The history of mathematical notation includes
the commencement, progress, and cultural diffusion
of mathematical symbols and the conflict of
the methods of notation confronted in a notation's
move to popularity or inconspicuousness. Mathematical
notation comprises the symbols used to write
mathematical equations and formulas. Notation
generally implies a set of well-defined representations
of quantities and symbols operators. The history
includes Hindu–Arabic numerals, letters
from the Roman, Greek, Hebrew, and German
alphabets, and a host of symbols invented
by mathematicians over the past several centuries.
The development of mathematical notation can
be divided in stages. The "rhetorical" stage
is where calculations are performed by words
and no symbols are used. The "syncopated"
stage is where frequently used operations
and quantities are represented by symbolic
syntactical abbreviations. From ancient times
through the post-classical age, bursts of
mathematical creativity were often followed
by centuries of stagnation. As the early modern
age opened and the worldwide spread of knowledge
began, written examples of mathematical developments
came to light. The "symbolic" stage is where
comprehensive systems of notation supersede
rhetoric. Beginning in Italy in the 16th century,
new mathematical developments, interacting
with new scientific discoveries, were made
at an increasing pace that continues through
the present day. This symbolic system was
in use by medieval Indian mathematicians and
in Europe since the middle of the 17th century,
and has continued to develop in the contemporary
era.
The area of study known as the history of
mathematics is primarily an investigation
into the origin of discoveries in mathematics
and, the focus here, the investigation into
the mathematical methods and notation of the
past.
== Rhetorical stage ==
Although the history commences with that of
the Ionian schools, there is no doubt that
those Ancient Greeks who paid attention to
it were largely indebted to the previous investigations
of the Ancient Egyptians and Ancient Phoenicians.
Numerical notation's distinctive feature,
i.e. symbols having local as well as intrinsic
values (arithmetic), implies a state of civilization
at the period of its invention. Our knowledge
of the mathematical attainments of these early
peoples, to which this section is devoted,
is imperfect and the following brief notes
be regarded as a summary of the conclusions
which seem most probable, and the history
of mathematics begins with the symbolic sections.
Many areas of mathematics began with the study
of real world problems, before the underlying
rules and concepts were identified and defined
as abstract structures. For example, geometry
has its origins in the calculation of distances
and areas in the real world; algebra started
with methods of solving problems in arithmetic.
There can be no doubt that most early peoples
which have left records knew something of
numeration and mechanics, and that a few were
also acquainted with the elements of land-surveying.
In particular, the Egyptians paid attention
to geometry and numbers, and the Phoenicians
to practical arithmetic, book-keeping, navigation,
and land-surveying. The results attained by
these people seem to have been accessible,
under certain conditions, to travelers. It
is probable that the knowledge of the Egyptians
and Phoenicians was largely the result of
observation and measurement, and represented
the accumulated experience of many ages.
=== Beginning of notation ===
Written mathematics began with numbers expressed
as tally marks, with each tally representing
a single unit. The numerical symbols consisted
probably of strokes or notches cut in wood
or stone, and intelligible alike to all nations.
For example, one notch in a bone represented
one animal, or person, or anything else. The
peoples with whom the Greeks of Asia Minor
(amongst whom notation in western history
begins) were likely to have come into frequent
contact were those inhabiting the eastern
littoral of the Mediterranean: and Greek tradition
uniformly assigned the special development
of geometry to the Egyptians, and that of
the science of numbers either to the Egyptians
or to the Phoenicians.
The Ancient Egyptians had a symbolic notation
which was the numeration by Hieroglyphics.
The Egyptian mathematics had a symbol for
one, ten, one-hundred, one-thousand, ten-thousand,
one-hundred-thousand, and one-million. Smaller
digits were placed on the left of the number,
as they are in Hindu–Arabic numerals. Later,
the Egyptians used hieratic instead of hieroglyphic
script to show numbers. Hieratic was more
like cursive and replaced several groups of
symbols with individual ones. For example,
the four vertical lines used to represent
four were replaced by a single horizontal
line. This is found in the Rhind Mathematical
Papyrus (c. 2000–1800 BC) and the Moscow
Mathematical Papyrus (c. 1890 BC). The system
the Egyptians used was discovered and modified
by many other civilizations in the Mediterranean.
The Egyptians also had symbols for basic operations:
legs going forward represented addition, and
legs walking backward to represent subtraction.
The Mesopotamians had symbols for each power
of ten. Later, they wrote their numbers in
almost exactly the same way done in modern
times. Instead of having symbols for each
power of ten, they would just put the coefficient
of that number. Each digit was at separated
by only a space, but by the time of Alexander
the Great, they had created a symbol that
represented zero and was a placeholder. The
Mesopotamians also used a sexagesimal system,
that is base sixty. It is this system that
is used in modern times when measuring time
and angles. 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 and the 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.The majority of Mesopotamian
clay tablets date from 1800 to 1600 BC, and
cover topics which 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 and quadratic equations.
The Babylonian tablet YBC 7289 gives an approximation
of √2 accurate to five decimal places. 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 minutes
and seconds of arc to denote fractions of
a degree. Babylonian advances in mathematics
were facilitated by the fact that 60 has many
divisors: the reciprocal of any integer which
is a multiple of divisors of 60 has a finite
expansion in base 60. (In decimal arithmetic,
only reciprocals of multiples of 2 and 5 have
finite decimal expansions.) 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. They
lacked, however, an equivalent of the decimal
point, and so the place value of a symbol
often had to be inferred from the context.
== Syncopated stage ==
The history of mathematics cannot with certainty
be traced back to any school or period before
that of the Ionian Greeks, but the subsequent
history may be divided into periods, the distinctions
between which are tolerably well marked. Greek
mathematics, which originated with the study
of geometry, tended from its commencement
to be deductive and scientific. Since the
fourth century AD, Pythagoras has commonly
been given credit for discovering the Pythagorean
theorem, a theorem in geometry that states
that in a right-angled triangle the area of
the square on the hypotenuse (the side opposite
the right angle) is equal to the sum of the
areas of the squares of the other two sides.
The ancient mathematical texts are available
with the prior mentioned Ancient Egyptians
notation and with Plimpton 322 (Babylonian
mathematics c. 1900 BC). The study of mathematics
as a subject in its own right begins in the
6th century BC with the Pythagoreans, who
coined the term "mathematics" from the ancient
Greek μάθημα (mathema), meaning "subject
of instruction".Plato's influence has been
especially strong in mathematics and the sciences.
He helped to distinguish between pure and
applied mathematics by widening the gap between
"arithmetic", now called number theory and
"logistic", now called arithmetic. 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. Aristotle
is credited with what later would be called
the law of excluded middle.
Abstract Mathematics is what treats of magnitude
or quantity, absolutely and generally conferred,
without regard to any species of particular
magnitude, such as Arithmetic and Geometry,
In this sense, abstract mathematics is opposed
to mixed mathematics; wherein simple and abstract
properties, and the relations of quantities
primitively considered in mathematics, are
applied to sensible objects, and by that means
become intermixed with physical considerations;
Such are Hydrostatics, Optics, Navigation,
&c.Archimedes is generally considered to be
the greatest mathematician of antiquity and
one of the greatest of all time. He used the
method of exhaustion to calculate the area
under the arc of a parabola with the summation
of an infinite series, and gave a remarkably
accurate approximation of pi. He also defined
the spiral bearing his name, formulae for
the volumes of surfaces of revolution and
an ingenious system for expressing very large
numbers.
In the historical development of geometry,
the steps in the abstraction of geometry were
made by the ancient Greeks. Euclid's Elements
being the earliest extant documentation of
the axioms of plane geometry— though Proclus
tells of an earlier axiomatisation by Hippocrates
of Chios. Euclid's Elements (c. 300 BC) is
one of the oldest extant Greek mathematical
treatises and consisted of 13 books written
in Alexandria; collecting theorems proven
by other mathematicians, supplemented by some
original work. The document is a successful
collection of definitions, postulates (axioms),
propositions (theorems and constructions),
and mathematical proofs of the propositions.
Euclid's first theorem is a lemma that possesses
properties of prime numbers. The influential
thirteen books cover Euclidean geometry, geometric
algebra, and the ancient Greek version of
algebraic systems and elementary number theory.
It was ubiquitous in the Quadrivium and is
instrumental in the development of logic,
mathematics, and science.
Diophantus of Alexandria was author of a series
of books called Arithmetica, many of which
are now lost. These texts deal with solving
algebraic equations. 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.
=== Acrophonic and Milesian numeration ===
The Greeks employed Attic numeration, which
was based on the system of the Egyptians and
was later adapted and used by the Romans.
Greek numerals one through four were vertical
lines, as in the hieroglyphics. The symbol
for five was the Greek letter Π (pi), which
is the letter of the Greek word for five,
pente. Numbers six through nine were pente
with vertical lines next to it. Ten was represented
by the letter (Δ) of the word for ten, deka,
one hundred by the letter from the word for
hundred, etc.
The Ionian numeration used their entire alphabet
including three archaic letters. The numeral
notation of the Greeks, though far less convenient
than that now in use, was formed on a perfectly
regular and scientific plan, and could be
used with tolerable effect as an instrument
of calculation, to which purpose the Roman
system was totally inapplicable. The Greeks
divided the twenty-four letters of their alphabet
into three classes, and, by adding another
symbol to each class, they had characters
to represent the units, tens, and hundreds.
(Jean Baptiste Joseph Delambre's Astronomie
Ancienne, t. ii.)
This system appeared in the third century
BC, before the letters digamma (Ϝ), koppa
(Ϟ), and sampi (Ϡ) became obsolete. When
lowercase letters became differentiated from
upper case letters, the lower case letters
were used as the symbols for notation. Multiples
of one thousand were written as the nine numbers
with a stroke in front of them: thus one thousand
was ",α", two-thousand was ",β", etc. M
(for μὐριοι, as in "myriad") was used
to multiply numbers by ten thousand. For example,
the number 88,888,888 would be written as
M,ηωπη*ηωπηGreek mathematical reasoning
was almost entirely geometric (albeit often
used to reason about non-geometric subjects
such as number theory), and hence the Greeks
had no interest in algebraic symbols. The
great exception was Diophantus of Alexandria,
the great algebraist. His Arithmetica was
one of the texts to use symbols in equations.
It was not completely symbolic, but was much
more so than previous books. An unknown number
was called s. The square of s was
Δ
y
{\displaystyle \Delta ^{y}}
; the cube was
K
y
{\displaystyle K^{y}}
; the fourth power was
Δ
y
Δ
{\displaystyle \Delta ^{y}\Delta }
; and the fifth power was
Δ
K
y
{\displaystyle \Delta K^{y}}
.
=== Chinese mathematical notation ===
The Chinese used numerals that look much like
the tally system. Numbers one through four
were horizontal lines. Five was an X between
two horizontal lines; it looked almost exactly
the same as the Roman numeral for ten. Nowadays,
the huāmǎ system is only used for displaying
prices in Chinese markets or on traditional
handwritten invoices.
In the history of the Chinese, there were
those who were familiar with the sciences
of arithmetic, geometry, mechanics, optics,
navigation, and astronomy. Mathematics in
China emerged independently by the 11th century
BC. It is almost certain that the Chinese
were acquainted with several geometrical or
rather architectural implements; with mechanical
machines; that they knew of the characteristic
property of the magnetic needle; and were
aware that astronomical events occurred in
cycles. Chinese of that time had made attempts
to classify or extend the rules of arithmetic
or geometry which they knew, and to explain
the causes of the phenomena with which they
were acquainted beforehand. The Chinese independently
developed very large and negative numbers,
decimals, a place value decimal system, a
binary system, algebra, geometry, and trigonometry.
Chinese mathematics made early contributions,
including a place value system. The geometrical
theorem known to the ancient Chinese were
acquainted was applicable in certain cases
(namely the ratio of sides). It is that geometrical
theorems which can be demonstrated in the
quasi-experimental way of superposition were
also known to them. In arithmetic their knowledge
seems to have been confined to the art of
calculation by means of the swan-pan, and
the power of expressing the results in writing.
Our knowledge of the early attainments of
the Chinese, slight though it is, is more
complete than in the case of most of their
contemporaries. It is thus instructive, and
serves to illustrate the fact, that it can
be known a nation may possess considerable
skill in the applied arts with but our knowledge
of the later mathematics on which those arts
are founded can be scarce. Knowledge of Chinese
mathematics before 254 BC is somewhat fragmentary,
and even after this date the manuscript traditions
are obscure. Dates centuries before the classical
period are generally considered conjectural
by Chinese scholars unless accompanied by
verified archaeological evidence.
As in other early societies the focus was
on astronomy in order to perfect the agricultural
calendar, and other practical tasks, and not
on establishing formal systems.The Chinese
Board of Mathematics duties were confined
to the annual preparation of an almanac, the
dates and predictions in which it regulated.
Ancient Chinese mathematicians did not develop
an axiomatic approach, but made advances in
algorithm development and algebra. The achievement
of Chinese algebra reached its zenith in the
13th century, when Zhu Shijie invented method
of four unknowns.
As a result of obvious linguistic and geographic
barriers, as well as content, Chinese mathematics
and that of the mathematics of the ancient
Mediterranean world are presumed to have developed
more or less independently up to the time
when The Nine Chapters on the Mathematical
Art reached its final form, while the Writings
on Reckoning and Huainanzi are roughly contemporary
with classical Greek mathematics. Some exchange
of ideas across Asia through known cultural
exchanges from at least Roman times is likely.
Frequently, elements of the mathematics of
early societies correspond to rudimentary
results found later in branches of modern
mathematics such as geometry or number theory.
The Pythagorean theorem for example, has been
attested to the time of the Duke of Zhou.
Knowledge of Pascal's triangle has also been
shown to have existed in China centuries before
Pascal, such as by Shen Kuo.
The state of trigonometry in China slowly
began to change and advance during the Song
Dynasty (960–1279), where Chinese mathematicians
began to express greater emphasis for the
need of spherical trigonometry in calendarical
science and astronomical calculations. The
polymath Chinese scientist, mathematician
and official Shen Kuo (1031–1095) used trigonometric
functions to solve mathematical problems of
chords and arcs. Sal Restivo writes that Shen's
work in the lengths of arcs of circles provided
the basis for spherical trigonometry developed
in the 13th century by the mathematician and
astronomer Guo Shoujing (1231–1316). As
the historians L. Gauchet and Joseph Needham
state, Guo Shoujing used spherical trigonometry
in his calculations to improve the calendar
system and Chinese astronomy. The mathematical
science of the Chinese would incorporate the
work and teaching of Arab missionaries with
knowledge of spherical trigonometry who had
come to China in the course of the thirteenth
century.
=== Indian mathematical notation ===
Although the origin of our present system
of numerical notation is ancient, there is
no doubt that it was in use among the Hindus
over two thousand years ago. The algebraic
notation of the Indian mathematician, Brahmagupta,
was syncopated. Addition was indicated by
placing the numbers side by side, subtraction
by placing a dot over the subtrahend (the
number to be subtracted), and division by
placing the divisor below the dividend, similar
to our notation but without the bar. Multiplication,
evolution, and unknown quantities were represented
by abbreviations of appropriate terms. The
Hindu–Arabic numeral system and the rules
for the use of its operations, in use throughout
the world today, likely evolved over the course
of the first millennium AD in India and was
transmitted to the west via Islamic mathematics.
==== Hindu–Arabic numerals and notations
====
Despite their name, Arabic numerals actually
started in India. The reason for this misnomer
is Europeans saw the numerals used in an Arabic
book, Concerning the Hindu Art of Reckoning,
by Mohommed ibn-Musa al-Khwarizmi. 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.
Al-Khwarizmi did not claim the numerals as
Arabic, but over several Latin translations,
the fact that the numerals were Indian in
origin was lost. The word algorithm is derived
from the Latinization of Al-Khwārizmī's
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).
Islamic mathematics developed and expanded
the mathematics known to Central Asian civilizations.
Al-Khwārizmī gave an exhaustive explanation
for the algebraic solution of quadratic equations
with positive roots, and Al-Khwārizmī was
to teach algebra in an elementary form and
for its own sake. Al-Khwārizmī 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." Al-Khwārizmī 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."Al-Karaji, in
his treatise al-Fakhri, extends the methodology
to incorporate integer powers and integer
roots of unknown quantities. 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 would develop
analytic geometry. Al-Haytham derived 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. Al-Haytham 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. In the late 11th
century, Omar Khayyam would develop algebraic
geometry, wrote Discussions of the Difficulties
in Euclid, and wrote on the general geometric
solution to cubic equations. Nasir al-Din
Tusi (Nasireddin) made advances in spherical
trigonometry. Muslim mathematicians during
this period include the addition of the decimal
point notation to the Arabic numerals.
Many Greek and Arabic texts on mathematics
were then translated into Latin, which led
to further development of mathematics in medieval
Europe. In the 12th century, scholars traveled
to Spain and Sicily seeking scientific Arabic
texts, including al-Khwārizmī's and the
complete text of Euclid's Elements. One of
the European books that advocated using the
numerals was Liber Abaci, by Leonardo of Pisa,
better known as Fibonacci. Liber Abaci is
better known for the mathematical problem
Fibonacci wrote in it about a population of
rabbits. The growth of the population ended
up being a Fibonacci sequence, where a term
is the sum of the two preceding terms.
Abū al-Hasan ibn Alī al-Qalasādī (1412–1482)
was the last major medieval Arab algebraist,
who improved on the algebraic notation earlier
used by Ibn al-Yāsamīn in the 12th century
and, in the Maghreb, by Ibn al-Banna in the
13th century. In contrast to the syncopated
notations of their predecessors, Diophantus
and Brahmagupta, which lacked symbols for
mathematical operations, al-Qalasadi's algebraic
notation was the first to have symbols for
these functions and was thus "the first steps
toward the introduction of algebraic symbolism."
He represented mathematical symbols using
characters from the Arabic alphabet.
== Symbolic stage ==
Symbols by popular introduction date
=== Early arithmetic and multiplication ===
The 14th century saw the development of new
mathematical concepts to investigate a wide
range of problems. The two widely used arithmetic
symbols are addition and subtraction, + and
−. The plus sign was used by 1360 by Nicole
Oresme in his work Algorismus proportionum.
It is thought an abbreviation for "et", meaning
"and" in Latin, in much the same way the ampersand
sign also began as "et". Oresme at the University
of Paris and the Italian Giovanni di Casali
independently provided graphical demonstrations
of the distance covered by a body undergoing
uniformly accelerated motion, asserting that
the area under the line depicting the constant
acceleration and represented the total distance
traveled. The minus sign was used in 1489
by Johannes Widmann in Mercantile Arithmetic
or Behende und hüpsche Rechenung auff allen
Kauffmanschafft,. Widmann used the minus symbol
with the plus symbol, to indicate deficit
and surplus, respectively. In Summa de arithmetica,
geometria, proportioni e proportionalità,
Luca Pacioli used symbols for plus and minus
symbols and contained algebra.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.
In 1533, Regiomontanus's table of sines and
cosines were published. 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.
The radical symbol for square root was introduced
by Christoph Rudolff. Michael Stifel's important
work Arithmetica integra contained important
innovations in mathematical notation. In 1556,
Niccolò Tartaglia used parentheses for precedence
grouping. In 1557 Robert Recorde published
The Whetstone of Witte which used the equal
sign (=) as well as plus and minus signs for
the English reader. In 1564, Gerolamo Cardano
analyzed games of chance beginning the early
stages of probability theory. 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'), published
in Dutch in 1585, contained a systematic treatment
of decimal notation, which influenced all
later work on the real number system. The
New algebra (1591) of François Viète introduced
the modern notational manipulation of algebraic
expressions. For navigation and accurate maps
of large areas, trigonometry grew to be a
major branch of mathematics. Bartholomaeus
Pitiscus coin the word "trigonometry", publishing
his Trigonometria in 1595.
John Napier is best known as the inventor
of logarithms and made common the use of the
decimal point in arithmetic and mathematics.
After Napier, Edmund Gunter created the logarithmic
scales (lines, or rules) upon which slide
rules are based, it was William Oughtred who
used two such scales sliding by one another
to perform direct multiplication and division;
and he is credited as the inventor of the
slide rule in 1622. In 1631 Oughtred introduced
the multiplication sign (×) his proportionality
sign, and abbreviations sin and cos for the
sine and cosine functions. Albert Girard also
used the abbreviations 'sin', 'cos' and 'tan'
for the trigonometric functions in his treatise.
Johannes Kepler was one of the pioneers of
the mathematical applications of infinitesimals.
René Descartes is credited as the father
of analytical geometry, the bridge between
algebra and geometry, crucial to the discovery
of infinitesimal calculus and analysis. In
the 17th century, Descartes introduced Cartesian
co-ordinates which allowed the development
of analytic geometry. Blaise Pascal influenced
mathematics throughout his life. His Traité
du triangle arithmétique ("Treatise on the
Arithmetical Triangle") of 1653 described
a convenient tabular presentation for binomial
coefficients. Pierre de Fermat and Blaise
Pascal would investigate probability. John
Wallis introduced the infinity symbol. He
similarly used this notation for infinitesimals.
In 1657, Christiaan Huygens published the
treatise on probability, On Reasoning in Games
of Chance.Johann Rahn introduced the division
symbol (obelus) and the therefore sign in
1659. William Jones used π in Synopsis palmariorum
mathesios in 1706 because it is the letter
of the Greek word perimetron (περιμετρον),
which means perimeter in Greek. This usage
was popularized in 1737 by Euler. In 1734,
Pierre Bouguer used double horizontal bar
below the inequality sign.
=== Derivatives notation: Leibniz and Newton
===
The study of linear algebra emerged from the
study of determinants, which were used to
solve systems of linear equations. Calculus
had two main systems of notation, each created
by one of the creators: that developed by
Isaac Newton and the notation developed by
Gottfried Leibniz. Leibniz's is the notation
used most often today. Newton's was simply
a dot or dash placed above the function. In
modern usage, this notation generally denotes
derivatives of physical quantities with respect
to time, and is used frequently in the science
of mechanics. Leibniz, on the other hand,
used the letter d as a prefix to indicate
differentiation, and introduced the notation
representing derivatives as if they were a
special type of fraction. This notation makes
explicit the variable with respect to which
the derivative of the function is taken. Leibniz
also created the integral symbol. The symbol
is an elongated S, representing the Latin
word Summa, meaning "sum". When finding areas
under curves, integration is often illustrated
by dividing the area into infinitely many
tall, thin rectangles, whose areas are added.
Thus, the integral symbol is an elongated
s, for sum.
=== High division operators and functions
===
Letters of the alphabet in this time were
to be used as symbols of quantity; and although
much diversity existed with respect to the
choice of letters, there were to be several
universally recognized rules in the following
history. Here thus in the history of equations
the first letters of the alphabet were indicatively
known as coefficients, the last letters the
unknown terms (an incerti ordinis). In algebraic
geometry, again, a similar rule was to be
observed, the last letters of the alphabet
there denoting the variable or current coordinates.
Certain letters, such as
π
{\displaystyle \pi }
,
e
{\displaystyle e}
, etc., were by universal consent appropriated
as symbols of the frequently occurring numbers
3.14159 ..., and 2.7182818 ...., etc., and
their use in any other acceptation was to
be avoided as much as possible. Letters, too,
were to be employed as symbols of operation,
and with them other previously mentioned arbitrary
operation characters. The letters
d
{\displaystyle d}
, elongated
S
{\displaystyle S}
were to be appropriated as operative symbols
in the differential calculus and integral
calculus,
Δ
{\displaystyle \Delta }
and ∑ in the calculus of differences. In
functional notation, a letter, as a symbol
of operation, is combined with another which
is regarded as a symbol of quantity.Beginning
in 1718, Thomas Twinin used the division slash
(solidus), deriving it from the earlier Arabic
horizontal fraction bar. Pierre-Simon, marquis
de Laplace developed the widely used Laplacian
differential operator. In 1750, Gabriel Cramer
developed "Cramer's Rule" for solving linear
systems.
==== Euler and prime notations ====
Leonhard Euler was one of the most prolific
mathematicians in history, and also a prolific
inventor of canonical notation. His contributions
include his use of e to represent the base
of natural logarithms. It is not known exactly
why
e
{\displaystyle e}
was chosen, but it was probably because the
four letters of the alphabet were already
commonly used to represent variables and other
constants. Euler used
π
{\displaystyle \pi }
to represent pi consistently. The use of
π
{\displaystyle \pi }
was suggested by William Jones, who used it
as shorthand for perimeter. Euler used
i
{\displaystyle i}
to represent the square root of negative one,
although he earlier used it as an infinite
number. For summation, Euler used sigma, Σ.
For functions, Euler used the notation
f
(
x
)
{\displaystyle f(x)}
to represent a function of
x
{\displaystyle x}
. In 1730, Euler wrote the gamma function.
In 1736, Euler produces his paper on the Seven
Bridges of Königsberg initiating the study
of graph theory.
The mathematician, William Emerson would develop
the proportionality sign. Much later in the
abstract expressions of the value of various
proportional phenomena, the parts-per notation
would become useful as a set of pseudo units
to describe small values of miscellaneous
dimensionless quantities. Marquis de Condorcet,
in 1768, advanced the partial differential
sign. In 1771, Alexandre-Théophile Vandermonde
deduced the importance of topological features
when discussing the properties of knots related
to the geometry of position. Between 1772
and 1788, Joseph-Louis Lagrange re-formulated
the formulas and calculations of Classical
"Newtonian" mechanics, called Lagrangian mechanics.
The prime symbol for derivatives was also
made by Lagrange.
==== Gauss, Hamilton, and Matrix notations
====
At the turn of the 19th century, Carl Friedrich
Gauss developed the identity sign for congruence
relation and, in Quadratic reciprocity, the
integral part. Gauss contributed functions
of complex variables, in geometry, and on
the convergence of series. He gave the satisfactory
proofs of the fundamental theorem of algebra
and of the quadratic reciprocity law. Gauss
developed the theory of solving linear systems
by using Gaussian elimination, which was initially
listed as an advancement in geodesy. He would
also develop the product sign. Also in this
time, Niels Henrik Abel and Évariste Galois
conducted their work on the solvability of
equations, linking group theory and field
theory.
After the 1800s, Christian Kramp would promote
factorial notation during his research in
generalized factorial function which applied
to non-integers. Joseph Diaz Gergonne introduced
the set inclusion signs. Peter Gustav Lejeune
Dirichlet developed Dirichlet L-functions
to give the proof of Dirichlet's theorem on
arithmetic progressions and began analytic
number theory. In 1828, Gauss proved his Theorema
Egregium (remarkable theorem in Latin), establishing
property of surfaces. In the 1830s, George
Green developed Green's function. In 1829.
Carl Gustav Jacob Jacobi publishes Fundamenta
nova theoriae functionum ellipticarum with
his elliptic theta functions. By 1841, Karl
Weierstrass, the "father of modern analysis",
elaborated on the concept of absolute value
and the determinant of a matrix.
Matrix notation would be more fully developed
by Arthur Cayley in his three papers, on subjects
which had been suggested by reading the Mécanique
analytique of Lagrange and some of the works
of Laplace. Cayley defined matrix multiplication
and matrix inverses. Cayley used a single
letter to denote a matrix, thus treating a
matrix as an aggregate object. He also realized
the connection between matrices and determinants,
and wrote "There would be many things to say
about this theory of matrices which should,
it seems to me, precede the theory of determinants".
William Rowan Hamilton would introduce the
nabla symbol for vector differentials. This
was previously used by Hamilton as a general-purpose
operator sign. Hamilton reformulated Newtonian
mechanics, now called Hamiltonian mechanics.
This work has proven central to the modern
study of classical field theories such as
electromagnetism. This was also important
to the development of quantum mechanics. In
mathematics, he is perhaps best known as the
inventor of quaternion notation and biquaternions.
Hamilton also introduced the word "tensor"
in 1846. James Cockle would develop the tessarines
and, in 1849, coquaternions. In 1848, James
Joseph Sylvester introduced into matrix algebra
the term matrix.
==== Maxwell, Clifford, and Ricci notations
====
In 1864 James Clerk Maxwell reduced all of
the then current knowledge of electromagnetism
into a linked set of differential equations
with 20 equations in 20 variables, contained
in A Dynamical Theory of the Electromagnetic
Field. (See Maxwell's equations.) The method
of calculation which it is necessary to employ
was given by Lagrange, and afterwards developed,
with some modifications, by Hamilton's equations.
It is usually referred to as Hamilton's principle;
when the equations in the original form are
used they are known as Lagrange's equations.
In 1871 Richard Dedekind called a set of real
or complex numbers which is closed under the
four arithmetic operations a field. In 1873
Maxwell presented A Treatise on Electricity
and Magnetism.
In 1878, William Kingdon Clifford published
his Elements of Dynamic. Clifford developed
split-biquaternions, which he called algebraic
motors. Clifford obviated quaternion study
by separating the dot product and cross product
of two vectors from the complete quaternion
notation. This approach made vector calculus
available to engineers and others working
in three dimensions and skeptical of the lead–lag
effect in the fourth dimension. The common
vector notations are used when working with
vectors which are spatial or more abstract
members of vector spaces, while angle notation
(or phasor notation) is a notation used in
electronics.
In 1881, Leopold Kronecker defined what he
called a "domain of rationality", which is
a field extension of the field of rational
numbers in modern terms. In 1882, Hüseyin
Tevfik Paşa wrote the book titled "Linear
Algebra". Lord Kelvin's aetheric atom theory
(1860s) led Peter Guthrie Tait, in 1885, to
publish a topological table of knots with
up to ten crossings known as the Tait conjectures.
In 1893, Heinrich M. Weber gave the clear
definition of an abstract field. Tensor calculus
was developed by Gregorio Ricci-Curbastro
between 1887–96, presented in 1892 under
the title absolute differential calculus,
and the contemporary usage of "tensor" was
stated by Woldemar Voigt in 1898. In 1895,
Henri Poincaré published Analysis Situs.
In 1897, Charles Proteus Steinmetz would publish
Theory and Calculation of Alternating Current
Phenomena, with the assistance of Ernst J.
Berg.
==== From formula mathematics to tensors ====
In 1895 Giuseppe Peano issued his Formulario
mathematico, an effort to digest mathematics
into terse text based on special symbols.
He would provide a definition of a vector
space and linear map. He would also introduce
the intersection sign, the union sign, the
membership sign (is an element of), and existential
quantifier (there exists). Peano would pass
to Bertrand Russell his work in 1900 at a
Paris conference; it so impressed Russell
that Russell too was taken with the drive
to render mathematics more concisely. The
result was Principia Mathematica written with
Alfred North Whitehead. This treatise marks
a watershed in modern literature where symbol
became dominant. Ricci-Curbastro and Tullio
Levi-Civita popularized the tensor index notation
around 1900.
=== Mathematical logic and abstraction ===
At the beginning of this period, Felix Klein's
"Erlangen program" identified the underlying
theme of various geometries, defining each
of them as the study of properties invariant
under a given group of symmetries. This level
of abstraction revealed connections between
geometry and abstract algebra. Georg Cantor
would introduce the aleph symbol for cardinal
numbers of transfinite sets. His notation
for the cardinal numbers was the Hebrew letter
ℵ
{\displaystyle \aleph }
(aleph) with a natural number subscript; for
the ordinals he employed the Greek letter
ω (omega). This notation is still in use
today in ordinal notation of a finite sequence
of symbols from a finite alphabet which names
an ordinal number according to some scheme
which gives meaning to the language. His theory
created a great deal of controversy. Cantor
would, in his study of Fourier series, consider
point sets in Euclidean space.
After the turn of the 20th century, Josiah
Willard Gibbs would in physical chemistry
introduce middle dot for dot product and the
multiplication sign for cross products. He
would also supply notation for the scalar
and vector products, which was introduced
in Vector Analysis. In 1904, Ernst Zermelo
promotes axiom of choice and his proof of
the well-ordering theorem. Bertrand Russell
would shortly afterward introduce logical
disjunction (OR) in 1906. Also in 1906, Poincaré
would publish On the Dynamics of the Electron
and Maurice Fréchet introduced metric space.
Later, Gerhard Kowalewski and Cuthbert Edmund
Cullis would successively introduce matrices
notation, parenthetical matrix and box matrix
notation respectively. After 1907, mathematicians
studied knots from the point of view of the
knot group and invariants from homology theory.
In 1908, Joseph Wedderburn's structure theorems
were formulated for finite-dimensional algebras
over a field. Also in 1908, Ernst Zermelo
proposed "definite" property and the first
axiomatic set theory, Zermelo set theory.
In 1910 Ernst Steinitz published the influential
paper Algebraic Theory of Fields. In 1911,
Steinmetz would publish Theory and Calculation
of Transient Electric Phenomena and Oscillations.
Albert Einstein, in 1916, introduced the Einstein
notation which summed over a set of indexed
terms in a formula, thus exerting notational
brevity. Arnold Sommerfeld would create the
contour integral sign in 1917. Also in 1917,
Dimitry Mirimanoff proposes axiom of regularity.
In 1919, Theodor Kaluza would solve general
relativity equations using five dimensions,
the results would have electromagnetic equations
emerge. This would be published in 1921 in
"Zum Unitätsproblem der Physik". In 1922,
Abraham Fraenkel and Thoralf Skolem independently
proposed replacing the axiom schema of specification
with the axiom schema of replacement. Also
in 1922, Zermelo–Fraenkel set theory was
developed. In 1923, Steinmetz would publish
Four Lectures on Relativity and Space. Around
1924, Jan Arnoldus Schouten would develop
the modern notation and formalism for the
Ricci calculus framework during the absolute
differential calculus applications to general
relativity and differential geometry in the
early twentieth century. In 1925, Enrico Fermi
would describe a system comprising many identical
particles that obey the Pauli exclusion principle,
afterwards developing a diffusion equation
(Fermi age equation). In 1926, Oskar Klein
would develop the Kaluza–Klein theory. In
1928, Emil Artin abstracted ring theory with
Artinian rings. In 1933, Andrey Kolmogorov
introduces the Kolmogorov axioms. In 1937,
Bruno de Finetti deduced the "operational
subjective" concept.
==== Mathematical symbolism ====
Mathematical abstraction began as a process
of extracting the underlying essence of a
mathematical concept, removing any dependence
on real world objects with which it might
originally have been connected, and generalizing
it so that it has wider applications or matching
among other abstract descriptions of equivalent
phenomena. Two abstract areas of modern mathematics
are category theory and model theory. Bertrand
Russell, said, "Ordinary language is totally
unsuited for expressing what physics really
asserts, since the words of everyday life
are not sufficiently abstract. Only mathematics
and mathematical logic can say as little as
the physicist means to say". Though, one can
substituted mathematics for real world objects,
and wander off through equation after equation,
and can build a concept structure which has
no relation to reality.Symbolic logic studies
the purely formal properties of strings of
symbols. The interest in this area springs
from two sources. First, the notation used
in symbolic logic can be seen as representing
the words used in philosophical logic. Second,
the rules for manipulating symbols found in
symbolic logic can be implemented on a computing
machine. Symbolic logic is usually divided
into two subfields, propositional logic and
predicate logic. Other logics of interest
include temporal logic, modal logic and fuzzy
logic. The area of symbolic logic called propositional
logic, also called propositional calculus,
studies the properties of sentences formed
from constants and logical operators. The
corresponding logical operations are known,
respectively, as conjunction, disjunction,
material conditional, biconditional, and negation.
These operators are denoted as keywords and
by symbolic notation.
Some of the introduced mathematical logic
notation during this time included the set
of symbols used in Boolean algebra. This was
created by George Boole in 1854. Boole himself
did not see logic as a branch of mathematics,
but it has come to be encompassed anyway.
Symbols found in Boolean algebra include
∧
{\displaystyle \land }
(AND),
∨
{\displaystyle \lor }
(OR), and
¬
{\displaystyle \lnot }
(not). With these symbols, and letters to
represent different truth values, one can
make logical statements such as
a
∨
¬
a
=
1
{\displaystyle a\lor \lnot a=1}
, that is "(a is true OR a is not true) is
true", meaning it is true that a is either
true or not true (i.e. false). Boolean algebra
has many practical uses as it is, but it also
was the start of what would be a large set
of symbols to be used in logic. Predicate
logic, originally called predicate calculus,
expands on propositional logic by the introduction
of variables and by sentences containing variables,
called predicates. In addition, predicate
logic allows quantifiers. With these logic
symbols and additional quantifiers from predicate
logic, valid proofs can be made that are irrationally
artificial, but syntactical.
==== Gödel incompleteness notation ====
While proving his incompleteness theorems,
Kurt Gödel created an alternative to the
symbols normally used in logic. He used Gödel
numbers, which were numbers that represented
operations with set numbers, and variables
with the prime numbers greater than 10. With
Gödel numbers, logic statements can be broken
down into a number sequence. Gödel then took
this one step farther, taking the n prime
numbers and putting them to the power of the
numbers in the sequence. These numbers were
then multiplied together to get the final
product, giving every logic statement its
own number.
=== Contemporary notation and 
topics ===
==== Early 20th-century notation ====
Abstraction of notation is an ongoing process
and the historical development of many mathematical
topics exhibits a progression from the concrete
to the abstract. Various set notations would
be developed for fundamental object sets.
Around 1924, David Hilbert and Richard Courant
published "Methods of mathematical physics.
Partial differential equations". In 1926,
Oskar Klein and Walter Gordon proposed the
Klein–Gordon equation to describe relativistic
particles. The first formulation of a quantum
theory describing radiation and matter interaction
is due to Paul Adrien Maurice Dirac, who,
during 1920, was first able to compute the
coefficient of spontaneous emission of an
atom. In 1928, the relativistic Dirac equation
was formulated by Dirac to explain the behavior
of the relativistically moving electron. Dirac
described the quantification of the electromagnetic
field as an ensemble of harmonic oscillators
with the introduction of the concept of creation
and annihilation operators of particles. In
the following years, with contributions from
Wolfgang Pauli, Eugene Wigner, Pascual Jordan,
and Werner Heisenberg, and an elegant formulation
of quantum electrodynamics due to Enrico Fermi,
physicists came to believe that, in principle,
it would be possible to perform any computation
for any physical process involving photons
and charged particles.
In 1931, Alexandru Proca developed the Proca
equation (Euler–Lagrange equation) for the
vector meson theory of nuclear forces and
the relativistic quantum field equations.
John Archibald Wheeler in 1937 develops S-matrix.
Studies by Felix Bloch with Arnold Nordsieck,
and Victor Weisskopf, in 1937 and 1939, revealed
that such computations were reliable only
at a first order of perturbation theory, a
problem already pointed out by Robert Oppenheimer.
At higher orders in the series infinities
emerged, making such computations meaningless
and casting serious doubts on the internal
consistency of the theory itself. With no
solution for this problem known at the time,
it appeared that a fundamental incompatibility
existed between special relativity and quantum
mechanics.
In the 1930s, the double-struck capital Z
for integer number sets was created by Edmund
Landau. Nicolas Bourbaki created the double-struck
capital Q for rational number sets. In 1935,
Gerhard Gentzen made universal quantifiers.
In 1936, Tarski's undefinability theorem is
stated by Alfred Tarski and proved. In 1938,
Gödel proposes the constructible universe
in the paper "The Consistency of the Axiom
of Choice and of the Generalized Continuum-Hypothesis".
André Weil and Nicolas Bourbaki would develop
the empty set sign in 1939. That same year,
Nathan Jacobson would coin the double-struck
capital C for complex number sets.
Around the 1930s, Voigt notation would be
developed for multilinear algebra as a way
to represent a symmetric tensor by reducing
its order. Schönflies notation became one
of two conventions used to describe point
groups (the other being Hermann–Mauguin
notation). Also in this time, van der Waerden
notation became popular for the usage of two-component
spinors (Weyl spinors) in four spacetime dimensions.
Arend Heyting would introduce Heyting algebra
and Heyting arithmetic.
The arrow, e.g., →, was developed for function
notation in 1936 by Øystein Ore to denote
images of specific elements. Later, in 1940,
it took its present form, e.g., f: X → Y,
through the work of Witold Hurewicz. Werner
Heisenberg, in 1941, proposed the S-matrix
theory of particle interactions.
Bra–ket notation (Dirac notation) is a standard
notation for describing quantum states, composed
of angle brackets and vertical bars. It can
also be used to denote abstract vectors and
linear functionals. It is so called because
the inner product (or dot product on a complex
vector space) of two states is denoted by
a ⟨bra|ket⟩ consisting of a left part,
⟨φ|, and a right part, |ψ⟩. The notation
was introduced in 1939 by Paul Dirac, though
the notation has precursors in Grassmann's
use of the notation [φ|ψ] for his inner
products nearly 100 years previously.Bra–ket
notation is widespread in quantum mechanics:
almost every phenomenon that is explained
using quantum mechanics—including a large
portion of modern physics—is usually explained
with the help of bra–ket notation. The notation
establishes an encoded abstract representation-independence,
producing a versatile specific representation
(e.g., x, or p, or eigenfunction base) without
much ado, or excessive reliance on, the nature
of the linear spaces involved. The overlap
expression ⟨φ|ψ⟩ is typically interpreted
as the probability amplitude for the state
ψ to collapse into the state ϕ. The Feynman
slash notation (Dirac slash notation) was
developed by Richard Feynman for the study
of Dirac fields in quantum field theory.
In 1948, Valentine Bargmann and Eugene Wigner
proposed the relativistic Bargmann–Wigner
equations to describe free particles and the
equations are in the form of multi-component
spinor field wavefunctions. In 1950, William
Vallance Douglas Hodge presented "The topological
invariants of algebraic varieties" at the
Proceedings of the International Congress
of Mathematicians. Between 1954 and 1957,
Eugenio Calabi worked on the Calabi conjecture
for Kähler metrics and the development of
Calabi–Yau manifolds. In 1957, Tullio Regge
formulated the mathematical property of potential
scattering in the Schrödinger equation. Stanley
Mandelstam, along with Regge, did the initial
development of the Regge theory of strong
interaction phenomenology. In 1958, Murray
Gell-Mann and Richard Feynman, along with
George Sudarshan and Robert Marshak, deduced
the chiral structures of the weak interaction
in physics. Geoffrey Chew, along with others,
would promote matrix notation for the strong
interaction, and the associated bootstrap
principle, in 1960. In the 1960s, set-builder
notation was developed for describing a set
by stating the properties that its members
must satisfy. Also in the 1960s, tensors are
abstracted within category theory by means
of the concept of monoidal category. Later,
multi-index notation eliminates conventional
notions used in multivariable calculus, partial
differential equations, and the theory of
distributions, by abstracting the concept
of an integer index to an ordered tuple of
indices.
==== Modern mathematical notation ====
In the modern mathematics of special relativity,
electromagnetism and wave theory, the d'Alembert
operator is the Laplace operator of Minkowski
space. The Levi-Civita symbol is used in tensor
calculus.
After the full Lorentz covariance formulations
that were finite at any order in a perturbation
series of quantum electrodynamics, Sin-Itiro
Tomonaga, Julian Schwinger and Richard Feynman
were jointly awarded with a Nobel prize in
physics in 1965. Their contributions, and
those of Freeman Dyson, were about covariant
and gauge invariant formulations of quantum
electrodynamics that allow computations of
observables at any order of perturbation theory.
Feynman's mathematical technique, based on
his diagrams, initially seemed very different
from the field-theoretic, operator-based approach
of Schwinger and Tomonaga, but Freeman Dyson
later showed that the two approaches were
equivalent. Renormalization, the need to attach
a physical meaning at certain divergences
appearing in the theory through integrals,
has subsequently become one of the fundamental
aspects of quantum field theory and has come
to be seen as a criterion for a theory's general
acceptability. Quantum electrodynamics has
served as the model and template for subsequent
quantum field theories. Peter Higgs, Jeffrey
Goldstone, and others, Sheldon Glashow, Steven
Weinberg and Abdus Salam independently showed
how the weak nuclear force and quantum electrodynamics
could be merged into a single electroweak
force. In the late 1960s, the particle zoo
was composed of the then known elementary
particles before the discovery of quarks.
A step towards the Standard Model was Sheldon
Glashow's discovery, in 1960, of a way to
combine the electromagnetic and weak interactions.
In 1967, Steven Weinberg and Abdus Salam incorporated
the Higgs mechanism into Glashow's electroweak
theory, giving it its modern form. The Higgs
mechanism is believed to give rise to the
masses of all the elementary particles in
the Standard Model. This includes the masses
of the W and Z bosons, and the masses of the
fermions - i.e. the quarks and leptons. Also
in 1967, Bryce DeWitt published his equation
under the name "Einstein–Schrödinger equation"
(later renamed the "Wheeler–DeWitt equation").
In 1969, Yoichiro Nambu, Holger Bech Nielsen,
and Leonard Susskind described space and time
in terms of strings. In 1970, Pierre Ramond
develop two-dimensional supersymmetries. Michio
Kaku and Keiji Kikkawa would afterwards formulate
string variations. In 1972, Michael Artin,
Alexandre Grothendieck, Jean-Louis Verdier
propose the Grothendieck universe.After the
neutral weak currents caused by Z boson exchange
were discovered at CERN in 1973, the electroweak
theory became widely accepted and Glashow,
Salam, and Weinberg shared the 1979 Nobel
Prize in Physics for discovering it. The theory
of the strong interaction, to which many contributed,
acquired its modern form around 1973–74.
With the establishment of quantum chromodynamics,
a finalized a set of fundamental and exchange
particles, which allowed for the establishment
of a "standard model" based on the mathematics
of gauge invariance, which successfully described
all forces except for gravity, and which remains
generally accepted within the domain to which
it is designed to be applied. In the late
1970s, William Thurston introduced hyperbolic
geometry into the study of knots with the
hyperbolization theorem. The orbifold notation
system, invented by Thurston, has been developed
for representing types of symmetry groups
in two-dimensional spaces of constant curvature.
In 1978, Shing-Tung Yau deduced that the Calabi
conjecture have Ricci flat metrics. In 1979,
Daniel Friedan showed that the equations of
motions of string theory are abstractions
of Einstein equations of General Relativity.
The first superstring revolution is composed
of mathematical equations developed between
1984 and 1986. In 1984, Vaughan Jones deduced
the Jones polynomial and subsequent contributions
from Edward Witten, Maxim Kontsevich, and
others, revealed deep connections between
knot theory and mathematical methods in statistical
mechanics and quantum field theory. According
to string theory, all particles in the "particle
zoo" have a common ancestor, namely a vibrating
string. In 1985, Philip Candelas, Gary Horowitz,
Andrew Strominger, and Edward Witten would
publish "Vacuum configurations for superstrings"
Later, the tetrad formalism (tetrad index
notation) would be introduced as an approach
to general relativity that replaces the choice
of a coordinate basis by the less restrictive
choice of a local basis for the tangent bundle.In
the 1990s, Roger Penrose would propose Penrose
graphical notation (tensor diagram notation)
as a, usually handwritten, visual depiction
of multilinear functions or tensors. Penrose
would also introduce abstract index notation.
In 1995, Edward Witten suggested M-theory
and subsequently used it to explain some observed
dualities, initiating the second superstring
revolution.
John Conway would further various notations,
including the Conway chained arrow notation,
the Conway notation of knot theory, and the
Conway polyhedron notation. The Coxeter notation
system classifies symmetry groups, describing
the angles between with fundamental reflections
of a Coxeter group. It uses a bracketed notation,
with modifiers to indicate certain subgroups.
The notation is named after H. S. M. Coxeter
and Norman Johnson more comprehensively defined
it.
Combinatorial LCF notation has been developed
for the representation of cubic graphs that
are Hamiltonian. The cycle notation is the
convention for writing down a permutation
in terms of its constituent cycles. This is
also called circular notation and the permutation
called a cyclic or circular permutation.
==== Computers and markup notation ====
In 1931, IBM produces the IBM 601 Multiplying
Punch; it is an electromechanical machine
that could read two numbers, up to 8 digits
long, from a card and punch their product
onto the same card. In 1934, Wallace Eckert
used a rigged IBM 601 Multiplying Punch to
automate the integration of differential equations.
In 1936, Alan Turing publishes "On Computable
Numbers, With an Application to the Entscheidungsproblem".
John von Neumann, pioneer of the digital computer
and of computer science, in 1945, writes the
incomplete First Draft of a Report on the
EDVAC. In 1962, Kenneth E. Iverson developed
an integral part notation that became known
as Iverson Notation for manipulating arrays
that he taught to his students, and described
in his book A Programming Language. In 1970,
E.F. Codd proposed relational algebra as a
relational model of data for database query
languages. In 1971, Stephen Cook publishes
"The complexity of theorem proving procedures"
In the 1970s within computer architecture,
Quote notation was developed for a representing
number system of rational numbers. Also in
this decade, the Z notation (just like the
APL language, long before it) uses many non-ASCII
symbols, the specification includes suggestions
for rendering the Z notation symbols in ASCII
and in LaTeX. There are presently various
C mathematical functions (Math.h) and numerical
libraries. They are libraries used in software
development for performing numerical calculations.
These calculations can be handled by symbolic
executions; analyzing a program to determine
what inputs cause each part of a program to
execute. Mathematica and SymPy are examples
of computational software programs based on
symbolic mathematics.
== Future of mathematical notation ==
In the history of mathematical notation, ideographic
symbol notation has come full circle with
the rise of computer visualization systems.
The notations can be applied to abstract visualizations,
such as for rendering some projections of
a Calabi-Yau manifold. Examples of abstract
visualization which properly belong to the
mathematical imagination can be found in computer
graphics. The need for such models abounds,
for example, when the measures for the subject
of study are actually random variables and
not really ordinary mathematical functions.
== See also ==
Main relevance
Abuse of notation, Well-formed formula, Big
O notation (L-notation), Dowker notation,
Hungarian notation, Infix notation, Positional
notation, Polish notation (Reverse Polish
notation), Sign-value notation, Subtractive
notation, infix notation, History of writing
numbersNumbers and quantities
List of numbers, Irrational and suspected
irrational numbers, γ, ζ(3), √2, √3,
√5, φ, ρ, δS, α, e, π, δ, Physical
constants, c, ε0, h, G, Greek letters used
in mathematics, science, and engineeringGeneral
relevance
Order of operations, Scientific notation (Engineering
notation), Actuarial notationDot notation
Chemical notation (Lewis dot notation (Electron
dot notation)), Dot-decimal notationArrow
notation
Knuth's up-arrow notation, infinitary combinatorics
(Arrow notation (Ramsey theory))Geometries
Projective geometry, Affine geometry, Finite
geometryLists and outlines
Outline of mathematics (Mathematics history
topics and Mathematics topics (Mathematics
categories)), Mathematical theories ( First-order
theories, Theorems and Disproved mathematical
ideas), Mathematical proofs (Incomplete proofs),
Mathematical identities, Mathematical series,
Mathematics reference tables, Mathematical
logic topics, Mathematics-based methods, Mathematical
functions, Transforms and Operators, Points
in mathematics, Mathematical shapes, Knots
(Prime knots and Mathematical knots and links),
Inequalities, Mathematical concepts named
after places, Mathematical topics in classical
mechanics, Mathematical topics in quantum
theory, Mathematical topics in relativity,
String theory topics, Unsolved problems in
mathematics, Mathematical jargon, Mathematical
examples, Mathematical abbreviations, List
of mathematical symbolsMisc.
Hilbert's problems, Mathematical coincidence,
Chess notation, Line notation, Musical notation
(Dotted note), Whyte notation, Dice notation,
recursive categorical syntaxPeople
Mathematicians (Amateur mathematicians and
Female mathematicians), Thomas Bradwardine,
Thomas Harriot, Felix Hausdorff, Gaston Julia,
Helge von Koch, Paul Lévy, Aleksandr Lyapunov,
Benoit Mandelbrot, Lewis Fry Richardson, Wacław
Sierpiński, Saunders Mac Lane, Paul Cohen,
Gottlob Frege, G. S. Carr, Robert Recorde,
Bartel Leendert van der Waerden, G. H. Hardy,
E. M. Wright, James R. Newman, Carl Gustav
Jacob Jacobi, Roger Joseph Boscovich, Eric
W. Weisstein, Mathematical probabilists, Statisticians
== 
Notes ==
== 
References and citations ==
GeneralFlorian Cajori (1929) A History of
Mathematical Notations, 2 vols. Dover reprint
in 1 vol., 1993. ISBN 0-486-67766-4.Citations
== Further reading ==
General
A Short Account of the History of Mathematics.
By Walter William Rouse Ball.
A Primer of the History of Mathematics. By
Walter William Rouse Ball.
A History of Elementary Mathematics: With
Hints on Methods of Teaching. By Florian Cajori.
A History of Elementary Mathematics. By Florian
Cajori.
A History of Mathematics. By Florian Cajori.
A Short History of Greek Mathematics. By James
Gow.
On the Development of Mathematical Thought
During the Nineteenth Century. By John Theodore
Merz.
A New Mathematical and Philosophical Dictionary.
By Peter Barlow.
Historical Introduction to Mathematical Literature.
By George Abram Miller
A Brief History of Mathematics. By Karl Fink,
Wooster Woodruff Beman, David Eugene Smith
History of Modern Mathematics. By David Eugene
Smith.
History of modern mathematics. By David Eugene
Smith, Mansfield Merriman.OtherPrincipia Mathematica,
Volume 1 & Volume 2. By Alfred North Whitehead,
Bertrand Russell.
The Mathematical Principles of Natural Philosophy,
Volume 1, Issue 1. By Sir Isaac Newton, Andrew
Motte, William Davis, John Machin, William
Emerson.
General investigations of curved surfaces
of 1827 and 1825. By Carl Friedrich Gaus.
== External links ==
Mathematical Notation: Past and Future
History of Mathematical Notation
Earliest Uses of Mathematical Notation
Finger counting. files.chem.vt.edu.
Some Common Mathematical Symbols and Abbreviations
(with History). Isaiah Lankham, Bruno Nachtergaele,
Anne Schilling.
