenged mathematicians to solve fundamental questions that included the the continuum hypothesis, the well ordering of the real, Goldbach's conjecture, the transcendence of powers of algebraic numbers, the Riemann hypothesis, the extension of Dirichlet's principle and many more. Many of the problems were solved during this century, and each time one of the problems was solved it was a major event for mathematics.

Hilbert contributed to many branches of mathematics, including invariants, algebraic number fields, functional analysis, integral equations, mathematical physics, and the calculus of variations. In 1905 the Hungarian Academy of Sciences gave a special citation for Hilbert. Hilbert’s work in integral equation in 1909 led to the research in functional analysis. His work also established the basis for his work on infinite dimensional space, later called Hilbert space, a concept that is useful in mathematical analysis and quantum mechanics. Making use of his results on integral equations, Hilbert contributed to the development of mathematical physics by his important memoirs on kinetic gas theory and the theory of radiations. In 1909 he proved the conjecture in number theory that for any n, all positive integers are sums of a certain fixed number of nth powers.

In 1930 Hilbert retired and the city of Königsberg made him an honorary citizen of the city; although he continued to lecture occasionally. In 1932 Gottingen celebrated his seventieth birthday with a torchlight processions. After 1934 he never again set foot in the Mathematical Institute.

Hilbert died on February 13, 1943, ten years after the Nazis came to power.it was the middle of the war and barely a dozen people attended his funeral. News of his death only became known to the wider world six months after he had died. The epitaph on his tombstone in Göttingen is the famous lines he had spoken at the conclusion of his retirement address to the Society of German Scientists and Physicians in the fall of 1930.

Wir müssen wissen.

Wir werden wissen.

In English:

We must know.

We will know.

C F Gauss

                                                            Carl Friedrich Gauss

Carl Friedrich Gauss was a German mathematician who was born on 30 April 1777 in Braunschweig (Germany) as the son of poor working-class parents. His mother was illiterate, but Gauss was one of those child prodigies whose natural aptitude for mathematics soon became apparent.

As a child of three, according to a well-authenticated story, he corrected an error in his father’s payroll calculations. His arithmetical powers so overwhelmed his schoolmasters that, by the time Gauss was 10 years old, they admitted that there was nothing more they could teach the boy. It is said that in his first arithmetical class Gauss astonished his teacher Butner and his assistant Martin Bartels by instantly solving what was intended to be a busy work problem: - Find the sum of all the numbers from 1 to 100. The young Gauss later confessed to having recognized the pattern: 1 + 100 = 2 + 99 =.... 50 + 51 =101. Since, there are 50 pairs of numbers, each of which adds up to 101, the sum of all the numbers must be 50 x 101 = 5050.

In 1788 Gauss began his education at the Gymnasium with the help of Buttner and Bartels, where he learnt German and Latin. Gauss's intellectual abilities attracted the attention of the Duke of Braunschweig, who sent him to the Collegium Carolinum from 1792 to 1795, and to the University of Göttingen from 1795 to 1798.

While in university, Gauss independently rediscovered several important theorems. Gauss went on to a succession of triumphs, each new discovery following on the heels of a previous one. The problem of constructing regular polygons with only “Euclidean Tools”, that is to say with ruler and compass alone, had long been laid aside in the belief that the ancients had exhausted all the possible constructions. In 1796, Gauss showed that the 17- sided regular polygon is so constructable, the first advance in this area since Euclid’s time and the discovery ultimately led Gauss to choose mathematics instead of philology as a career. Gauss was so pleased by this result that he requested that a regular heptadecagon be inscribed on his tombstone. The year 1796 was most productive for both Gauss and number theory.

He discovered a construction of the heptadecagon on 30 March. He further advanced modular arithmetic, greatly simplifying manipulations in number theory. He became the first to prove the quadratic reciprocity law. This remarkably general law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic. Gauss returned to Brunswick where he received a degree in 1799.

After the Duke of Brunswick had agreed to continue Gauss's stipend, he requested that Gauss submit a doctoral dissertation to the University of Helmsted. Gauss's dissertation was a discussion of the Fundamental Theorem of Algebra, which has been stated first by Girard in 1629 and then proved imperfectly by d’Alembert in 1746 and later by Euler in 1749. The theorem states that an algebraic equation of degree n has exactly n complex roots. This theorem was always a favourite with Gauss, and he gave, in all, four distinct demonstration of it. He published the book Disquisitiones Arithmeticae in the summer of 1801. This book placed Gauss in the front rank of mathematician.

The most extraordinary achievement of Gauss was more in the realm of theoretical astronomy than of mathematics. On the opening night of the 19^th century, January 1,1801, the Italian astronomer Piazzi discovered the first of the so called minor planets Geres. But after the course of this newly found body, visible only by telescope, passed the sun, neither Piazzi nor any other astronomer could locate it again. Piazzi’s observations extended over a period of 41 days, during which the orbit swept out an angle of only 9 degrees of its orbit before it disappeared behind the Sun. Zach published several predictions of its position, including one by Gauss which differed greatly from the others. When Ceres was rediscovered by Zach on 7 December 1801 it was almost exactly where Gauss had predicted. 

Gauss married Johanna Ostoff on 9 October, 1805. Despite having a happy personal life for the first time, his benefactor, the Duke of Brunswick, was killed fighting for the Prussian army. In 1807 Gauss left Brunswick to take up the position of director of the Göttingen observatory.Gauss arrived in Göttingen in late 1807. In 1808 his father died, and a year later Gauss's wife Johanna died after giving birth to their second son. Gauss's work never seemed to suffer from his personal tragedy.

He published his second book, Theoria motus corporum coelestium in sectionibus conicis Solem ambientium, in 1809, a major two volume treatise on the motion of celestial bodies. In the first volume he discussed differential equations, conic sections and elliptic orbits, while in the second volume, the main part of the work, he showed how to estimate and then to refine the estimation of a planet's orbit. Gauss's contributions to theoretical astronomy stopped after 1817, although he went on making observations until the age of 70.

Gauss had been asked in 1818 to carry out a geodesic survey of the state of Hanover to link up with the existing Danish grid. Gauss was pleased to accept and took personal charge of the survey. Because of the survey, Gauss invented the heliotrope which worked by reflecting the Sun's rays using a design of mirrors and a small telescope. The survey of Hanover fueled Gauss's interest in differential geometry, a field of mathematics dealing with curves and surfaces.

Among other things he came up with the notion of Gaussian curvature. Gauss also claimed to have discovered the possibility of non-Euclidean geometries but never published it. This discovery was a major paradigm shift in mathematics, as it freed mathematicians from the mistaken belief that Euclid's axioms were the only way to make geometry consistent and non-contradictory.

In 1831 Gauss developed a fruitful collaboration with the physics professor Wilhelm Weber, leading to new knowledge in magnetism and the discovery of Kirchhoff's circuit laws in electricity. His health deteriorated slowly, and Gauss died in his sleep early in the morning of 23 February, 1855.

ARYABHATA THE INDIAN MATHEMATICIAN

Aryabhata

Born- 476 AD in Kusumpur Patna

Died -550 AD

Contribution:- Gave exact value of Pi up to 4 decimal places, formula for Area and Volume of    

                        Sphere, Trigonometry table of Sine etc .

Aryabhata , the greatest mathematical genius of India was born in Kusumpur now in Patna in Bihar around 476 AD at the time when Patna was the capital of Gupta Empire. Patna was the major centre of learning.  The exact date and timing of his birth  is not confirmed but from the sloka of Aryabhatiya it is evident that he was born in 476 AD.  He writes that he was twenty three years of age when he wrote Aryabhatiya which he finished in 499.

“k”V~;Cnkuka “kf”V;Znk O;rhrkL=;Üp~ ;qxiknk% A

«;kf/kdk foa’kfrjCnkLrnsg ee tUeuks·rhrk %AA

Aryabhata had written at least three books on astronomy but the surviving one is the Aryabhata’s masterpiece Aryabhatiya which is a small astronomical treatise written in 118 verses giving a summary of Hindu mathematics up to that time. The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry and spherical trigonometry. It also contains continued fractions, quadratic equations, sums of power series and a table of sines.

Aryabhata also invented a numeral system for representing numbers that consists of giving numerical 90, 100. The higher numbers are denoted by these consonants followed by a vowel to obtain 100, 10000,  .... In fact the system allows numbers up to 10¹⁸ to be represented with an alphabetical notation.

oxkZ{kjkf.k oxsZ·oxsZ·oxkZ{kjkf.k dkr~³~ekS ;% A

[kf}uods Lojk uo oxsZ·oxsZ uokUR;oxsZ ok AA

In the Devnagri alphabets, the twenty five varghiya vyanjannas (calssified consonants) from k to m constituting the five vargas (calsses) viz., ka- varga (d &oxZ) etc—represents the number 1 to 25 as shown in the given table---

Varga	Letters representing numbers
d oxZ	d 1	[k 2	Xk 3	?k 4	³  5
Pk oxZ	Pk  6	N 7	Tk 8	>  9	´ 10
V oxZ	V  11	B 12	M 13	< 14	.k 15
r oxZ	r  16	Fk 17	n 18	/k 19	Uk 20
i oxZ	i  21	Q 22	c 23	Hk 24	e 25

Ifrah argues that ---it is extremely likely that Aryabhata knew the sign for zero and the numerals of the place value system. This supposition is based on the following two facts: first, the invention of his alphabetical counting system would have been impossible without zero or the place-value system; secondly, he carries out calculations on square and cubic roots which are impossible if the numbers in question are not written according to the place-value system and zero.

Aryabhata gave an accurate approximation for pi

prqjf/kda ‘kre”Vxq.ka }k”kf”VLrFkk lglzk.kke~ A

v;qr};fo”dEHkL;kléks o`rifj.kkga AA

He wrote in the Aryabhatiya the following:-

Add four to one hundred, multiply by eight and then add sixty-two thousand. the result is approximately the circumference of a circle of diameter twenty thousand. By this rule the relation of the circumference to diameter is given.

This gives pi = ⁶²⁸³²/₂₀₀₀₀ = 3.1416 which is a surprisingly accurate value. 

He gave a table of sines calculating the approximate values at intervals of 90/24 = 3 45'. In order to do this he used a formula for sin(n+1)x - sin nx in terms of sin nx and sin (n-1)x.

Aryabhata gives a systematic treatment of the position of the planets in space. He gave 62832 miles as the circumference of the earth, which is an excellent approximation. He believed that the apparent rotation of the heavens was due to the axial rotation of the Earth. He correctly explains the causes of eclipses of the Sun and the Moon. The Indian belief up to that time was that eclipses were caused by a demon called Rahu. His value for the length of the year at 365 days 6 hours 12 minutes 30 seconds is an overestimate since the true value is less than 365 days 6 hours.

For the system of equationons of the form by = ax + c and by = ax - c, where a, b, care integers, Aryabhata used the kuttaka method. The word kuttaka means "to pulverise" and the method consisted of breaking the problem down into new problems where the coefficients became smaller and smaller with each step. 

The importance of Aryabhata in Indian mathematical world can be summed up with the following quotes of Bhaskaracharya-

 Aryabhata is the master who, after reaching the furthest shores and plumbing the inmost depths of the sea of ultimate knowledge of mathematics, kinematics and spherics, handed over the three sciences to the learned world.

Archimedes

 Archimedes

                                                                       

Born: - 287 BC

Died: - 212 BC

Books: - Floating Bodies, Spirals, The Sand Reckoner, Measurement of the  Circle,

Main Contribution:- Found that the value of pi lies between 22/7 and 223/71 , Archimedes' principle , Archimedes screw (a device for raising water) etc.

Archimedes was a native of Syracuse, Sicily where he was born in 287 BC. His father Phidias was an astronomer. There is no more information available about his family background. He studied at Euclid's school (probably after Euclid's death), but his work far surpassed the works of Euclid.

 Archimedes is universally acknowledged to be the greatest of ancient mathematicians. He had gained the reputation in his own time which few other mathematicians of this period achieved. Some math historians consider Archimedes to be one of history's greatest mathematicians, along with possibly Newton, Gauss, and Euler. He is therefore called – the wise man, the master, and the great geometer.  Most of the facts about his life come from a biography about the Roman soldier Marcellus written by the Roman biographer Plutarch.

His achievements are particularly impressive given the lack of good mathematical notation in his day. His proofs are noted not only for brilliance but for their "awesome" clarity. Archimedes made advances in number theory, algebra, and analysis, but is most renowned for his many theorems of plane and solid geometry.  His excellent approximation to √3 indicates that he'd partially anticipated the method of continued fractions. He found a method to trisect an arbitrary angle.  His original achievements in physics include the principles of leverage, the first law of hydrostatics, and inventions like the compound pulley, the hydraulic screw, and war machines. He had invented more machines which were used as engines of war that were particularly effective in the defence of Syracuse when it was attacked by the Romans.

                  Archimedes Screw

His books include Floating Bodies, Spirals, The Sand Reckoner, Measurement of the Circle, and Sphere and Cylinder.  Archimedes proved that the volume of a sphere is two-thirds the volume of a circumscribing cylinder. He requested that a representation of such a sphere and cylinder be inscribed on his tomb. Archimedes discovered formulae for the volume and surface area of a sphere, and may even have been first to notice and prove the simple relationship between a circle's circumference and area. For these reasons, π is often called Archimedes' constant. His approximation 223/71 < π < 22/7 was the best of his day.

His fascination with geometry is beautifully described by Plutarch:-

Of times Archimedes' servants got him against his will to the baths, to wash and anoint him, and yet being there, he would ever be drawing out of the geometrical figures, even in the very embers of the chimney. And while they were anointing of him with oils and sweet savours, with his fingers he drew lines upon his naked body, so far was he taken from himself, and brought into ecstasy or trance, with the delight he had in the study of geometry.

His most famous theorem which gives the weight of a body immersed in liquid called Archimedes principle states-- an object immersed in a fluid experiences a buoyant force that is equal in magnitude to the force of gravity on the displaced fluid. 

Legend has it that Archimedes discovered his famous theory of buoyancy - Archimedes Principle - while taking a bath. He was so excited that he ran naked through the streets of Syracuse shouting "Eureka, eureka (I have found it)!” He used this principle to unearth the truth of the Golden crown of the king Hieron II of Syracuse.

His famous quote is -"Give me a place to stand and rest my lever on, and I can move the Earth."

The mathematical genius of such repute was beheaded mercilessly by the Roman soldier in 212 BC while Archimedes was busy solving a sum on sand.  He was buried at Syracuse, where he was born, grew up, worked and died. On his grave there is an inscription of pi, his most famous discovery. His followers also placed on his tombstone the figure of a sphere inscribed inside a cylinder and the 2:3 ratios of the volumes between them, the solution to the problem he considered his greatest achievement.

Life of Al- Khwarizmi

Al- Khwarizmi

           

Born: - around 780 in Baghdad (Now in Iraq)

            Died:- about 850

            Important contribution: - Hisab al-jabr w'al-muqabala (the book on Algebra).

           

alkhwarizmi

Al- Khwarizmi alias- Abu- Jafar- Muhammad- ibn- Musa al Khwarizmi was a Persian who worked as a mathematician, astronomer and geographer in the Golden Age of Islamic science.There is not much knowledge about the life of this  Math genius and founder of Algebra but what little knowledge do we have about Al- Khwarizmi is due to the historian Al- Tabari.

Al- Tabari writes- Harun al – Rashid became the fifth Caliph of the Abbasid dynasty on 14^th September 786, about the time that al- Khwarizmi was born. After the death of the Caliph his elder son Al- Maamun became the Caliph and he founded an academy called the House of Wisdom where Greek philosophical and scientific works were translted. 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.

He introduced the Hindu decimal system to the Islamic world and Europe; invented the horary quadrant; improved the sundial; developed trigonometry tables; and improved on Ptolemy's astronomy and geography.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. In his book he introduced the natural numbers, solution of equation such as linear and quadratic. The reduction of a quadratic equation carries two operations of al-jabr and al-muqabala. The word al-jabr means- completion and al-muqabal means- balancing. Let us understand the two words with the example-

For the quation-

x² = 40 – 4x²  the al- jabr transforms into 5 x² = 40 x and two applications of al-muqabala reduces 50 + 3x +x² = 29 +10 x into 21 + x² = 7x.

He furthers explain the process of multiplication of expression like- (a+ bx) (c + dx). Gandz writes about the al-Khwarizmi’ algebra –

Al-Khwarizmi's algebra is regarded as the foundation and cornerstone of the sciences. In a sense, al-Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because al-Khwarizmi is the first to teach algebra in an elementary form and for its own sake,  Diophantus  is primarily concerned with the theory of numbers.

                              

                                                Book Hisab al-jabr w'al-muqabala 

In the next part of his book al- Khwarizmi writes about application of algebra in finding the area of circle, volume of sphere, cone and pyramid. Al- Khwarizmi also wrote a treatise on Hindu- Arabic Numerals which describes about the the Hindu place value system of numerals based on 1,,2,3,4,5,6,7,8,9,0. The first use of zero as pace holder in positional base notation was probably due to al- Khwarizmi.  Besides Algebra, Al-Khwarizmi wrote on geography, astronomy and geometry. Mohammed Khan quoted in praise of al- Khwarizmi’s algebra in the following word-

In the foremost rank of mathematicians of all time stands Al-Khwarizmi. He composed the oldest works on arithmetic and algebra. They were the principal source of mathematical knowledge for centuries to come in the East and the West. The work on arithmetic first introduced the Hindu numbers to Europe, as the very name algorism signifies; and the work on algebra ... gave the name to this important branch of mathematics in the European world...

He introduced the Hindu decimal system to the Islamic world and Europe; invented the horary quadrant; improved the sundial; developed trigonometry tables; and improved on Ptolemy's astronomy and geography. He wrote the book Al-Jabr, which demonstrated simple algebra and geometry, and several other influential books. Unlike Diophantus' work, which dealt in specific examples, Al-Khowârizmi presented general methods. The word algorithm is borrowed from Al-Khowârizmi's name. There were several Muslim mathematicians who contributed to the development of Islamic science, and indirectly to Europe's later Renaissance, but Al-Khowârizmi was one of the earliest and most influential.