Mathematics is the study of
quantity,
structure,
space, and
change.
Mathematicians seek out
patterns, formulate new
conjectures, and establish truth by
rigorous deduction from appropriately chosen
axioms and
definitions.
There is debate over whether mathematical objects such as
numbers and points exist naturally or are human
creations. The mathematician
Benjamin
Peirce called mathematics "the science that draws necessary
conclusions".
Albert Einstein, on
the other hand, stated that "as far as the laws of mathematics
refer to reality, they are not certain; and as far as they are
certain, they do not refer to reality."
Through the use of
abstraction and
logical reasoning,
mathematics evolved from
counting,
calculation,
measurement, and the systematic study of the
shapes and
motions of physical objects. Although
incorrectly considered part of mathematics by many, calculations
and measurement are features of
accountancy and
arithmetic. Practical mathematics has been a
human activity for as far back as
written records exist.
Rigorous arguments first appeared in
Greek mathematics, most notably in
Euclid's
Elements. Being an open intellectual
system, mathematics continued to develop, in fitful bursts, until
the
Renaissance, when mathematical
innovations interacted with new
scientific discoveries,
leading to an acceleration in research that continues to the
present day.
Today, mathematics is used throughout the world as an essential
tool in many fields, including
natural
science,
engineering,
medicine, and the
social
sciences.
Applied
mathematics, the branch of mathematics concerned with
application of mathematical knowledge to other fields, inspires and
makes use of new mathematical discoveries and sometimes leads to
the development of entirely new disciplines.
Numerology is considered an application of
mathematics by many but differs from mathematics in that it holds a
mystical view of numbers. Mathematicians also engage in
pure mathematics, or mathematics for its
own sake, without having any application in mind, although
practical applications for what began as pure mathematics are often
discovered later.
Etymology
The word "mathematics" comes from the
Greek μάθημα (
máthēma),
which means
learning,
study,
science,
and additionally came to have the narrower and more technical
meaning "mathematical study", even in Classical times. Its
adjective is μαθηματικός (
mathēmatikós),
related to
learning, or
studious, which likewise further came to
mean
mathematical. In particular, (
mathēmatikḗ
tékhnē), in
Latin ars
mathematica, meant
the mathematical art.
The apparent plural form in
English, like the
French plural form
les
mathématiques (and the less commonly used singular derivative
la mathématique), goes back to the Latin neuter plural
mathematica (
Cicero), based on the
Greek plural τα μαθηματικά (
ta mathēmatiká), used by
Aristotle, and meaning roughly "all things
mathematical"; although it is plausible that English borrowed only
the adjective
mathematic(al) and formed the noun
mathematics anew, after the pattern of
physics and
metaphysics,
which were inherited from the Greek. In English, the noun
mathematics takes singular verb forms. It is often
shortened to
maths, or
math in Englishspeaking
North America.
History
The evolution of mathematics might be seen as an everincreasing
series of
abstractions, or
alternatively an expansion of subject matter. The first
abstraction, which is shared by many animals, was probably that of
numbers: the realization that two apples and
two oranges (for example) have something in common.
In addition to recognizing how to
count
physical objects,
prehistoric
peoples also recognized how to count
abstract quantities,
like
time –
days,
seasons,
years.
Elementary arithmetic (
addition,
subtraction,
multiplication and
division) naturally followed.
Further steps needed
writing or some other
system for recording numbers such as
tallies or the knotted strings called
quipu used by the
Inca to store
numerical data.
Numeral systems have
been many and diverse, with the first known written numerals
created by
Egyptians in
Middle Kingdom texts such as the
Rhind Mathematical
Papyrus. The
Indus Valley
civilization developed the modern
decimal system, including the concept of
zero.
The earliest uses of mathematics were in
trading,
land
measurement,
painting and
weaving patterns and the recording of
time and nothing much more advanced until around 3000BC
onwards when the
Babylonians and
Egyptians began using arithmetic, algebra and
geometry for
taxation and other financial
calculations, building and construction and
astronomy. The systematic study of mathematics in
its own right began with the Ancient Greeks between 600 and
300BC.
Mathematics has since been greatly extended, and there has been a
fruitful interaction between mathematics and
science, to the benefit of both. Mathematical
discoveries have been made throughout history and continue to be
made today. According to Mikhail B. Sevryuk, in the January 2006
issue of the
Bulletin of the
American Mathematical Society, "The number of papers and books
included in the
Mathematical
Reviews database since 1940 (the first year of operation of MR)
is now more than 1.9 million, and more than 75 thousand items are
added to the database each year. The overwhelming majority of works
in this ocean contain new mathematical
theorems and their
proofs."
Inspiration, pure and applied mathematics, and aesthetics
Mathematics arises from many different kinds of problems. At first
these were found in
commerce,
land measurement,
architecture and later
astronomy; nowadays, all sciences suggest problems
studied by mathematicians, and many problems arise within
mathematics itself. For example, the
physicist Richard
Feynman invented the
path
integral formulation of
quantum
mechanics using a combination of mathematical reasoning and
physical insight, and today's
string
theory, a stilldeveloping scientific theory which attempts to
unify the four
fundamental
forces of nature, continues to inspire new mathematics. Some
mathematics is only relevant in the area that inspired it, and is
applied to solve further problems in that area. But often
mathematics inspired by one area proves useful in many areas, and
joins the general stock of mathematical concepts. A distinction is
often made between
pure mathematics
and
applied mathematics. However
pure mathematics topics often turn out to have applications, e.g.
number theory in
cryptography. This remarkable fact that even
the "purest" mathematics often turns out to have practical
applications is what
Eugene Wigner has
called "
the
unreasonable effectiveness of mathematics."As in most areas of
study, the explosion of knowledge in the scientific age has led to
specialization: there are now hundreds of specialized areas in
mathematics and the latest
Mathematics Subject
Classification runs to 46 pages. Several areas of applied
mathematics have merged with related traditions outside of
mathematics and become disciplines in their own right, including
statistics,
operations research, and
computer science.
For those who are mathematically inclined, there is often a
definite aesthetic aspect to much of mathematics. Many
mathematicians talk about the
elegance of mathematics, its
intrinsic
aesthetics and inner
beauty.
Simplicity and
generality are valued. There is beauty in a simple and elegant
proof, such as
Euclid's proof that there are infinitely many
prime numbers, and in an elegant
numerical method that speeds
calculation, such as the
fast
Fourier transform.
G. H. Hardy in
A Mathematician's
Apology expressed the belief that these aesthetic
considerations are, in themselves, sufficient to justify the study
of pure mathematics. He identified criteria such as significance,
unexpectedness, inevitability, and economy as factors that
contribute to a mathematical aesthetic. Mathematicians often strive
to find proofs of theorems that are particularly elegant, a quest
Paul Erdős often referred to as
finding proofs from "The Book" in which God had written down his
favorite proofs. The popularity of
recreational mathematics is another
sign of the pleasure many find in solving mathematical
questions.
Notation, language, and rigor
Most of the mathematical notation in use today was not invented
until the 16th century. Before that, mathematics was written out in
words, a painstaking process that limited mathematical discovery.
Euler (1707–1783) was responsible for
many of the notations in use today. Modern notation makes
mathematics much easier for the professional, but beginners often
find it daunting. It is extremely compressed: a few symbols contain
a great deal of information. Like
musical notation, modern mathematical
notation has a strict syntax and encodes information that would be
difficult to write in any other way.
Mathematical
language can also be hard for
beginners. Words such as
or and
only have more
precise meanings than in everyday speech. Additionally, words such
as
open and
field have been given specialized
mathematical meanings.
Mathematical
jargon includes technical terms such as
homeomorphism and
integrable. But there is a reason for
special notation and technical jargon: mathematics requires more
precision than everyday speech. Mathematicians refer to this
precision of language and logic as "rigor".
Mathematical proof is
fundamentally a matter of
rigor.
Mathematicians want their theorems to follow from axioms by means
of systematic reasoning. This is to avoid mistaken "
theorems", based on fallible intuitions, of which
many instances have occurred in the history of the subject. The
level of rigor expected in mathematics has varied over time: the
Greeks expected detailed arguments, but at the time of
Isaac Newton the methods employed were less
rigorous. Problems inherent in the definitions used by Newton would
lead to a resurgence of careful analysis and formal proof in the
19th century. Misunderstanding the rigor is a cause for some of the
common misconceptions of mathematics. Today, mathematicians
continue to argue among themselves about
computerassisted proofs. Since
large computations are hard to verify, such proofs may not be
sufficiently rigorous.
Axioms in traditional thought were
"selfevident truths", but that conception is problematic. At a
formal level, an axiom is just a string of
symbols, which has an intrinsic meaning only
in the context of all derivable formulas of an
axiomatic system. It was the goal of
Hilbert's program to put all of
mathematics on a firm axiomatic basis, but according to
Gödel's incompleteness
theorem every (sufficiently powerful) axiomatic system has
undecidable
formulas; and so a final
axiomatization of mathematics is impossible.
Nonetheless mathematics is often imagined to be (as far as its
formal content) nothing but
set theory in
some axiomatization, in the sense that every mathematical statement
or proof could be cast into formulas within set theory.
Mathematics as science
Carl Friedrich Gauss referred
to mathematics as "the Queen of the Sciences". In the original
Latin
Regina Scientiarum, as well as in
German Königin der Wissenschaften,
the word corresponding to
science means (field of)
knowledge. Indeed, this is also the original meaning in English,
and there is no doubt that mathematics is in this sense a science.
The specialization restricting the meaning to
natural
science is of later date. If one considers
science to be strictly about the physical world,
then mathematics, or at least
pure
mathematics, is not a science.
Albert Einstein stated that
"as far as
the laws of mathematics refer to reality, they are not certain; and
as far as they are certain, they do not refer to
reality."
Many philosophers believe that mathematics is not experimentally
falsifiable, and thus not a science
according to the definition of
Karl
Popper. However, in the 1930s important work in mathematical
logic showed that mathematics cannot be reduced to logic, and Karl
Popper concluded that "most mathematical theories are, like those
of
physics and
biology,
hypothetico
deductive:
pure mathematics therefore turns out to be much closer to the
natural sciences whose hypotheses are conjectures, than it seemed
even recently." Other thinkers, notably
Imre Lakatos, have applied a version of
falsificationism to mathematics
itself.
An alternative view is that certain scientific fields (such as
theoretical physics) are
mathematics with axioms that are intended to correspond to reality.
In fact, the theoretical physicist,
J.
M. Ziman,
proposed that science is
public knowledge and thus
includes mathematics. In any case, mathematics shares much in
common with many fields in the physical sciences, notably the
exploration of the logical consequences of assumptions.
Intuition and
experimentation also play a role in the
formulation of
conjectures in both
mathematics and the (other) sciences.
Experimental mathematics continues
to grow in importance within mathematics, and computation and
simulation are playing an increasing role in both the sciences and
mathematics, weakening the objection that mathematics does not use
the
scientific method. In his 2002
book
A New Kind of
Science,
Stephen Wolfram
argues that computational mathematics deserves to be explored
empirically as a scientific field in its own right.
The opinions of mathematicians on this matter are varied. Many
mathematicians feel that to call their area a science is to
downplay the importance of its aesthetic side, and its history in
the traditional seven
liberal arts;
others feel that to ignore its connection to the sciences is to
turn a blind eye to the fact that the interface between mathematics
and its applications in science and
engineering has driven much development in
mathematics. One way this difference of viewpoint plays out is in
the philosophical debate as to whether mathematics is
created (as in art) or
discovered (as in
science). It is common to see
universities divided into sections that include a
division of
Science and Mathematics, indicating that the
fields are seen as being allied but that they do not coincide. In
practice, mathematicians are typically grouped with scientists at
the gross level but separated at finer levels. This is one of many
issues considered in the
philosophy of mathematics.
Mathematical awards are generally kept separate from their
equivalents in science. The most prestigious award in mathematics
is the , established in 1936 and now awarded every 4 years. It is
often considered the equivalent of science's
Nobel Prizes. The
Wolf Prize in Mathematics,
instituted in 1978, recognizes lifetime achievement, and another
major international award, the
Abel
Prize, was introduced in 2003. These are awarded for a
particular body of work, which may be innovation, or resolution of
an outstanding problem in an established field. A famous list of 23
such
open problems, called "
Hilbert's problems", was compiled in 1900
by German mathematician
David Hilbert.
This list achieved great celebrity among mathematicians, and at
least nine of the problems have now been solved. A new list of
seven important problems, titled the "
Millennium Prize Problems", was
published in 2000. Solution of each of these problems carries a $1
million reward, and only one (the
Riemann hypothesis) is duplicated in
Hilbert's problems.
Fields of mathematics
Mathematics can, broadly speaking, be subdivided into the study of
quantity, structure, space, and change (i.e.
arithmetic,
algebra,
geometry, and
analysis). In addition to these main
concerns, there are also subdivisions dedicated to exploring links
from the heart of mathematics to other fields: to
logic, to
set
theory (
foundations),
to the empirical mathematics of the various sciences (
applied mathematics), and more recently
to the rigorous study of
uncertainty.
Quantity
The study of quantity starts with
numbers,
first the familiar
natural numbers
and
integers ("whole numbers") and
arithmetical operations on them, which are characterized in
arithmetic. The deeper properties of
integers are studied in
number theory,
from which come such popular results as
Fermat's Last Theorem. Number theory
also holds two problems widely considered unsolved: the
twin prime conjecture and
Goldbach's conjecture.
As the number system is further developed, the integers are
recognized as a
subset of the
rational numbers ("
fractions"). These, in turn, are
contained within the
real numbers, which
are used to represent
continuous
quantities. Real numbers are generalized to
complex numbers. These are the first steps of
a hierarchy of numbers that goes on to include
quarternions and
octonions. Consideration of the natural numbers
also leads to the
transfinite
numbers, which formalize the concept of "
infinity". Another area of study is size, which
leads to the
cardinal numbers and
then to another conception of infinity: the
aleph numbers, which allow meaningful
comparison of the size of infinitely large sets.
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Space
The study of space originates with
geometry– in particular,
Euclidean geometry.
Trigonometryis the branch of mathematics that
deals with relationships between the sides and the angles of
triangles and with the trigonometric functions; it combines space
and numbers, and encompasses the wellknown
Pythagorean theorem. The modern study of
space generalizes these ideas to include higherdimensional
geometry,
nonEuclidean
geometries(which play a central role in
general relativity) and
topology. Quantity and space both play a role in
analytic geometry,
differential geometry, and
algebraic geometry. Within differential
geometry are the concepts of
fiber
bundlesand calculus on
manifolds, in
particular,
vectorand
tensor calculus. Within algebraic geometry
is the description of geometric objects as solution sets of
polynomialequations, combining the
concepts of quantity and space, and also the study of
topological groups, which combine
structure and space.
Lie groupsare used to
study space, structure, and change.
Topologyin all its many ramifications may have been
the greatest growth area in 20th century mathematics; it includes
pointset topology,
settheoretic topology,
algebraic topologyand
differential topology. In particular,
instances of modern day topology are
metrizability theory,
axiomatic set theory,
homotopy theory, and
Morse theory. Topology also includes the now
solved
Poincaré
conjectureand the controversial
four color theorem, whose only proof, by
computer, has never been verified by a human.
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Change
Understanding and describing change is a common theme in the
natural sciences, and
calculuswas developed as a powerful tool to
investigate it.
Functionsarise here, as a central
concept describing a changing quantity. The rigorous study of
real numbersand functions of a real
variable is known as
real analysis,
with
complex analysisthe equivalent
field for the
complex numbers.
Functional analysisfocuses
attention on (typically infinitedimensional)
spacesof functions. One of many
applications of functional analysis is
quantum mechanics. Many problems lead
naturally to relationships between a quantity and its rate of
change, and these are studied as
differential equations. Many phenomena
in nature can be described by
dynamical
systems;
chaos theorymakes precise
the ways in which many of these systems exhibit unpredictable yet
still
deterministicbehavior.
Structure
Many mathematical objects, such as
setsof numbers and
functions, exhibit internal
structure. The structural properties of these objects are
investigated in the study of
groups,
rings,
fieldsand other abstract systems, which
are themselves such objects. This is the field of
abstract algebra. An important concept here
is that of
vectors, generalized
to
vector spaces, and studied in
linear algebra. The study of vectors
combines three of the fundamental areas of mathematics: quantity,
structure, and space. A number of ancient problems concerning
Compass and
straightedge constructionswere finally solved using
Galois theory.
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Foundations and philosophy
In order to clarify the
foundations of mathematics, the
fields of
mathematical logicand
set theorywere developed. Mathematical
logic includes the mathematical study of
logicand the applications of formal logic to other
areas of mathematics; set theory is the branch of mathematics that
studies
setsor collections of
objects.
Category theory, which
deals in an abstract way with
mathematical structuresand
relationships between them, is still in development. The phrase
"crisis of foundations" describes the search for a rigorous
foundation for mathematics that took place from approximately 1900
to 1930. Some disagreement about the foundations of mathematics
continues to present day. The crisis of foundations was stimulated
by a number of controversies at the time, including the
controversy over Cantor's set
theoryand the
BrouwerHilbert
controversy.
Mathematical logic is concerned with setting mathematics on a
rigorous
axiomaticframework, and studying the
results of such a framework. As such, it is home to
Gödel's second incompleteness theorem, perhaps the most widely
celebrated result in logic, which (informally) implies that any
formal systemthat contains basic
arithmetic, if
sound(meaning that all theorems that can be
proven are true), is necessarily
incomplete(meaning that
there are true theorems which cannot be proved
in that
system). Gödel showed how to construct, whatever the given
collection of numbertheoretical axioms, a formal statement in the
logic that is a true numbertheoretical fact, but which does not
follow from those axioms. Therefore no formal system is a true
axiomatization of full number theory. Modern logic is divided into
recursion theory,
model theory, and
proof
theory, and is closely linked to
theoreticalcomputer science.
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Discrete mathematics
Discrete mathematicsis the
common name for the fields of mathematics most generally useful in
theoretical computer
science. This includes, on the computer science side,
computability theory,
computational complexity
theory, and
information
theory. Computability theory examines the limitations of
various theoretical models of the computer, including the most
powerful known model – the
Turing
machine. Complexity theory is the study of tractability by
computer; some problems, although theoretically solvable by
computer, are so expensive in terms of time or space that solving
them is likely to remain practically unfeasible, even with rapid
advance of computer hardware. Finally, information theory is
concerned with the amount of data that can be stored on a given
medium, and hence deals with concepts such as
compressionand
entropy.
On the purely mathematical side, this field includes
combinatoricsand
graph
theory.
As a relatively new field, discrete mathematics has a number of
fundamental open problems. The most famous of these is the
"
P=NP?" problem, one of the
Millennium Prize Problems.
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Applied mathematics
Applied mathematicsconsiders the
use of abstract mathematical tools in solving concrete problems in
the
sciences,
business, and other areas.
Applied mathematics has significant overlap with the discipline of
statistics, whose theory is formulated
mathematically, especially with
probability theory. Statisticians
(working as part of a research project) "create data that makes
sense" with
random samplingand with
randomized
experiments; the
design of a statistical sample or experiment specifies the analysis
of the data (before the data be available). When reconsidering data
from experiments and samples or when analyzing data from
observational studies, statisticians
"make sense of the data" using the art of
modellingand the theory of
inference– with
modelselectionand
estimation; the estimated models and
consequential
predictionsshould
be
testedon
new data.
Computational
mathematicsproposes and studies methods for solving
mathematical problems that are typically too large for human
numerical capacity.
Numerical
analysisstudies methods for problems in
analysisusing ideas of
functional analysisand techniques of
approximation theory; numerical
analysis includes the study of
approximationand
discretizationbroadly with special concern
for
rounding errors. Other areas of
computational mathematics include
computer algebraand
symbolic computation.
Image:Gravitation space source.png 
Mathematical
physics
Image:BernoullisLawDerivationDiagram.svg 
Fluid dynamics
Image:Composite trapezoidal rule illustration small.svg 
Numerical
analysis
Image:Maximum boxed.png 
Optimization
Image:Two red dice 01.svg 
Probability
theory
Image:Oldfaithful3.png 
Statistics
Image:Market Data Index NYA on 20050726 202628 UTC.png 
Financial
mathematics
Image:Arbitrarygametreesolved.svg 
Game theory
Image:Signal transduction v1.png 
Mathematical
biology
Image:Ch4structure.png 
Mathematical
chemistry
Image:GDP PPP Per Capita IMF 2008.png 
Mathematical
economics
Image:Simple feedback control loop2.png 
Control theory
See also
Notes
 No likeness or description of Euclid's physical appearance made
during his lifetime survived antiquity. Therefore, Euclid's
depiction in works of art depends on the artist's imagination
(see Euclid).
 Steen, L.A.
(April 29, 1988). The Science of Patterns.
Science, 240: 611–616. and summarized at
Association for Supervision and Curriculum
Development.
 Devlin,
Keith, Mathematics: The Science of Patterns: The Search for
Order in Life, Mind and the Universe (Scientific American
Paperback Library) 1996, ISBN 9780716750475
 Jourdain
 Peirce, p.97
 Eves
 Peterson
 Both senses can be found in Plato. Liddell and Scott,
s.voceμαθηματικός
 The Oxford
Dictionary of English Etymology, Oxford English Dictionary,
sub "mathematics", "mathematic", "mathematics"
 See, for example, Raymond L. Wilder, Evolution of
Mathematical Concepts; an Elementary Study,
passim
 Kline 1990, Chapter 1.
 Sevryuk
 Eugene
Wigner, 1960, " The Unreasonable Effectiveness of Mathematics in
the Natural Sciences," Communications on
Pure and Applied Mathematics 13(1):
1–14.
 Mathematics Subject Classification 2010
 Earliest Uses of Various Mathematical Symbols
(Contains many further references)
 Kline, pp. 140 (on Diophantus; pp.261, on Vieta.
 See false
proof for simple examples of what can go wrong in a formal
proof. The history of the Four Color Theorem
contains examples of false proofs accidentally accepted by other
mathematicians at the time.
 Ivars Peterson, The Mathematical Tourist, Freeman,
1988, ISBN 0716719533. p. 4 "A few complain that the computer
program can't be verified properly," (in reference to the
HakenApple proof of the Four Color Theorem).
 Patrick Suppes, Axiomatic Set Theory, Dover, 1972,
ISBN 0486616304. p. 1, "Among the many branches of modern
mathematics set theory occupies a unique place: with a few rare
exceptions the entities which are studied and analyzed in
mathematics may be regarded as certain particular sets or classes
of objects."
 Waltershausen
 Einstein, p. 28. The quote is Einstein's answer to the
question: "how can it be that mathematics, being after all a
product of human thought which is independent of experience, is so
admirably appropriate to the objects of reality?" He, too, is
concerned with
The Unreasonable Effectiveness of Mathematics in the Natural
Sciences.
 Popper 1995, p. 56
 Ziman
 "The Fields Medal is now indisputably the best known and
most influential award in mathematics." Monastyrsky
 Riehm
 Luke Howard Hodgkin & Luke Hodgkin, A History of
Mathematics, Oxford University Press, 2005.
 Clay Mathematics Institute P=NP
 Like other mathematical sciences such as physics and computer science, statistics is an autonomous
discipline rather than a branch of applied mathematics. Like
research physicists and computer scientists, research statisticians
are mathematical scientists. Many statisticians have a degree in
mathematics, and some statisticians are also mathematicians.
References
 Benson, Donald C., The Moment of Proof: Mathematical
Epiphanies, Oxford
University Press, USA; New Ed edition (December 14, 2000). ISBN
0195139194.
 Boyer, Carl B., A History of
Mathematics, Wiley; 2 edition (March 6, 1991). ISBN
0471543977. — A concise history of mathematics from the Concept
of Number to contemporary Mathematics.
 Courant, R. and H. Robbins, What Is Mathematics? :
An Elementary Approach to Ideas and Methods, Oxford University
Press, USA; 2 edition (July 18, 1996). ISBN 0195105192.
 Davis, Philip J. and Hersh, Reuben, The Mathematical
Experience. Mariner Books; Reprint edition (January 14,
1999). ISBN 0395929687. — A gentle introduction to the world of
mathematics.
 Eves, Howard, An Introduction to the History of
Mathematics, Sixth Edition, Saunders, 1990, ISBN
0030295580.
 Gullberg, Jan, Mathematics — From the Birth of
Numbers. W. W. Norton & Company; 1st edition
(October 1997). ISBN 039304002X. — An encyclopedic overview of
mathematics presented in clear, simple language.
 Hazewinkel, Michiel (ed.), Encyclopaedia of
Mathematics. Kluwer
Academic Publishers 2000. — A translated and expanded version
of a Soviet mathematics encyclopedia, in ten (expensive) volumes,
the most complete and authoritative work available. Also in
paperback and on CDROM, and online.
 Jourdain, Philip E. B., The Nature of Mathematics, in
The World of Mathematics, James R. Newman, editor,
Dover Publications, 2003, ISBN
0486432688.
 Kline, Morris, Mathematical
Thought from Ancient to Modern Times, Oxford University Press,
USA; Paperback edition (March 1, 1990). ISBN 0195061357.
 Oxford English
Dictionary, second edition, ed. John Simpson and Edmund Weiner,
Clarendon Press, 1989, ISBN
0198611862.
 The
Oxford Dictionary of English Etymology, 1983 reprint. ISBN
0198611129.
 Pappas, Theoni, The Joy Of Mathematics, Wide World
Publishing; Revised edition (June 1989). ISBN 0933174659.
 JSTOR.
 Peterson, Ivars, Mathematical Tourist, New and Updated
Snapshots of Modern Mathematics, Owl Books, 2001, ISBN
0805071598.
External links
 Free Mathematics books Free Mathematics books
collection.
 Applications of High School Algebra
 Encyclopaedia of
Mathematics online encyclopadia from Springer,
Graduatelevel reference work with over 8,000 entries, illuminating
nearly 50,000 notions in mathematics.
 HyperMath site at Georgia State University
 FreeScience Library The mathematics section of
FreeScience library
 Rusin, Dave: The Mathematical Atlas. A guided tour through the
various branches of modern mathematics. (Can also be found here.)
 Polyanin, Andrei: EqWorld: The World of Mathematical
Equations. An online resource focusing on algebraic,
ordinary differential, partial differential (mathematical physics), integral, and
other mathematical equations.
 Cain, George: Online Mathematics Textbooks available free
online.
 Tricki,
Wikistyle site that is intended to develop into a large store of
useful mathematical problemsolving techniques.
 Mathematical Structures, list information about
classes of mathematical structures.
 Math & Logic: The history of formal
mathematical, logical, linguistic and methodological ideas. In
The Dictionary of the History of Ideas.
 Mathematician Biographies. The MacTutor History of
Mathematics archive Extensive history and quotes from all
famous mathematicians.
 Metamath. A site and a language, that formalize
mathematics from its foundations.
 Nrich, a prizewinning site for students from age five
from Cambridge
University
 Open
Problem Garden, a wiki of open problems in
mathematics
 Planet
Math. An online mathematics encyclopedia under
construction, focusing on modern mathematics. Uses the AttributionShareAlike license, allowing article
exchange with Wikipedia. Uses TeX markup.
 Some
mathematics applets, at MIT
 Weisstein, Eric et al.: MathWorld: World of Mathematics. An online
encyclopedia of mathematics.
 Patrick Jones' Video Tutorials on Mathematics
1, 2, 3\,\! 
2, 1, 0, 1, 2\,\! 
2, \frac{2}{3}, 1.21\,\! 
e, \sqrt{2}, 3, \pi\,\! 
2, i, 2+3i, 2e^{i\frac{4\pi}{3}}\,\! 

Natural numbers 
Integers 
Rational numbers 
Real numbers 
Complex numbers 







Geometry 
Trigonometry 
Differential geometry 
Topology 
Fractal geometry 
Measure Theory 








Calculus 
Vector calculus 
Differential
equations 
Dynamical systems 
Chaos theory 
Complex analysis 





Number theory 
Abstract algebra 
Group theory 
Order theory 
p \Rightarrow q \, 



Mathematical logic 
Set theory 
Category theory 

\begin{matrix} (1,2,3) & (1,3,2) \\ (2,1,3) & (2,3,1)
\\ (3,1,2) & (3,2,1) \end{matrix} 




Combinatorics 
Theory of computation 
Cryptography 
Graph theory 