The Story of Maths
Encyclopedia
The Story of Maths is a British
United Kingdom
The United Kingdom of Great Britain and Northern IrelandIn the United Kingdom and Dependencies, other languages have been officially recognised as legitimate autochthonous languages under the European Charter for Regional or Minority Languages...

 television
Television
Television is a telecommunication medium for transmitting and receiving moving images that can be monochrome or colored, with accompanying sound...

 series outlining aspects of the history of mathematics
History of mathematics
The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past....

. The series was a co-production between the Open University
Open University
The Open University is a distance learning and research university founded by Royal Charter in the United Kingdom...

 and the BBC
BBC
The British Broadcasting Corporation is a British public service broadcaster. Its headquarters is at Broadcasting House in the City of Westminster, London. It is the largest broadcaster in the world, with about 23,000 staff...

 and aired in October 2008 on BBC Four
BBC Four
BBC Four is a British television network operated by the British Broadcasting Corporation and available to digital television viewers on Freeview, IPTV, satellite and cable....

. The material was written and presented by Oxford professor Marcus du Sautoy
Marcus du Sautoy
Marcus Peter Francis du Sautoy OBE is the Simonyi Professor for the Public Understanding of Science and a Professor of Mathematics at the University of Oxford. Formerly a Fellow of All Souls College, and Wadham College, he is now a Fellow of New College...

. The consultants were the Open University academics Robin Wilson
Robin Wilson (mathematician)
Robin James Wilson is a professor in the Department of Mathematics at the Open University, a Stipendiary Lecturer at Pembroke College, Oxford and, , Professor of Geometry at Gresham College, London, where he has also been a visiting professor...

, Professor Jeremy Gray
Jeremy Gray
Jeremy Gray is an English mathematician primarily interested in the history of mathematics.He studied mathematics at Oxford University from 1966 to 1969, and then at Warwick University, obtaining his Ph.D. in 1980 under the supervision of Ian Stewart and David Fowler. He has worked at the Open...

 and Dr June Barrow-Green. Kim Duke is credited as series producer.

The series comprised four programmes respectively titled: The Language of the Universe; The Genius of the East; The Frontiers of Space; and To Infinity and Beyond. Du Sautoy documents the development of mathematics covering subjects such as the invention of zero and the unproven Riemann hypothesis
Riemann hypothesis
In mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...

, a 150 year old problem for whose solution the Clay Mathematics Institute
Clay Mathematics Institute
The Clay Mathematics Institute is a private, non-profit foundation, based in Cambridge, Massachusetts. The Institute is dedicated to increasing and disseminating mathematical knowledge. It gives out various awards and sponsorships to promising mathematicians. The institute was founded in 1998...

 has offered a $1,000,000 prize. He escorts viewers through the subject's history and geography. He examines the development of key mathematical ideas and shows how mathematical ideas underpin the science, technology, and culture that shape our world.

He starts his journey in ancient Egypt
Egyptian mathematics
Egyptian mathematics is the mathematics that was developed and used in Ancient Egypt from ca. 3000 BC to ca. 300 BC.-Overview:Written evidence of the use of mathematics dates back to at least 3000 BC with the ivory labels found at Tomb Uj at Abydos. These labels appear to have been used as tags for...

 and finishes it by looking at current mathematics. But in between he travels through Babylon
Babylonian mathematics
Babylonian mathematics refers to any mathematics of the people of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in 539 BC. Babylonian mathematical texts are plentiful and well edited...

, Greece
Greek mathematics
Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the Eastern shores of the Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to...

, India
Indian mathematics
Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics , important contributions were made by scholars like Aryabhata, Brahmagupta, and Bhaskara II. The decimal number system in use today was first...

, China
Chinese mathematics
Mathematics in China emerged independently by the 11th century BC. The Chinese independently developed very large and negative numbers, decimals, a place value decimal system, a binary system, algebra, geometry, and trigonometry....

, and the medieval Middle East. He also looks at mathematics in Europe and then in America and takes the viewers inside the lives of many of the greatest mathematicians.

"The Language of the Universe"

In this opening programme Marcus du Sautoy looks at how important and fundamental mathematics is to our lives before looking at the mathematics of ancient Egypt
Ancient Egypt
Ancient Egypt was an ancient civilization of Northeastern Africa, concentrated along the lower reaches of the Nile River in what is now the modern country of Egypt. Egyptian civilization coalesced around 3150 BC with the political unification of Upper and Lower Egypt under the first pharaoh...

, Mesopotamia
Mesopotamia
Mesopotamia is a toponym for the area of the Tigris–Euphrates river system, largely corresponding to modern-day Iraq, northeastern Syria, southeastern Turkey and southwestern Iran.Widely considered to be the cradle of civilization, Bronze Age Mesopotamia included Sumer and the...

, and Greece
Ancient Greece
Ancient Greece is a civilization belonging to a period of Greek history that lasted from the Archaic period of the 8th to 6th centuries BC to the end of antiquity. Immediately following this period was the beginning of the Early Middle Ages and the Byzantine era. Included in Ancient Greece is the...

.

Du Sautoy commences in Egypt
Egyptian mathematics
Egyptian mathematics is the mathematics that was developed and used in Ancient Egypt from ca. 3000 BC to ca. 300 BC.-Overview:Written evidence of the use of mathematics dates back to at least 3000 BC with the ivory labels found at Tomb Uj at Abydos. These labels appear to have been used as tags for...

 where recording the patterns of the seasons and in particular the flooding of the Nile
Nile
The Nile is a major north-flowing river in North Africa, generally regarded as the longest river in the world. It is long. It runs through the ten countries of Sudan, South Sudan, Burundi, Rwanda, Democratic Republic of the Congo, Tanzania, Kenya, Ethiopia, Uganda and Egypt.The Nile has two major...

 was essential to their economy. There was a need to solve practical problems such as land area for taxation purposes. Du Sautoy discovers the use of a decimal system based on the fingers on the hands, the unusual method for multiplication and division. He examines the Rhind Papyrus, the Moscow Papyrus and explores their understanding of binary numbers, fractions and solid shapes.

He then travels to Babylon
Babylonian mathematics
Babylonian mathematics refers to any mathematics of the people of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in 539 BC. Babylonian mathematical texts are plentiful and well edited...

 and discovered that the way we tell the time today is based on the Babylonian 60 base number system
Babylonian numerals
Babylonian numerals were written in cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record....

. So because of the Babylonians we have 60 seconds in a minute, and 60 minutes in an hour. He then shows how the Babylonians used quadratic equation
Quadratic equation
In mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formax^2+bx+c=0,\,...

s to measure their land. He deals briefly with Plimpton 322
Plimpton 322
Plimpton 322 is a Babylonian clay tablet, notable as containing an example of Babylonian mathematics. It has number 322 in the G.A. Plimpton Collection at Columbia University...

.

In Greece, the home of ancient Greek mathematics
Greek mathematics
Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the Eastern shores of the Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to...

, he looks at the contributions of some of its greatest and well known mathematicians including Pythagoras
Pythagoras
Pythagoras of Samos was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him...

, Plato
Plato
Plato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...

, Euclid
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...

, and Archimedes
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...

, who are some of the people who are credited with beginning the transformation of mathematics from a tool for counting into the analytical subject we know today. A controversial figure, Pythagoras’ teachings were considered suspect and his followers seen as social outcasts and a little be strange and not in the norm. There is a legend going around that one of his followers, Hippasus
Hippasus
Hippasus of Metapontum in Magna Graecia, was a Pythagorean philosopher. Little is known about his life or his beliefs, but he is sometimes credited with the discovery of the existence of irrational numbers.-Life:...

, was drowned when he announced his discovery of irrational numbers. As well as his work on the properties of right angled triangles, Pythagoras developed another important theory after observing musical instruments. He discovered that the intervals between harmonious musical notes are always in whole number intervals. It deals briefly with Hypatia of Alexandria
Hypatia of Alexandria
Hypatia was an Egyptian Neoplatonist philosopher who was the first notable woman in mathematics. As head of the Platonist school at Alexandria, she also taught philosophy and astronomy...

.

"The Genius of the East"

With the decline of ancient Greece, the development of maths stagnated in Europe. However the progress of mathematics continued in the East. Du Sautoy describes both the Chinese use of maths
Chinese mathematics
Mathematics in China emerged independently by the 11th century BC. The Chinese independently developed very large and negative numbers, decimals, a place value decimal system, a binary system, algebra, geometry, and trigonometry....

 in engineering projects
History of science and technology in China
The history of science and technology in China is both long and rich with many contributions to science and technology. In antiquity, independently of other civilizations, ancient Chinese philosophers made significant advances in science, technology, mathematics, and astronomy...

 and their belief in the mystical powers of numbers. He mentions Qin Jiushao.

He describes Indian mathematicians
Indian mathematics
Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics , important contributions were made by scholars like Aryabhata, Brahmagupta, and Bhaskara II. The decimal number system in use today was first...

’ invention of trigonometry
Trigonometry
Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...

; their introduction of a symbol for the number zero
0 (number)
0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...

 and their contribution to the new concepts of infinity
Infinity
Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...

 and negative numbers. It shows Gwalior Fort
Gwalior Fort
ċċċċċt̪--122.177.251.15 13:02, 20 November 2011 --122.177.251.15 13:02, 20 November 2011 --122.177.251.15 13:02, 20 November 2011 Gwalior Fort in Gwalior, in the central Indian state of Madhya Pradesh, stands on an isolated rock, overlooking the Gwalior town, and contains a number of historic...

 where zero is inscribed on its walls. It mentions the work of Brahmagupta
Brahmagupta
Brahmagupta was an Indian mathematician and astronomer who wrote many important works on mathematics and astronomy. His best known work is the Brāhmasphuṭasiddhānta , written in 628 in Bhinmal...

 and Bhāskara II on the subject of zero. He mentions Madhava of Sangamagrama
Madhava of Sangamagrama
Mādhava of Sañgamāgrama was a prominent Kerala mathematician-astronomer from the town of Irińńālakkuţa near Cochin, Kerala, India. He is considered the founder of the Kerala School of Astronomy and Mathematics...

 and Aryabhata
Aryabhata
Aryabhata was the first in the line of great mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy...

.

Du Sautoy then considers the Middle East: the invention of the new language of algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

 and the evolution of a solution to cubic equations
Cubic function
In mathematics, a cubic function is a function of the formf=ax^3+bx^2+cx+d,\,where a is nonzero; or in other words, a polynomial of degree three. The derivative of a cubic function is a quadratic function...

. He talks about the House of Wisdom
House of Wisdom
The House of Wisdom was a library and translation institute established in Abbassid-era Baghdad, Iraq. It was a key institution in the Translation Movement and considered to have been a major intellectual centre during the Islamic Golden Age...

 with Muhammad ibn Mūsā al-Khwārizmī
Muhammad ibn Musa al-Khwarizmi
'There is some confusion in the literature on whether al-Khwārizmī's full name is ' or '. Ibn Khaldun notes in his encyclopedic work: "The first who wrote upon this branch was Abu ʿAbdallah al-Khowarizmi, after whom came Abu Kamil Shojaʿ ibn Aslam." . 'There is some confusion in the literature on...

 and he visits University of Al-Karaouine. He mentions Omar Khayyám
Omar Khayyám
Omar Khayyám was aPersian polymath: philosopher, mathematician, astronomer and poet. He also wrote treatises on mechanics, geography, mineralogy, music, climatology and theology....

.

Finally he examines the spread of Eastern knowledge to the West through mathematicians such as Leonardo Fibonacci, famous for the Fibonacci sequence
Fibonacci number
In mathematics, the Fibonacci numbers are the numbers in the following integer sequence:0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots\; ....

. He mentions Niccolò Fontana Tartaglia
Niccolò Fontana Tartaglia
Niccolò Fontana Tartaglia was a mathematician, an engineer , a surveyor and a bookkeeper from the then-Republic of Venice...

.

"The Frontiers of Space"

From the seventeenth century, Europe replaced the Middle East as the engine house of mathematical ideas. Du Sautoy visits Urbino
Urbino
Urbino is a walled city in the Marche region of Italy, south-west of Pesaro, a World Heritage Site notable for a remarkable historical legacy of independent Renaissance culture, especially under the patronage of Federico da Montefeltro, duke of Urbino from 1444 to 1482...

 to introduce Perspective
Perspective (graphical)
Perspective in the graphic arts, such as drawing, is an approximate representation, on a flat surface , of an image as it is seen by the eye...

 using mathematician and artist, Piero della Francesca
Piero della Francesca
Piero della Francesca was a painter of the Early Renaissance. As testified by Giorgio Vasari in his Lives of the Artists, to contemporaries he was also known as a mathematician and geometer. Nowadays Piero della Francesca is chiefly appreciated for his art. His painting was characterized by its...

's The Flagellation of Christ
Flagellation of Christ (Piero della Francesca)
The Flagellation of Christ is a painting by Piero della Francesca in the Galleria Nazionale delle Marche in Urbino, Italy. Called by one writer an "enigmatic little painting," the composition is complex and unusual, and its iconography has been the subject of widely differing theories...

.

Du Sautoy proceeds to describes René Descartes
René Descartes
René Descartes ; was a French philosopher and writer who spent most of his adult life in the Dutch Republic. He has been dubbed the 'Father of Modern Philosophy', and much subsequent Western philosophy is a response to his writings, which are studied closely to this day...

 realisation that it was possible to describe curved lines as equations and thus link algebra and geometry. He talks with Henk Bos about Descartes. He shows how one of Pierre de Fermat
Pierre de Fermat
Pierre de Fermat was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to infinitesimal calculus, including his adequality...

’s theorems is now the basis for the codes that protect credit card transactions on the internet. He describes Isaac Newton’s development of math and physics crucial to understanding the behaviour of moving objects in engineering. He covers the Leibniz and Newton calculus controversy and the Bernoulli family. He further covers Leonard Euler, the father of topology, and Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...

' invention of a new way of handling equations, modular arithmetic. He mentions János Bolyai
János Bolyai
János Bolyai was a Hungarian mathematician, known for his work in non-Euclidean geometry.Bolyai was born in the Transylvanian town of Kolozsvár , then part of the Habsburg Empire , the son of Zsuzsanna Benkő and the well-known mathematician Farkas Bolyai.-Life:By the age of 13, he had mastered...

.

The further contribution of Gauss to our understanding of how prime numbers are distributed is covered thus providing the platform for Bernhard Riemann
Bernhard Riemann
Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....

's theories on prime numbers. In addition Riemann worked on the properties of objects, which he saw as manifolds that could exist in multi-dimensional space.

Hilbert's first problem

The final episode considers the great unsolved problems that confronted mathematicians in the 20th century. On 8 August 1900 David Hilbert
David Hilbert
David Hilbert was a German mathematician. 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...

 gave a historic talk at the International Congress of Mathematicians
International Congress of Mathematicians
The International Congress of Mathematicians is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union ....

 in Paris. Hilbert posed twenty-three then unsolved problems
Hilbert's problems
Hilbert's problems form a list of twenty-three problems in mathematics published by German mathematician David Hilbert in 1900. The problems were all unsolved at the time, and several of them were very influential for 20th century mathematics...

 in mathematics which he believed were of the most immediate importance. Hilbert succeeded in setting the agenda for 20thC mathematics and the programme commenced with Hilbert's first problem.

Georg Cantor
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets,...

 considered the infinite set of whole numbers 1, 2, 3 ... ∞ which he compared with the smaller set of numbers 10, 20, 30 ... ∞. Cantor showed that these two infinite sets of numbers actually had the same size as it was possible to pair each number up; 1 - 10, 2 - 20, 3 - 30 ... etc.

If fractions now are considered there are an infinite number of fractions between any of the two whole numbers, suggesting that the infinity of fractions is bigger than the infinity of whole numbers. Yet Cantor was still able to pair each such fraction to a whole number 1 - 1/1; 2 - 2/1; 3 - 1/2 ... etc through to ∞; ie the infinities of both fractions and whole numbers were shown to have the same size.

But when the set of all infinite decimal numbers was considered, Cantor was able to prove that this produced a bigger infinity. This was because, no matter how one tried to construct such a list, Cantor was able to provide a new decimal number that was missing from that list. Thus he showed that there were different infinities, some bigger than others.

However there was a problem that Cantor was unable to solve: Is there an infinity sitting between the smaller infinity of all the fractions and the larger infinity of the decimals? Cantor believed, in what became known as the Continuum Hypothesis
Continuum hypothesis
In mathematics, the continuum hypothesis is a hypothesis, advanced by Georg Cantor in 1874, about the possible sizes of infinite sets. It states:Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's 23 problems presented in the year 1900...

, that there is no such set. This would be the first problem listed by Hilbert.

Poincaré conjecture

Next Marcus discusses Henri Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...

's work on the discipline of 'Bendy geometry'. If two shapes can be moulded or morphed to each other's shape then they have the same topology. Poincaré was able to identify all possible two-dimensional topological surfaces; however in 1904 he came up with a topological problem, the Poincaré conjecture
Poincaré conjecture
In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere , which is the hypersphere that bounds the unit ball in four-dimensional space...

, that he could not solve; namely what are all the possible shapes for a 3D universe.

According to the programme, the question was solved in 2002 by Grigori Perelman
Grigori Perelman
Grigori Yakovlevich Perelman is a Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology.In 1992, Perelman proved the soul conjecture. In 2002, he proved Thurston's geometrization conjecture...

 who linked the problem to a different area of mathematics. Perelman looked at the dynamics of the way things can flow over the shape. This enabled him to find all the ways that 3D space could be wrapped up in higher dimensions.

David Hilbert

The achievements of David Hilbert were now considered. In addition to Hilbert's problems
Hilbert's problems
Hilbert's problems form a list of twenty-three problems in mathematics published by German mathematician David Hilbert in 1900. The problems were all unsolved at the time, and several of them were very influential for 20th century mathematics...

, Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

, Hilbert Classification and the Hilbert Inequality, du Sautoy highlights Hilbert's early work on equations as marking him out as a mathematician able to think in new ways. Hilbert showed that, while there were an infinity of equations, these equations could be constructed from a finite number of building block like sets. Ironically Hilbert could not construct that list of sets; he simply proved that it existed. In effect Hilbert had created a new more abstract style of Mathematics.

Hilbert's second problem

For 30 years Hilbert believed that mathematics was a universal language powerful enough to unlock all the truths and solve each of his 23 Problems. Yet, even as Hilbert was stating We must know, we will know, Kurt Gödel
Kurt Gödel
Kurt Friedrich Gödel was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the...

 had shattered this belief; he had formulated the Incompleteness Theorem based on his study of Hilbert's second problem
Hilbert's second problem
In mathematics, Hilbert's second problem was posed by David Hilbert in 1900 as one of his 23 problems. It asks for a proof that arithmetic is consistent – free of any internal contradictions....

:
This statement cannot be proved


Using a code based on prime numbers, Gödel was able to transform the above into a pure statement of arithmetic. Logically, the above cannot be false and hence Gödel had discovered the existence of mathematical statements that were true but were incapable of being proved.

Hilbert's first problem revisited

In 1950s America Paul Cohen
Paul Cohen (mathematician)
Paul Joseph Cohen was an American mathematician best known for his proof of the independence of the continuum hypothesis and the axiom of choice from Zermelo–Fraenkel set theory, the most widely accepted axiomatization of set theory.-Early years:Cohen was born in Long Branch, New Jersey, into a...

 took up the challenge of Cantor's Continuum Hypothesis which asks "is there is or isn't there an infinite set of number bigger than the set of whole numbers but smaller than the set of all decimals". Cohen found that there existed two equally consistent mathematical worlds. In one world the Hypothesis was true and there did not exist such a set. Yet there existed a mutually exclusive but equally consistent mathematical proof that Hypothesis was false and there was such a set. Cohen would subsequently work on Hilbert's eighth problem
Hilbert's eighth problem
Hilbert's eighth problem is one of David Hilbert's list of open mathematical problems posed in 1900. It concerns number theory, and in particular the Riemann hypothesis, although it is also concerned with the Goldbach Conjecture...

, the Riemann hypothesis
Riemann hypothesis
In mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...

, although without the success of his earlier work.

Hilbert's tenth problem

Hilbert's tenth problem
Hilbert's tenth problem
Hilbert's tenth problem is the tenth on the list of Hilbert's problems of 1900. Its statement is as follows:Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite...

 asked if there was some universal method that could tell whether any equation had whole number solutions or not. The growing belief was that no so such method was possible yet the question remained, how could you prove that, no matter how ingenious you were, you would never come up with such a method. He mentions Paul Cohen
Paul Cohen (mathematician)
Paul Joseph Cohen was an American mathematician best known for his proof of the independence of the continuum hypothesis and the axiom of choice from Zermelo–Fraenkel set theory, the most widely accepted axiomatization of set theory.-Early years:Cohen was born in Long Branch, New Jersey, into a...

. To answer this Julia Robinson
Julia Robinson
Julia Hall Bowman Robinson was an American mathematician best known for her work on decision problems and Hilbert's Tenth Problem.-Background and education:...

, who created the Robinson Hypothesis which stated that to show that there was no such method all you had to do was cook up one equation whose solutions were a very specific set of numbers: The set of numbers needed to grow exponentially yet still be captured by the equations at the heart of Hilbert's problem. Robinson was unable to find this set. This part of the solution fell to Yuri Matiyasevich
Yuri Matiyasevich
Yuri Vladimirovich Matiyasevich, is a Russian mathematician and computer scientist. He is best known for his negative solution of Hilbert's tenth problem, presented in his doctoral thesis, at LOMI .- Biography :* In 1962-1963 studied at Saint Petersburg Lyceum 239...

 who saw how to capture the Fibonacci sequence using the equations at the heart of Hilbert's tenth.

Algebraic geometry

The final section briefly covers algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

. Évariste Galois
Évariste Galois
Évariste Galois was a French mathematician born in Bourg-la-Reine. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long-standing problem...

 had refined a new language for mathematics. Galois believed mathematics should be the study of structure as opposed to number and shape. Galois had discovered new techniques to tell whether certain equations could have solutions or not. The symmetry of certain geometric objects was the key. Galois' work was picked up by André Weil
André Weil
André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...

 who built Algebraic Geometry, a whole new language. Weil's work connected number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, algebra, topology and geometry.

Finally du Sautoy mentions Weil's part in the creation of the fictional mathematician Nicolas Bourbaki
Nicolas Bourbaki
Nicolas Bourbaki is the collective pseudonym under which a group of 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. With the goal of founding all of mathematics on set theory, the group strove for rigour and generality...

 and another contributor to Bourbaki's output - Alexander Grothendieck
Alexander Grothendieck
Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...

.

See also

  • Chemistry: A Volatile History
    Chemistry: A Volatile History
    Chemistry: A Volatile History is a 2010 BBC documentary on the history of chemistry presented by Jim Al-Khalili. It was nominated for the 2010 British Academy Television Awards in the category Specialist Factual.-Introduction:...

  • The Story of Science: Power, Proof and Passion
    The Story of Science: Power, Proof and Passion
    The Story of Science: Power, Proof and Passion is a 2010 BBC documentary on the history of science presented by Michael J. Mosley.- Episodes :- External links :*...


External links

The source of this article is wikipedia, the free encyclopedia.  The text of this article is licensed under the GFDL.
 
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