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.
Before the modern age and the worldwide spread
of knowledge, written examples of new mathematical
developments have come to light only in a
few locales.
From 3000 BC the Mesopotamian states of Sumer,
Akkad and Assyria, together with Ancient Egypt
and Ebla began using arithmetic, algebra and
geometry for purposes of taxation, commerce,
trade and also in the field of astronomy and
to formulate calendars and record time.
The most ancient mathematical texts available
are from Mesopotamia and Egypt - Plimpton
322 (Babylonian c. 1900 BC), the Rhind Mathematical
Papyrus (Egyptian c. 2000–1800 BC) and the
Moscow Mathematical Papyrus (Egyptian c. 1890
BC).
All of these texts mention the so-called Pythagorean
triples and so, by inference, the Pythagorean
theorem, seems to be the most ancient and
widespread mathematical development after
basic arithmetic and geometry.
The study of mathematics as a "demonstrative
discipline" begins in the 6th century BC with
the Pythagoreans, who coined the term "mathematics"
from the ancient Greek μάθημα (mathema),
meaning "subject of instruction".
Greek mathematics greatly refined the methods
(especially through the introduction of deductive
reasoning and mathematical rigor in proofs)
and expanded the subject matter of mathematics.
Although they made virtually no contributions
to theoretical mathematics, the ancient Romans
used applied mathematics in surveying, structural
engineering, mechanical engineering, bookkeeping,
creation of lunar and solar calendars, and
even arts and crafts.
Chinese mathematics made early contributions,
including a place value system and the first
use of negative numbers.
The Hindu–Arabic numeral system and the
rules for the use of its operations, in use
throughout the world today evolved over the
course of the first millennium AD in India
and were transmitted to the Western world
via Islamic mathematics through the work of
Muḥammad ibn Mūsā al-Khwārizmī.
Islamic mathematics, in turn, developed and
expanded the mathematics known to these civilizations.
Contemporaneous with but independent of these
traditions were the mathematics developed
by the Maya civilization of Mexico and Central
America, where the concept of zero was given
a standard symbol in Maya numerals.
Many Greek and Arabic texts on mathematics
were translated into Latin from the 12th century
onward, leading to further development of
mathematics in Medieval Europe.
From ancient times through the Middle Ages,
periods of mathematical discovery were often
followed by centuries of stagnation.
Beginning in Renaissance Italy in the 15th
century, new mathematical developments, interacting
with new scientific discoveries, were made
at an increasing pace that continues through
the present day.
This includes the groundbreaking work of both
Isaac Newton and Gottfried Wilhelm Leibniz
in the development of infinitesimal calculus
during the course of the 17th century.
At the end of the 19th century the International
Congress of Mathematicians was founded and
continues to spearhead advances in the field.
== Prehistoric ==
The origins of mathematical thought lie in
the concepts of number, magnitude, and form.
Modern studies of animal cognition have shown
that these concepts are not unique to humans.
Such concepts would have been part of everyday
life in hunter-gatherer societies.
The idea of the "number" concept evolving
gradually over time is supported by the existence
of languages which preserve the distinction
between "one", "two", and "many", but not
of numbers larger than two.Prehistoric artifacts
discovered in Africa, dated 20,000 years old
or more suggest early attempts to quantify
time.
The Ishango bone, found near the headwaters
of the Nile river (northeastern Congo), may
be more than 20,000 years old and consists
of a series of marks carved in three columns
running the length of the bone.
Common interpretations are that the Ishango
bone shows either a tally of the earliest
known demonstration of sequences of prime
numbers or a six-month lunar calendar.
Peter Rudman argues that the development of
the concept of prime numbers could only have
come about after the concept of division,
which he dates to after 10,000 BC, with prime
numbers probably not being understood until
about 500 BC.
He also writes that "no attempt has been made
to explain why a tally of something should
exhibit multiples of two, prime numbers between
10 and 20, and some numbers that are almost
multiples of 10."
The Ishango bone, according to scholar Alexander
Marshack, may have influenced the later development
of mathematics in Egypt as, like some entries
on the Ishango bone, Egyptian arithmetic also
made use of multiplication by 2; this however,
is disputed.Predynastic Egyptians of the 5th
millennium BC pictorially represented geometric
designs.
It has been claimed that megalithic monuments
in England and Scotland, dating from the 3rd
millennium BC, incorporate geometric ideas
such as circles, ellipses, and Pythagorean
triples in their design.
All of the above are disputed however, and
the currently oldest undisputed mathematical
documents are from Babylonian and dynastic
Egyptian sources.
== Babylonian ==
Babylonian mathematics refers to any mathematics
of the peoples of Mesopotamia (modern Iraq)
from the days of the early Sumerians through
the Hellenistic period almost to the dawn
of Christianity.
The majority of Babylonian mathematical work
comes from two widely separated periods: The
first few hundred years of the second millennium
BC (Old Babylonian period), and the last few
centuries of the first millennium BC (Seleucid
period).
It is named Babylonian mathematics due to
the central role of Babylon as a place of
study.
Later under the Arab Empire, Mesopotamia,
especially Baghdad, once again became an important
center of study for Islamic mathematics.
In contrast to the sparsity of sources in
Egyptian mathematics, our knowledge of Babylonian
mathematics is derived from more than 400
clay tablets unearthed since the 1850s.
Written in Cuneiform script, tablets were
inscribed whilst the clay was moist, and baked
hard in an oven or by the heat of the sun.
Some of these appear to be graded homework.The
earliest evidence of written mathematics dates
back to the ancient Sumerians, who built the
earliest civilization in Mesopotamia.
They developed a complex system of metrology
from 3000 BC.
From around 2500 BC onwards, the Sumerians
wrote multiplication tables on clay tablets
and dealt with geometrical exercises and division
problems.
The earliest traces of the Babylonian numerals
also date back to this period.
Babylonian mathematics were written using
a sexagesimal (base-60) numeral system.
From this derives the modern day usage of
60 seconds in a minute, 60 minutes in an hour,
and 360 (60 x 6) degrees in a circle, as well
as the use of seconds and minutes of arc to
denote fractions of a degree.
It is likely the sexagesimal system was chosen
because 60 can be evenly divided by 2, 3,
4, 5, 6, 10, 12, 15, 20 and 30.
Also, unlike the Egyptians, Greeks, and Romans,
the Babylonians had a true place-value system,
where digits written in the left column represented
larger values, much as in the decimal system.
The power of the Babylonian notational system
lay in that it could be used to represent
fractions as easily as whole numbers; thus
multiplying two numbers that contained fractions
was no different than multiplying integers,
similar to our modern notation.
The notational system of the Babylonians was
the best of any civilization until the Renaissance,
and its power allowed it to achieve remarkable
computation accuracy and power; for example,
the Babylonian tablet YBC 7289 gives an approximation
of √2 accurate to five decimal places.
The Babylonians lacked, however, an equivalent
of the decimal point, and so the place value
of a symbol often had to be inferred from
the context.
By the Seleucid period, the Babylonians had
developed a zero symbol as a placeholder for
empty positions; however it was only used
for intermediate positions.
This zero sign does not appear in terminal
positions, thus the Babylonians came close
but did not develop a true place value system.Other
topics covered by Babylonian mathematics include
fractions, algebra, quadratic and cubic equations,
and the calculation of regular reciprocal
pairs.
The tablets also include multiplication tables
and methods for solving linear, quadratic
equations and cubic equations, a remarkable
achievement for the time.
Tablets from the Old Babylonian period also
contain the earliest known statement of the
Pythagorean theorem.
However, as with Egyptian mathematics, Babylonian
mathematics shows no awareness of the difference
between exact and approximate solutions, or
the solvability of a problem, and most importantly,
no explicit statement of the need for proofs
or logical principles.
== Egyptian ==
Egyptian mathematics refers to mathematics
written in the Egyptian language.
From the Hellenistic period, Greek replaced
Egyptian as the written language of Egyptian
scholars.
Mathematical study in Egypt later continued
under the Arab Empire as part of Islamic mathematics,
when Arabic became the written language of
Egyptian scholars.
The most extensive Egyptian mathematical text
is the Rhind papyrus (sometimes also called
the Ahmes Papyrus after its author), dated
to c. 1650 BC but likely a copy of an older
document from the Middle Kingdom of about
2000–1800 BC.
It is an instruction manual for students in
arithmetic and geometry.
In addition to giving area formulas and methods
for multiplication, division and working with
unit fractions, it also contains evidence
of other mathematical knowledge, including
composite and prime numbers; arithmetic, geometric
and harmonic means; and simplistic understandings
of both the Sieve of Eratosthenes and perfect
number theory (namely, that of the number
6).
It also shows how to solve first order linear
equations as well as arithmetic and geometric
series.Another significant Egyptian mathematical
text is the Moscow papyrus, also from the
Middle Kingdom period, dated to c. 1890 BC.
It consists of what are today called word
problems or story problems, which were apparently
intended as entertainment.
One problem is considered to be of particular
importance because it gives a method for finding
the volume of a frustum (truncated pyramid).
Finally, the Berlin Papyrus 6619 (c. 1800
BC) shows that ancient Egyptians could solve
a second-order algebraic equation.
== Greek ==
Greek mathematics refers to the mathematics
written in the Greek language from the time
of Thales of Miletus (~600 BC) to the closure
of the Academy of Athens in 529 AD.
Greek mathematicians lived in cities spread
over the entire Eastern Mediterranean, from
Italy to North Africa, but were united by
culture and language.
Greek mathematics of the period following
Alexander the Great is sometimes called Hellenistic
mathematics.Greek mathematics was much more
sophisticated than the mathematics that had
been developed by earlier cultures.
All surviving records of pre-Greek mathematics
show the use of inductive reasoning, that
is, repeated observations used to establish
rules of thumb.
Greek mathematicians, by contrast, used deductive
reasoning.
The Greeks used logic to derive conclusions
from definitions and axioms, and used mathematical
rigor to prove them.Greek mathematics is thought
to have begun with Thales of Miletus (c. 624–c.546
BC) and Pythagoras of Samos (c. 582–c. 507
BC).
Although the extent of the influence is disputed,
they were probably inspired by Egyptian and
Babylonian mathematics.
According to legend, Pythagoras traveled to
Egypt to learn mathematics, geometry, and
astronomy from Egyptian priests.
Thales used geometry to solve problems such
as calculating the height of pyramids and
the distance of ships from the shore.
He is credited with the first use of deductive
reasoning applied to geometry, by deriving
four corollaries to Thales' Theorem.
As a result, he has been hailed as the first
true mathematician and the first known individual
to whom a mathematical discovery has been
attributed.
Pythagoras established the Pythagorean School,
whose doctrine it was that mathematics ruled
the universe and whose motto was "All is number".
It was the Pythagoreans who coined the term
"mathematics", and with whom the study of
mathematics for its own sake begins.
The Pythagoreans are credited with the first
proof of the Pythagorean theorem, though the
statement of the theorem has a long history,
and with the proof of the existence of irrational
numbers.
Although he was preceded by the Babylonians
and the Chinese, the Neopythagorean mathematician
Nicomachus (60–120 AD) provided one of the
earliest Greco-Roman multiplication tables,
whereas the oldest extant Greek multiplication
table is found on a wax tablet dated to the
1st century AD (now found in the British Museum).
The association of the Neopythagoreans with
the Western invention of the multiplication
table is evident in its later Medieval name:
the mensa Pythagorica.Plato (428/427 BC – 348/347
BC) is important in the history of mathematics
for inspiring and guiding others.
His Platonic Academy, in Athens, became the
mathematical center of the world in the 4th
century BC, and it was from this school that
the leading mathematicians of the day, such
as Eudoxus of Cnidus, came.
Plato also discussed the foundations of mathematics,
clarified some of the definitions (e.g. that
of a line as "breadthless length"), and reorganized
the assumptions.
The analytic method is ascribed to Plato,
while a formula for obtaining Pythagorean
triples bears his name.Eudoxus (408–c. 355
BC) developed the method of exhaustion, a
precursor of modern integration and a theory
of ratios that avoided the problem of incommensurable
magnitudes.
The former allowed the calculations of areas
and volumes of curvilinear figures, while
the latter enabled subsequent geometers to
make significant advances in geometry.
Though he made no specific technical mathematical
discoveries, Aristotle (384–c. 322 BC) contributed
significantly to the development of mathematics
by laying the foundations of logic.
In the 3rd century BC, the premier center
of mathematical education and research was
the Musaeum of Alexandria.
It was there that Euclid (c. 300 BC) taught,
and wrote the Elements, widely considered
the most successful and influential textbook
of all time.
The Elements introduced mathematical rigor
through the axiomatic method and is the earliest
example of the format still used in mathematics
today, that of definition, axiom, theorem,
and proof.
Although most of the contents of the Elements
were already known, Euclid arranged them into
a single, coherent logical framework.
The Elements was known to all educated people
in the West until the middle of the 20th century
and its contents are still taught in geometry
classes today.
In addition to the familiar theorems of Euclidean
geometry, the Elements was meant as an introductory
textbook to all mathematical subjects of the
time, such as number theory, algebra and solid
geometry, including proofs that the square
root of two is irrational and that there are
infinitely many prime numbers.
Euclid also wrote extensively on other subjects,
such as conic sections, optics, spherical
geometry, and mechanics, but only half of
his writings survive.
Archimedes (c. 287–212 BC) of Syracuse,
widely considered the greatest mathematician
of antiquity, used the method of exhaustion
to calculate the area under the arc of a parabola
with the summation of an infinite series,
in a manner not too dissimilar from modern
calculus.
He also showed one could use the method of
exhaustion to calculate the value of π with
as much precision as desired, and obtained
the most accurate value of π then known,
310/71 < π < 310/70.
He also studied the spiral bearing his name,
obtained formulas for the volumes of surfaces
of revolution (paraboloid, ellipsoid, hyperboloid),
and an ingenious method of exponentiation
for expressing very large numbers.
While he is also known for his contributions
to physics and several advanced mechanical
devices, Archimedes himself placed far greater
value on the products of his thought and general
mathematical principles.
He regarded as his greatest achievement his
finding of the surface area and volume of
a sphere, which he obtained by proving these
are 2/3 the surface area and volume of a cylinder
circumscribing the sphere.
Apollonius of Perga (c. 262–190 BC) made
significant advances to the study of conic
sections, showing that one can obtain all
three varieties of conic section by varying
the angle of the plane that cuts a double-napped
cone.
He also coined the terminology in use today
for conic sections, namely parabola ("place
beside" or "comparison"), "ellipse" ("deficiency"),
and "hyperbola" ("a throw beyond").
His work Conics is one of the best known and
preserved mathematical works from antiquity,
and in it he derives many theorems concerning
conic sections that would prove invaluable
to later mathematicians and astronomers studying
planetary motion, such as Isaac Newton.
While neither Apollonius nor any other Greek
mathematicians made the leap to coordinate
geometry, Apollonius' treatment of curves
is in some ways similar to the modern treatment,
and some of his work seems to anticipate the
development of analytical geometry by Descartes
some 1800 years later.Around the same time,
Eratosthenes of Cyrene (c. 276–194 BC) devised
the Sieve of Eratosthenes for finding prime
numbers.
The 3rd century BC is generally regarded as
the "Golden Age" of Greek mathematics, with
advances in pure mathematics henceforth in
relative decline.
Nevertheless, in the centuries that followed
significant advances were made in applied
mathematics, most notably trigonometry, largely
to address the needs of astronomers.
Hipparchus of Nicaea (c. 190–120 BC) is
considered the founder of trigonometry for
compiling the first known trigonometric table,
and to him is also due the systematic use
of the 360 degree circle.
Heron of Alexandria (c. 10–70 AD) is credited
with Heron's formula for finding the area
of a scalene triangle and with being the first
to recognize the possibility of negative numbers
possessing square roots.
Menelaus of Alexandria (c. 100 AD) pioneered
spherical trigonometry through Menelaus' theorem.
The most complete and influential trigonometric
work of antiquity is the Almagest of Ptolemy
(c.
AD 90–168), a landmark astronomical treatise
whose trigonometric tables would be used by
astronomers for the next thousand years.
Ptolemy is also credited with Ptolemy's theorem
for deriving trigonometric quantities, and
the most accurate value of π outside of China
until the medieval period, 3.1416.
Following a period of stagnation after Ptolemy,
the period between 250 and 350 AD is sometimes
referred to as the "Silver Age" of Greek mathematics.
During this period, Diophantus made significant
advances in algebra, particularly indeterminate
analysis, which is also known as "Diophantine
analysis".
The study of Diophantine equations and Diophantine
approximations is a significant area of research
to this day.
His main work was the Arithmetica, a collection
of 150 algebraic problems dealing with exact
solutions to determinate and indeterminate
equations.
The Arithmetica had a significant influence
on later mathematicians, such as Pierre de
Fermat, who arrived at his famous Last Theorem
after trying to generalize a problem he had
read in the Arithmetica (that of dividing
a square into two squares).
Diophantus also made significant advances
in notation, the Arithmetica being the first
instance of algebraic symbolism and syncopation.
Among the last great Greek mathematicians
is Pappus of Alexandria (4th century AD).
He is known for his hexagon theorem and centroid
theorem, as well as the Pappus configuration
and Pappus graph.
His Collection is a major source of knowledge
on Greek mathematics as most of it has survived.
Pappus is considered the last major innovator
in Greek mathematics, with subsequent work
consisting mostly of commentaries on earlier
work.
The first woman mathematician recorded by
history was Hypatia of Alexandria (AD 350–415).
She succeeded her father (Theon of Alexandria)
as Librarian at the Great Library and wrote
many works on applied mathematics.
Because of a political dispute, the Christian
community in Alexandria had her stripped publicly
and executed.
Her death is sometimes taken as the end of
the era of the Alexandrian Greek mathematics,
although work did continue in Athens for another
century with figures such as Proclus, Simplicius
and Eutocius.
Although Proclus and Simplicius were more
philosophers than mathematicians, their commentaries
on earlier works are valuable sources on Greek
mathematics.
The closure of the neo-Platonic Academy of
Athens by the emperor Justinian in 529 AD
is traditionally held as marking the end of
the era of Greek mathematics, although the
Greek tradition continued unbroken in the
Byzantine empire with mathematicians such
as Anthemius of Tralles and Isidore of Miletus,
the architects of the Hagia Sophia.
Nevertheless, Byzantine mathematics consisted
mostly of commentaries, with little in the
way of innovation, and the centers of mathematical
innovation were to be found elsewhere by this
time.
== Roman ==
Although ethnic Greek mathematicians continued
to live under the rule of the late Roman Republic
and subsequent Roman Empire, there were no
noteworthy native Latin mathematicians in
comparison.
Ancient Romans such as Cicero (106–43 BC),
an influential Roman statesman who studied
mathematics in Greece, believed that Roman
surveyors and calculators were far more interested
in applied mathematics than the theoretical
mathematics and geometry that were prized
by the Greeks.
It is unclear if the Romans first derived
their numerical system directly from the Greek
precedent or from Etruscan numerals used by
the Etruscan civilization centered in what
is now Tuscany, central Italy.Using calculation,
Romans were adept at both instigating and
detecting financial fraud, as well as managing
taxes for the treasury.
Siculus Flaccus, one of the Roman gromatici
(i.e. land surveyor), wrote the Categories
of Fields, which aided Roman surveyors in
measuring the surface areas of allotted lands
and territories.
Aside from managing trade and taxes, the Romans
also regularly applied mathematics to solve
problems in engineering, including the erection
of architecture such as bridges, road-building,
and preparation for military campaigns.
Arts and crafts such as Roman mosaics, inspired
by previous Greek designs, created illusionist
geometric patterns and rich, detailed scenes
that required precise measurements for each
tessera tile, the opus tessellatum pieces
on average measuring eight millimeters square
and the finer opus vermiculatum pieces having
an average surface of four millimeters square.The
creation of the Roman calendar also necessitated
basic mathematics.
The first calendar allegedly dates back to
8th century BC during the Roman Kingdom and
included 356 days plus a leap year every other
year.
In contrast, the lunar calendar of the Republican
era contained 355 days, roughly ten-and-one-fourth
days shorter than the solar year, a discrepancy
that was solved by adding an extra month into
the calendar after the 23rd of February.
This calendar was supplanted by the Julian
calendar, a solar calendar organized by Julius
Caesar (100–44 BC) and devised by Sosigenes
of Alexandria to include a leap day every
four years in a 365-day cycle.
This calendar, which contained an error of
11 minutes and 14 seconds, was later corrected
by the Gregorian calendar organized by Pope
Gregory XIII (r. 1572–1585), virtually the
same solar calendar used in modern times as
the international standard calendar.At roughly
the same time, the Han Chinese and the Romans
both invented the wheeled odometer device
for measuring distances traveled, the Roman
model first described by the Roman civil engineer
and architect Vitruvius (c. 80 BC - c. 15
BC).
The device was used at least until the reign
of emperor Commodus (r. 177 – 192 AD), but
its design seems to have been lost until experiments
were made during the 15th century in Western
Europe.
Perhaps relying on similar gear-work and technology
found in the Antikythera mechanism, the odometer
of Vitruvius featured chariot wheels measuring
4 feet (1.2 m) in diameter turning four-hundred
times in one Roman mile (roughly 4590 ft/1400
m).
With each revolution, a pin-and-axle device
engaged a 400-tooth cogwheel that turned a
second gear responsible for dropping pebbles
into a box, each pebble representing one mile
traversed.
== Chinese ==
An analysis of early Chinese mathematics has
demonstrated its unique development compared
to other parts of the world, leading scholars
to assume an entirely independent development.
The oldest extant mathematical text from China
is the Zhoubi Suanjing, variously dated to
between 1200 BC and 100 BC, though a date
of about 300 BC during the Warring States
Period appears reasonable.
However, the Tsinghua Bamboo Slips, containing
the earliest known decimal multiplication
table (although ancient Babylonians had ones
with a base of 60), is dated around 305 BC
and is perhaps the oldest surviving mathematical
text of China.
Of particular note is the use in Chinese mathematics
of a decimal positional notation system, the
so-called "rod numerals" in which distinct
ciphers were used for numbers between 1 and
10, and additional ciphers for powers of ten.
Thus, the number 123 would be written using
the symbol for "1", followed by the symbol
for "100", then the symbol for "2" followed
by the symbol for "10", followed by the symbol
for "3".
This was the most advanced number system in
the world at the time, apparently in use several
centuries before the common era and well before
the development of the Indian numeral system.
Rod numerals allowed the representation of
numbers as large as desired and allowed calculations
to be carried out on the suan pan, or Chinese
abacus.
The date of the invention of the suan pan
is not certain, but the earliest written mention
dates from AD 190, in Xu Yue's Supplementary
Notes on the Art of Figures.
The oldest existent work on geometry in China
comes from the philosophical Mohist canon
c. 330 BC, compiled by the followers of Mozi
(470–390 BC).
The Mo Jing described various aspects of many
fields associated with physical science, and
provided a small number of geometrical theorems
as well.
It also defined the concepts of circumference,
diameter, radius, and volume.
In 212 BC, the Emperor Qin Shi Huang commanded
all books in the Qin Empire other than officially
sanctioned ones be burned.
This decree was not universally obeyed, but
as a consequence of this order little is known
about ancient Chinese mathematics before this
date.
After the book burning of 212 BC, the Han
dynasty (202 BC–220 AD) produced works of
mathematics which presumably expanded on works
that are now lost.
The most important of these is The Nine Chapters
on the Mathematical Art, the full title of
which appeared by AD 179, but existed in part
under other titles beforehand.
It consists of 246 word problems involving
agriculture, business, employment of geometry
to figure height spans and dimension ratios
for Chinese pagoda towers, engineering, surveying,
and includes material on right triangles.
It created mathematical proof for the Pythagorean
theorem, and a mathematical formula for Gaussian
elimination.
The treatise also provides values of π, which
Chinese mathematicians originally approximated
as 3 until Liu Xin (d.
23 AD) provided a figure of 3.1457 and subsequently
Zhang Heng (78–139) approximated pi as 3.1724,
as well as 3.162 by taking the square root
of 10.
Liu Hui commented on the Nine Chapters in
the 3rd century AD and gave a value of π
accurate to 5 decimal places (i.e. 3.14159).
Though more of a matter of computational stamina
than theoretical insight, in the 5th century
AD Zu Chongzhi computed the value of π to
seven decimal places (i.e. 3.141592), which
remained the most accurate value of π for
almost the next 1000 years.
He also established a method which would later
be called Cavalieri's principle to find the
volume of a sphere.The high-water mark of
Chinese mathematics occurred in the 13th century
during the latter half of the Song dynasty
(960–1279), with the development of Chinese
algebra.
The most important text from that period is
the Precious Mirror of the Four Elements by
Zhu Shijie (1249–1314), dealing with the
solution of simultaneous higher order algebraic
equations using a method similar to Horner's
method.
The Precious Mirror also contains a diagram
of Pascal's triangle with coefficients of
binomial expansions through the eighth power,
though both appear in Chinese works as early
as 1100.
The Chinese also made use of the complex combinatorial
diagram known as the magic square and magic
circles, described in ancient times and perfected
by Yang Hui (AD 1238–1298).Even after European
mathematics began to flourish during the Renaissance,
European and Chinese mathematics were separate
traditions, with significant Chinese mathematical
output in decline from the 13th century onwards.
Jesuit missionaries such as Matteo Ricci carried
mathematical ideas back and forth between
the two cultures from the 16th to 18th centuries,
though at this point far more mathematical
ideas were entering China than leaving.Japanese
mathematics, Korean mathematics, and Vietnamese
mathematics are traditionally viewed as stemming
from Chinese mathematics and belonging to
the Confucian-based East Asian cultural sphere.
Korean and Japanese mathematics were heavily
influenced by the algebraic works produced
during China's Song dynasty, whereas Vietnamese
mathematics was heavily indebted to popular
works of China's Ming dynasty (1368–1644).
For instance, although Vietnamese mathematical
treatises were written in either Chinese or
the native Vietnamese Chữ Nôm script, all
of them followed the Chinese format of presenting
a collection of problems with algorithms for
solving them, followed by numerical answers.
Mathematics in Vietnam and Korea were mostly
associated with the professional court bureaucracy
of mathematicians and astronomers, whereas
in Japan it was more prevalent in the realm
of private schools.
== Indian ==
The earliest civilization on the Indian subcontinent
is the Indus Valley Civilization (mature phase:
2600 to 1900 BC) that flourished in the Indus
river basin.
Their cities were laid out with geometric
regularity, but no known mathematical documents
survive from this civilization.The oldest
extant mathematical records from India are
the Sulba Sutras (dated variously between
the 8th century BC and the 2nd century AD),
appendices to religious texts which give simple
rules for constructing altars of various shapes,
such as squares, rectangles, parallelograms,
and others.
As with Egypt, the preoccupation with temple
functions points to an origin of mathematics
in religious ritual.
The Sulba Sutras give methods for constructing
a circle with approximately the same area
as a given square, which imply several different
approximations of the value of π.
In addition, they compute the square root
of 2 to several decimal places, list Pythagorean
triples, and give a statement of the Pythagorean
theorem.
All of these results are present in Babylonian
mathematics, indicating Mesopotamian influence.
It is not known to what extent the Sulba Sutras
influenced later Indian mathematicians.
As in China, there is a lack of continuity
in Indian mathematics; significant advances
are separated by long periods of inactivity.Pāṇini
(c. 5th century BC) formulated the rules for
Sanskrit grammar.
His notation was similar to modern mathematical
notation, and used metarules, transformations,
and recursion.
Pingala (roughly 3rd–1st centuries BC) in
his treatise of prosody uses a device corresponding
to a binary numeral system.
His discussion of the combinatorics of meters
corresponds to an elementary version of the
binomial theorem.
Pingala's work also contains the basic ideas
of Fibonacci numbers (called mātrāmeru).The
next significant mathematical documents from
India after the Sulba Sutras are the Siddhantas,
astronomical treatises from the 4th and 5th
centuries AD (Gupta period) showing strong
Hellenistic influence.
They are significant in that they contain
the first instance of trigonometric relations
based on the half-chord, as is the case in
modern trigonometry, rather than the full
chord, as was the case in Ptolemaic trigonometry.
Through a series of translation errors, the
words "sine" and "cosine" derive from the
Sanskrit "jiya" and "kojiya".
Around 500 AD, Aryabhata wrote the Aryabhatiya,
a slim volume, written in verse, intended
to supplement the rules of calculation used
in astronomy and mathematical mensuration,
though with no feeling for logic or deductive
methodology.
Though about half of the entries are wrong,
it is in the Aryabhatiya that the decimal
place-value system first appears.
Several centuries later, the Muslim mathematician
Abu Rayhan Biruni described the Aryabhatiya
as a "mix of common pebbles and costly crystals".In
the 7th century, Brahmagupta identified the
Brahmagupta theorem, Brahmagupta's identity
and Brahmagupta's formula, and for the first
time, in Brahma-sphuta-siddhanta, he lucidly
explained the use of zero as both a placeholder
and decimal digit, and explained the Hindu–Arabic
numeral system.
It was from a translation of this Indian text
on mathematics (c. 770) that Islamic mathematicians
were introduced to this numeral system, which
they adapted as Arabic numerals.
Islamic scholars carried knowledge of this
number system to Europe by the 12th century,
and it has now displaced all older number
systems throughout the world.
Various symbol sets are used to represent
numbers in the Hindu–Arabic numeral system,
all of which evolved from the Brahmi numerals.
Each of the roughly dozen major scripts of
India has its own numeral glyphs.
In the 10th century, Halayudha's commentary
on Pingala's work contains a study of the
Fibonacci sequence and Pascal's triangle,
and describes the formation of a matrix.In
the 12th century, Bhāskara II lived in southern
India and wrote extensively on all then known
branches of mathematics.
His work contains mathematical objects equivalent
or approximately equivalent to infinitesimals,
derivatives, the mean value theorem and the
derivative of the sine function.
To what extent he anticipated the invention
of calculus is a controversial subject among
historians of mathematics.In the 14th century,
Madhava of Sangamagrama, the founder of the
so-called Kerala School of Mathematics, found
the Madhava–Leibniz series, and, using 21
terms, computed the value of π as 3.14159265359.
Madhava also found the Madhava-Gregory series
to determine the arctangent, the Madhava-Newton
power series to determine sine and cosine
and the Taylor approximation for sine and
cosine functions.
In the 16th century, Jyesthadeva consolidated
many of the Kerala School's developments and
theorems in the Yukti-bhāṣā.
However, the Kerala School did not formulate
a systematic theory of differentiation and
integration, nor is there any direct evidence
of their results being transmitted outside
Kerala.
== Islamic empire ==
The Islamic Empire established across Persia,
the Middle East, Central Asia, North Africa,
Iberia, and in parts of India in the 8th century
made significant contributions towards mathematics.
Although most Islamic texts on mathematics
were written in Arabic, most of them were
not written by Arabs, since much like the
status of Greek in the Hellenistic world,
Arabic was used as the written language of
non-Arab scholars throughout the Islamic world
at the time.
Persians contributed to the world of Mathematics
alongside Arabs.
In the 9th century, the Persian mathematician
Muḥammad ibn Mūsā al-Khwārizmī wrote
several important books on the Hindu–Arabic
numerals and on methods for solving equations.
His book On the Calculation with Hindu Numerals,
written about 825, along with the work of
Al-Kindi, were instrumental in spreading Indian
mathematics and Indian numerals to the West.
The word algorithm is derived from the Latinization
of his name, Algoritmi, and the word algebra
from the title of one of his works, Al-Kitāb
al-mukhtaṣar fī hīsāb al-ğabr wa’l-muqābala
(The Compendious Book on Calculation by Completion
and Balancing).
He gave an exhaustive explanation for the
algebraic solution of quadratic equations
with positive roots, and he was the first
to teach algebra in an elementary form and
for its own sake.
He also discussed the fundamental method of
"reduction" and "balancing", referring to
the transposition of subtracted terms to the
other side of an equation, that is, the cancellation
of like terms on opposite sides of the equation.
This is the operation which al-Khwārizmī
originally described as al-jabr.
His algebra was also no longer concerned "with
a series of problems to be resolved, but an
exposition which starts with primitive terms
in which the combinations must give all possible
prototypes for equations, which henceforward
explicitly constitute the true object of study."
He also studied an equation for its own sake
and "in a generic manner, insofar as it does
not simply emerge in the course of solving
a problem, but is specifically called on to
define an infinite class of problems."In Egypt,
Abu Kamil extended algebra to the set of irrational
numbers, accepting square roots and fourth
roots as solutions and coefficients to quadratic
equations.
He also developed techniques used to solve
three non-linear simultaneous equations with
three unknown variables.
One unique feature of his works was trying
to find all the possible solutions to some
of his problems, including one where he found
2676 solutions.
His works formed an important foundation for
the development of algebra and influenced
later mathematicians, such as al-Karaji and
Fibonacci.
Further developments in algebra were made
by Al-Karaji in his treatise al-Fakhri, where
he extends the methodology to incorporate
integer powers and integer roots of unknown
quantities.
Something close to a proof by mathematical
induction appears in a book written by Al-Karaji
around 1000 AD, who used it to prove the binomial
theorem, Pascal's triangle, and the sum of
integral cubes.
The historian of mathematics, F. Woepcke,
praised Al-Karaji for being "the first who
introduced the theory of algebraic calculus."
Also in the 10th century, Abul Wafa translated
the works of Diophantus into Arabic.
Ibn al-Haytham was the first mathematician
to derive the formula for the sum of the fourth
powers, using a method that is readily generalizable
for determining the general formula for the
sum of any integral powers.
He performed an integration in order to find
the volume of a paraboloid, and was able to
generalize his result for the integrals of
polynomials up to the fourth degree.
He thus came close to finding a general formula
for the integrals of polynomials, but he was
not concerned with any polynomials higher
than the fourth degree.In the late 11th century,
Omar Khayyam wrote Discussions of the Difficulties
in Euclid, a book about what he perceived
as flaws in Euclid's Elements, especially
the parallel postulate.
He was also the first to find the general
geometric solution to cubic equations.
He was also very influential in calendar reform.In
the 13th century, Nasir al-Din Tusi (Nasireddin)
made advances in spherical trigonometry.
He also wrote influential work on Euclid's
parallel postulate.
In the 15th century, Ghiyath al-Kashi computed
the value of π to the 16th decimal place.
Kashi also had an algorithm for calculating
nth roots, which was a special case of the
methods given many centuries later by Ruffini
and Horner.
Other achievements of Muslim mathematicians
during this period include the addition of
the decimal point notation to the Arabic numerals,
the discovery of all the modern trigonometric
functions besides the sine, al-Kindi's introduction
of cryptanalysis and frequency analysis, the
development of analytic geometry by Ibn al-Haytham,
the beginning of algebraic geometry by Omar
Khayyam and the development of an algebraic
notation by al-Qalasādī.During the time
of the Ottoman Empire and Safavid Empire from
the 15th century, the development of Islamic
mathematics became stagnant.
== Maya ==
In the Pre-Columbian Americas, the Maya civilization
that flourished in Mexico and Central America
during the 1st millennium AD developed a unique
tradition of mathematics that, due to its
geographic isolation, was entirely independent
of existing European, Egyptian, and Asian
mathematics.
Maya numerals utilized a base of 20, the vigesimal
system, instead of a base of ten that forms
the basis of the decimal system used by most
modern cultures.
The Mayas used mathematics to create the Maya
calendar as well as to predict astronomical
phenomena in their native Maya astronomy.
While the concept of zero had to be inferred
in the mathematics of many contemporary cultures,
the Mayas developed a standard symbol for
it.
== Medieval European ==
Medieval European interest in mathematics
was driven by concerns quite different from
those of modern mathematicians.
One driving element was the belief that mathematics
provided the key to understanding the created
order of nature, frequently justified by Plato's
Timaeus and the biblical passage (in the Book
of Wisdom) that God had ordered all things
in measure, and number, and weight.Boethius
provided a place for mathematics in the curriculum
in the 6th century when he coined the term
quadrivium to describe the study of arithmetic,
geometry, astronomy, and music.
He wrote De institutione arithmetica, a free
translation from the Greek of Nicomachus's
Introduction to Arithmetic; De institutione
musica, also derived from Greek sources; and
a series of excerpts from Euclid's Elements.
His works were theoretical, rather than practical,
and were the basis of mathematical study until
the recovery of Greek and Arabic mathematical
works.In the 12th century, European scholars
traveled to Spain and Sicily seeking scientific
Arabic texts, including al-Khwārizmī's The
Compendious Book on Calculation by Completion
and Balancing, translated into Latin by Robert
of Chester, and the complete text of Euclid's
Elements, translated in various versions by
Adelard of Bath, Herman of Carinthia, and
Gerard of Cremona.
These and other new sources sparked a renewal
of mathematics.
Leonardo of Pisa, now known as Fibonacci,
serendipitously learned about the Hindu–Arabic
numerals on a trip to what is now Béjaïa,
Algeria with his merchant father.
(Europe was still using Roman numerals.)
There, he observed a system of arithmetic
(specifically algorism) which due to the positional
notation of Hindu–Arabic numerals was much
more efficient and greatly facilitated commerce.
Leonardo wrote Liber Abaci in 1202 (updated
in 1254) introducing the technique to Europe
and beginning a long period of popularizing
it.
The book also brought to Europe what is now
known as the Fibonacci sequence (known to
Indian mathematicians for hundreds of years
before that) which was used as an unremarkable
example within the text.
The 14th century saw the development of new
mathematical concepts to investigate a wide
range of problems.
One important contribution was development
of mathematics of local motion.
Thomas Bradwardine proposed that speed (V)
increases in arithmetic proportion as the
ratio of force (F) to resistance (R) increases
in geometric proportion.
Bradwardine expressed this by a series of
specific examples, but although the logarithm
had not yet been conceived, we can express
his conclusion anachronistically by writing:
V = log (F/R).
Bradwardine's analysis is an example of transferring
a mathematical technique used by al-Kindi
and Arnald of Villanova to quantify the nature
of compound medicines to a different physical
problem.
One of the 14th-century Oxford Calculators,
William Heytesbury, lacking differential calculus
and the concept of limits, proposed to measure
instantaneous speed "by the path that would
be described by [a body] if... it were moved
uniformly at the same degree of speed with
which it is moved in that given instant".Heytesbury
and others mathematically determined the distance
covered by a body undergoing uniformly accelerated
motion (today solved by integration), stating
that "a moving body uniformly acquiring or
losing that increment [of speed] will traverse
in some given time a [distance] completely
equal to that which it would traverse if it
were moving continuously through the same
time with the mean degree [of speed]".Nicole
Oresme at the University of Paris and the
Italian Giovanni di Casali independently provided
graphical demonstrations of this relationship,
asserting that the area under the line depicting
the constant acceleration, represented the
total distance traveled.
In a later mathematical commentary on Euclid's
Elements, Oresme made a more detailed general
analysis in which he demonstrated that a body
will acquire in each successive increment
of time an increment of any quality that increases
as the odd numbers.
Since Euclid had demonstrated the sum of the
odd numbers are the square numbers, the total
quality acquired by the body increases as
the square of the time.
== Renaissance ==
During the Renaissance, the development of
mathematics and of accounting were intertwined.
While there is no direct relationship between
algebra and accounting, the teaching of the
subjects and the books published often intended
for the children of merchants who were sent
to reckoning schools (in Flanders and Germany)
or abacus schools (known as abbaco in Italy),
where they learned the skills useful for trade
and commerce.
There is probably no need for algebra in performing
bookkeeping operations, but for complex bartering
operations or the calculation of compound
interest, a basic knowledge of arithmetic
was mandatory and knowledge of algebra was
very useful.
Piero della Francesca (c. 1415–1492) wrote
books on solid geometry and linear perspective,
including De Prospectiva Pingendi (On Perspective
for Painting), Trattato d’Abaco (Abacus
Treatise), and De corporibus regularibus (Regular
Solids).
Luca Pacioli's Summa de Arithmetica, Geometria,
Proportioni et Proportionalità (Italian:
"Review of Arithmetic, Geometry, Ratio and
Proportion") was first printed and published
in Venice in 1494.
It included a 27-page treatise on bookkeeping,
"Particularis de Computis et Scripturis" (Italian:
"Details of Calculation and Recording").
It was written primarily for, and sold mainly
to, merchants who used the book as a reference
text, as a source of pleasure from the mathematical
puzzles it contained, and to aid the education
of their sons.
In Summa Arithmetica, Pacioli introduced symbols
for plus and minus for the first time in a
printed book, symbols that became standard
notation in Italian Renaissance mathematics.
Summa Arithmetica was also the first known
book printed in Italy to contain algebra.
Pacioli obtained many of his ideas from Piero
Della Francesca whom he plagiarized.
In Italy, during the first half of the 16th
century, Scipione del Ferro and Niccolò Fontana
Tartaglia discovered solutions for cubic equations.
Gerolamo Cardano published them in his 1545
book Ars Magna, together with a solution for
the quartic equations, discovered by his student
Lodovico Ferrari.
In 1572 Rafael Bombelli published his L'Algebra
in which he showed how to deal with the imaginary
quantities that could appear in Cardano's
formula for solving cubic equations.
Simon Stevin's book De Thiende ('the art of
tenths'), first published in Dutch in 1585,
contained the first systematic treatment of
decimal notation, which influenced all later
work on the real number system.
Driven by the demands of navigation and the
growing need for accurate maps of large areas,
trigonometry grew to be a major branch of
mathematics.
Bartholomaeus Pitiscus was the first to use
the word, publishing his Trigonometria in
1595.
Regiomontanus's table of sines and cosines
was published in 1533.During the Renaissance
the desire of artists to represent the natural
world realistically, together with the rediscovered
philosophy of the Greeks, led artists to study
mathematics.
They were also the engineers and architects
of that time, and so had need of mathematics
in any case.
The art of painting in perspective, and the
developments in geometry that involved, were
studied intensely.
== Mathematics during the Scientific Revolution
==
=== 17th century ===
The 17th century saw an unprecedented increase
of mathematical and scientific ideas across
Europe.
Galileo observed the moons of Jupiter in orbit
about that planet, using a telescope based
on a toy imported from Holland.
Tycho Brahe had gathered an enormous quantity
of mathematical data describing the positions
of the planets in the sky.
By his position as Brahe's assistant, Johannes
Kepler was first exposed to and seriously
interacted with the topic of planetary motion.
Kepler's calculations were made simpler by
the contemporaneous invention of logarithms
by John Napier and Jost Bürgi.
Kepler succeeded in formulating mathematical
laws of planetary motion.
The analytic geometry developed by René Descartes
(1596–1650) allowed those orbits to be plotted
on a graph, in Cartesian coordinates.
Building on earlier work by many predecessors,
Isaac Newton discovered the laws of physics
explaining Kepler's Laws, and brought together
the concepts now known as calculus.
Independently, Gottfried Wilhelm Leibniz,
who is arguably one of the most important
mathematicians of the 17th century, developed
calculus and much of the calculus notation
still in use today.
Science and mathematics had become an international
endeavor, which would soon spread over the
entire world.In addition to the application
of mathematics to the studies of the heavens,
applied mathematics began to expand into new
areas, with the correspondence of Pierre de
Fermat and Blaise Pascal.
Pascal and Fermat set the groundwork for the
investigations of probability theory and the
corresponding rules of combinatorics in their
discussions over a game of gambling.
Pascal, with his wager, attempted to use the
newly developing probability theory to argue
for a life devoted to religion, on the grounds
that even if the probability of success was
small, the rewards were infinite.
In some sense, this foreshadowed the development
of utility theory in the 18th–19th century.
=== 18th century ===
The most influential mathematician of the
18th century was arguably Leonhard Euler.
His contributions range from founding the
study of graph theory with the Seven Bridges
of Königsberg problem to standardizing many
modern mathematical terms and notations.
For example, he named the square root of minus
1 with the symbol i, and he popularized the
use of the Greek letter
π
{\displaystyle \pi }
to stand for the ratio of a circle's circumference
to its diameter.
He made numerous contributions to the study
of topology, graph theory, calculus, combinatorics,
and complex analysis, as evidenced by the
multitude of theorems and notations named
for him.
Other important European mathematicians of
the 18th century included Joseph Louis Lagrange,
who did pioneering work in number theory,
algebra, differential calculus, and the calculus
of variations, and Laplace who, in the age
of Napoleon, did important work on the foundations
of celestial mechanics and on statistics.
== Modern ==
=== 19th century ===
Throughout the 19th century mathematics became
increasingly abstract.
Carl Friedrich Gauss (1777–1855) epitomizes
this trend.
He did revolutionary work on functions of
complex variables, in geometry, and on the
convergence of series, leaving aside his many
contributions to science.
He also gave the first satisfactory proofs
of the fundamental theorem of algebra and
of the quadratic reciprocity law.
This century saw the development of the two
forms of non-Euclidean geometry, where the
parallel postulate of Euclidean geometry no
longer holds.
The Russian mathematician Nikolai Ivanovich
Lobachevsky and his rival, the Hungarian mathematician
János Bolyai, independently defined and studied
hyperbolic geometry, where uniqueness of parallels
no longer holds.
In this geometry the sum of angles in a triangle
add up to less than 180°.
Elliptic geometry was developed later in the
19th century by the German mathematician Bernhard
Riemann; here no parallel can be found and
the angles in a triangle add up to more than
180°.
Riemann also developed Riemannian geometry,
which unifies and vastly generalizes the three
types of geometry, and he defined the concept
of a manifold, which generalizes the ideas
of curves and surfaces.
The 19th century saw the beginning of a great
deal of abstract algebra.
Hermann Grassmann in Germany gave a first
version of vector spaces, William Rowan Hamilton
in Ireland developed noncommutative algebra.
The British mathematician George Boole devised
an algebra that soon evolved into what is
now called Boolean algebra, in which the only
numbers were 0 and 1.
Boolean algebra is the starting point of mathematical
logic and has important applications in computer
science.
Augustin-Louis Cauchy, Bernhard Riemann, and
Karl Weierstrass reformulated the calculus
in a more rigorous fashion.
Also, for the first time, the limits of mathematics
were explored.
Niels Henrik Abel, a Norwegian, and Évariste
Galois, a Frenchman, proved that there is
no general algebraic method for solving polynomial
equations of degree greater than four (Abel–Ruffini
theorem).
Other 19th-century mathematicians utilized
this in their proofs that straightedge and
compass alone are not sufficient to trisect
an arbitrary angle, to construct the side
of a cube twice the volume of a given cube,
nor to construct a square equal in area to
a given circle.
Mathematicians had vainly attempted to solve
all of these problems since the time of the
ancient Greeks.
On the other hand, the limitation of three
dimensions in geometry was surpassed in the
19th century through considerations of parameter
space and hypercomplex numbers.
Abel and Galois's investigations into the
solutions of various polynomial equations
laid the groundwork for further developments
of group theory, and the associated fields
of abstract algebra.
In the 20th century physicists and other scientists
have seen group theory as the ideal way to
study symmetry.
In the later 19th century, Georg Cantor established
the first foundations of set theory, which
enabled the rigorous treatment of the notion
of infinity and has become the common language
of nearly all mathematics.
Cantor's set theory, and the rise of mathematical
logic in the hands of Peano, L.E.J. Brouwer,
David Hilbert, Bertrand Russell, and A.N.
Whitehead, initiated a long running debate
on the foundations of mathematics.
The 19th century saw the founding of a number
of national mathematical societies: the London
Mathematical Society in 1865, the Société
Mathématique de France in 1872, the Circolo
Matematico di Palermo in 1884, the Edinburgh
Mathematical Society in 1883, and the American
Mathematical Society in 1888.
The first international, special-interest
society, the Quaternion Society, was formed
in 1899, in the context of a vector controversy.
In 1897, Hensel introduced p-adic numbers.
=== 20th century ===
The 20th century saw mathematics become a
major profession.
Every year, thousands of new Ph.D.s in mathematics
were awarded, and jobs were available in both
teaching and industry.
An effort to catalogue the areas and applications
of mathematics was undertaken in Klein's encyclopedia.
In a 1900 speech to the International Congress
of Mathematicians, David Hilbert set out a
list of 23 unsolved problems in mathematics.
These problems, spanning many areas of mathematics,
formed a central focus for much of 20th-century
mathematics.
Today, 10 have been solved, 7 are partially
solved, and 2 are still open.
The remaining 4 are too loosely formulated
to be stated as solved or not.
Notable historical conjectures were finally
proven.
In 1976, Wolfgang Haken and Kenneth Appel
proved the four color theorem, controversial
at the time for the use of a computer to do
so.
Andrew Wiles, building on the work of others,
proved Fermat's Last Theorem in 1995.
Paul Cohen and Kurt Gödel proved that the
continuum hypothesis is independent of (could
neither be proved nor disproved from) the
standard axioms of set theory.
In 1998 Thomas Callister Hales proved the
Kepler conjecture.
Mathematical collaborations of unprecedented
size and scope took place.
An example is the classification of finite
simple groups (also called the "enormous theorem"),
whose proof between 1955 and 1983 required
500-odd journal articles by about 100 authors,
and filling tens of thousands of pages.
A group of French mathematicians, including
Jean Dieudonné and André Weil, publishing
under the pseudonym "Nicolas Bourbaki", attempted
to exposit all of known mathematics as a coherent
rigorous whole.
The resulting several dozen volumes has had
a controversial influence on mathematical
education.
Differential geometry came into its own when
Einstein used it in general relativity.
Entirely new areas of mathematics such as
mathematical logic, topology, and John von
Neumann's game theory changed the kinds of
questions that could be answered by mathematical
methods.
All kinds of structures were abstracted using
axioms and given names like metric spaces,
topological spaces etc.
As mathematicians do, the concept of an abstract
structure was itself abstracted and led to
category theory.
Grothendieck and Serre recast algebraic geometry
using sheaf theory.
Large advances were made in the qualitative
study of dynamical systems that Poincaré
had begun in the 1890s.
Measure theory was developed in the late 19th
and early 20th centuries.
Applications of measures include the Lebesgue
integral, Kolmogorov's axiomatisation of probability
theory, and ergodic theory.
Knot theory greatly expanded.
Quantum mechanics led to the development of
functional analysis.
Other new areas include Laurent Schwartz's
distribution theory, fixed point theory, singularity
theory and René Thom's catastrophe theory,
model theory, and Mandelbrot's fractals.
Lie theory with its Lie groups and Lie algebras
became one of the major areas of study.
Non-standard analysis, introduced by Abraham
Robinson, rehabilitated the infinitesimal
approach to calculus, which had fallen into
disrepute in favour of the theory of limits,
by extending the field of real numbers to
the Hyperreal numbers which include infinitesimal
and infinite quantities.
An even larger number system, the surreal
numbers were discovered by John Horton Conway
in connection with combinatorial games.
The development and continual improvement
of computers, at first mechanical analog machines
and then digital electronic machines, allowed
industry to deal with larger and larger amounts
of data to facilitate mass production and
distribution and communication, and new areas
of mathematics were developed to deal with
this: Alan Turing's computability theory;
complexity theory; Derrick Henry Lehmer's
use of ENIAC to further number theory and
the Lucas-Lehmer test; Rózsa Péter's recursive
function theory; Claude Shannon's information
theory; signal processing; data analysis;
optimization and other areas of operations
research.
In the preceding centuries much mathematical
focus was on calculus and continuous functions,
but the rise of computing and communication
networks led to an increasing importance of
discrete concepts and the expansion of combinatorics
including graph theory.
The speed and data processing abilities of
computers also enabled the handling of mathematical
problems that were too time-consuming to deal
with by pencil and paper calculations, leading
to areas such as numerical analysis and symbolic
computation.
Some of the most important methods and algorithms
of the 20th century are: the simplex algorithm,
the Fast Fourier Transform, error-correcting
codes, the Kalman filter from control theory
and the RSA algorithm of public-key cryptography.
At the same time, deep insights were made
about the limitations to mathematics.
In 1929 and 1930, it was proved the truth
or falsity of all statements formulated about
the natural numbers plus one of addition and
multiplication, was decidable, i.e. could
be determined by some algorithm.
In 1931, Kurt Gödel found that this was not
the case for the natural numbers plus both
addition and multiplication; this system,
known as Peano arithmetic, was in fact incompletable.
(Peano arithmetic is adequate for a good deal
of number theory, including the notion of
prime number.)
A consequence of Gödel's two incompleteness
theorems is that in any mathematical system
that includes Peano arithmetic (including
all of analysis and geometry), truth necessarily
outruns proof, i.e. there are true statements
that cannot be proved within the system.
Hence mathematics cannot be reduced to mathematical
logic, and David Hilbert's dream of making
all of mathematics complete and consistent
needed to be reformulated.
One of the more colorful figures in 20th-century
mathematics was Srinivasa Aiyangar Ramanujan
(1887–1920), an Indian autodidact who conjectured
or proved over 3000 theorems, including properties
of highly composite numbers, the partition
function and its asymptotics, and mock theta
functions.
He also made major investigations in the areas
of gamma functions, modular forms, divergent
series, hypergeometric series and prime number
theory.
Paul Erdős published more papers than any
other mathematician in history, working with
hundreds of collaborators.
Mathematicians have a game equivalent to the
Kevin Bacon Game, which leads to the Erdős
number of a mathematician.
This describes the "collaborative distance"
between a person and Paul Erdős, as measured
by joint authorship of mathematical papers.
Emmy Noether has been described by many as
the most important woman in the history of
mathematics.
She studied the theories of rings, fields,
and algebras.
As in most areas of study, the explosion of
knowledge in the scientific age has led to
specialization: by the end of the century
there were hundreds of specialized areas in
mathematics and the Mathematics Subject Classification
was dozens of pages long.
More and more mathematical journals were published
and, by the end of the century, the development
of the World Wide Web led to online publishing.
=== 21st century ===
In 2000, the Clay Mathematics Institute announced
the seven Millennium Prize Problems, and in
2003 the Poincaré conjecture was solved by
Grigori Perelman (who declined to accept an
award, as he was critical of the mathematics
establishment).
Most mathematical journals now have online
versions as well as print versions, and many
online-only journals are launched.
There is an increasing drive towards open
access publishing, first popularized by the
arXiv.
== Future ==
There are many observable trends in mathematics,
the most notable being that the subject is
growing ever larger, computers are ever more
important and powerful, the application of
mathematics to bioinformatics is rapidly expanding,
and the volume of data being produced by science
and industry, facilitated by computers, is
explosively expanding.
== See also ==
== 
Notes
