History of mathematical notation
Encyclopedia
Mathematical notation comprises the symbol
s used to write mathematical equation
s and formula
s. It includes Hindu-Arabic numerals, letters from the Roman, Greek
, Hebrew
, and German
alphabet
s, and a host of symbols invented by mathematicians over the past several centuries.
The development of mathematical notation for algebra can be divided in three stages. The first is "rhetorical", where all calculations are performed by words and no symbols are used. Most medieval Islamic mathematicians belonged to this stage. The second is "syncopated", where frequently used operations and quantities are represented by symbolic abbreviations. To this stage Diophantus
belonged. The third is "symbolic", which is a complete system of notation replacing all rhetoric. This system was in use by medieval Indian mathematicians and in Europe since the middle of the 17th century.
(See table of mathematical symbols
for a list of symbols.)
, with each tally representing a single unit. For example, one notch in a bone represented one animal, or person, or anything else. The symbolic notation was that of the Egyptians
. They 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 first found in the Rhind Mathematical Papyrus
. 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.
Like the Egyptians, 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 first 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.
at first employed Attic numeration, which was based on the system of the Egyptians and was later adapted and used by the Romans
. Numbers one through four were vertical lines, like in the hieroglyphics. The symbol for five was the Greek letter pi, which is the first letter of the Greek word for five, pente. (This is not to be confused with the modern constant also denoted by, and referred to as "pi," which is a constant equal to the ratio of the diameter of a circle to its circumference. Greek mathematicians did not have a formal name for what we now know as "pi," and certainly did not associate their letter pi, equivalent to "P" in English, with it.) Numbers six through nine were pente with vertical lines next to it. Ten was represented by the first letter of the word for ten, deka, one-hundred by the first letter from the word for one-hundred, etc.
The Ionian numeration used the entire alphabet and three archaic letters.
|-
|Α (α)
|Β (β)
|Г (γ)
|Δ (δ)
|Ε (ε)
|Ϝ (ϝ)
|Z
(ζ)
|H
(η)
|θ (θ)
|I
(ι)
|K
(κ)
|Λ (λ)
|Μ (μ)
|Ν (ν)
|Ξ (ξ)
|Ο (ο)
|Π (π)
|Ϟ (ϟ)
|Ρ (ρ)
|Σ (σ)
|Τ (τ)
|Υ (υ)
|Ф (φ)
|Χ (χ)
|Ψ (ψ)
|Ω (ω)
|Ϡ (ϡ)
|-
|1
|2
|3
|4
|5
|6
|7
|8
|9
|10
|20
|30
|40
|50
|60
|70
|80
|90
|100
|200
|300
|400
|500
|600
|700
|800
|900>
This system appeared in the third century BC, before the letters digamma (Ϝ), koppa (Ϟ), and sampi (Ϡ) became archaic. When lowercase letters appeared, these replaced the uppercase ones as the symbols for notation. Multiples of one-thousand were written as the first nine numbers with a stroke in front of them; thus one-thousand was, α, two-thousand was, β, etc. M was used to multiply numbers by ten-thousand. The number 88,888,888 would be written as M,ηωπη*ηωπη
Greek 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 algebra
ic symbols. The great exception was Diophantus
of Alexandria
the first great algebraists. His Arithmetica
was one of the first texts to use symbols in equations. It was not completely symbolic, but was much more than previous books. An unknown number was called s. The square of s was
; the cube was 
; the fourth power was 
; and the fifth power was 
. The expression 
would be written as SS2 C3 x5 M S4 u6.
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.
, Brahmagupta
, was syncopated. Addition was indicated by placing the numbers side by side, subtraction by placing a dot over the subtrahend, 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.
is Europeans first saw the numerals used in an Arabic book, Concerning the Hindu Art of Reckoning, by Mohommed ibn-Musa al-Khwarizmi. 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.
One of the first 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.
(1412–1482) was the last major medieval Arab algebraist, who improved on the algebraic notation
earlier used in the Maghreb
by Ibn al-Banna
in the 13th century and by Ibn al-Yāsamīn in the 12th 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
.
began as "et"). The minus sign was first used by Johannes Widmann
in Mercantile Arithmetic. Widmann used the minus symbol with the plus symbol, to indicate deficit and surplus, respectively. The radical
symbol
for square root was introduced by Christoph Rudolff
because it resembled a lowercase "r" (for "radix").
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. The New algebra
(1591) of François Viète
introduced the modern notational manipulation of algebraic expressions. In 1631 William Oughtred
introduced the multiplication sign (×) and abbreviations sin and cos for the trigonometric functions.
William Jones used π in Synopsis palmariorum mathesios in 1706 because it is the first letter of the Greek word perimetron (περιμετρον), which means perimeter
in Greek. This usage was popularized by Euler in 1737.
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. For example, the derivative of the function x would be written as
. The second derivative of x would be written as 
, etc. 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. For example, the derivative of the function x with respect to the variable t in Leibniz's notation would be written as
. 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.
was one of the most prolific mathematicians in history, and perhaps was also the most prolific inventor of canonical notation. His contributions include his use of e
to represent the base of natural logarithm
s. It is not known exactly why
was chosen, but it was probably because the first four letters of the alphabet were already commonly used to represent variables and other constants. Euler was also one of the first to use 
to represent pi
consistently. The use of
was first suggested by William Jones, who used it as shorthand for perimeter
. Euler was also the first to use
to represent the square root of negative one, 
, although he earlier used it as an infinite number. (Nowadays the symbol created by John Wallis, 
, is used for infinity.) For summation
, Euler was the first to use sigma
, Σ, as in
. For functions
, Euler was the first to use the notation
to represent a function of 
.
issued the first edition of his Formulario mathematico
, an effort to digest mathematics into terse text based on special symbols. The copy he passed to Bertrand Russell
in 1900 at a Paris conference so impressed Russell that he 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. Peano's Formulario Mathematico, though less popular than Russell's work, continued through five editions. The fifth appeared in 1908 and included 4200 formulas and theorems.
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
(AND), 
(OR), and 
(NOT). With these symbols, and letters to represent different truth values, one can make logical statements such as 
, 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. Most of these symbols can be found in propositional calculus
, a formal system
described as
. 
is the set of elements, such as the a in the example with Boolean algebra above. 
is the set that contains the subsets that contain operations, such as 
or 
. 
contains the inference rules, which are the rules dictating how inferences may be logically made, and 
contains the axioms. (See also: Basic and Derived Argument Forms). With these symbols, proofs
can be made that are completely artificial.
While proving his incompleteness theorems, Kurt Gödel
created an alternative to the symbols normally used in logic. He used Gödel number
s, which were numbers that represented operations with set numbers, and variables with the first 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 first 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. For example, take the statement "There exists a number x such that it is not y". Using the symbols of propositional calculus, this would become
. If the Gödel numbers replace the symbols, it becomes {8, 4, 11, 9, 8, 11, 5, 1, 13, 9}. There are ten numbers, so the first ten prime numbers are found and these are {2, 3, 5, 7, 11, 13, 17, 19, 23, 29}. Then, the Gödel numbers are made the powers of the respective primes and multiplied, giving 
. The resulting number is approximately 
.
Symbol
A symbol is something which represents an idea, a physical entity or a process but is distinct from it. The purpose of a symbol is to communicate meaning. For example, a red octagon may be a symbol for "STOP". On a map, a picture of a tent might represent a campsite. Numerals are symbols for...
s used to write mathematical equation
Equation
An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...
s and formula
Formula
In mathematics, a formula is an entity constructed using the symbols and formation rules of a given logical language....
s. It includes Hindu-Arabic numerals, letters from the Roman, Greek
Greek alphabet
The Greek alphabet is the script that has been used to write the Greek language since at least 730 BC . The alphabet in its classical and modern form consists of 24 letters ordered in sequence from alpha to omega...
, Hebrew
Hebrew alphabet
The Hebrew alphabet , known variously by scholars as the Jewish script, square script, block script, or more historically, the Assyrian script, is used in the writing of the Hebrew language, as well as other Jewish languages, most notably Yiddish, Ladino, and Judeo-Arabic. There have been two...
, and German
German alphabet
The modern German alphabet is an extended Latin alphabet consisting of 30 letters – the same letters that are found in the Basic modern Latin alphabet plus four extra letters.In German, the individual letters have neuter gender: das A, das B etc....
alphabet
Alphabet
An alphabet is a standard set of letters—basic written symbols or graphemes—each of which represents a phoneme in a spoken language, either as it exists now or as it was in the past. There are other systems, such as logographies, in which each character represents a word, morpheme, or semantic...
s, and a host of symbols invented by mathematicians over the past several centuries.
The development of mathematical notation for algebra can be divided in three stages. The first is "rhetorical", where all calculations are performed by words and no symbols are used. Most medieval Islamic mathematicians belonged to this stage. The second is "syncopated", where frequently used operations and quantities are represented by symbolic abbreviations. To this stage Diophantus
Diophantus
Diophantus of Alexandria , sometimes called "the father of algebra", was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica. These texts deal with solving algebraic equations, many of which are now lost...
belonged. The third is "symbolic", which is a complete system of notation replacing all rhetoric. This system was in use by medieval Indian mathematicians and in Europe since the middle of the 17th century.
(See table of mathematical symbols
Table of mathematical symbols
This is a listing of common symbols found within all branches of mathematics. Each symbol is listed in both HTML, which depends on appropriate fonts being installed, and in , as an image.-Symbols:-Variations:...
for a list of symbols.)
Beginning of notation
Written mathematics began with numbers expressed as tally marksTally marks
Tally marks, or hash marks, are a unary numeral system. They are a form of numeral used for counting. They allow updating written intermediate results without erasing or discarding anything written down...
, with each tally representing a single unit. For example, one notch in a bone represented one animal, or person, or anything else. The symbolic notation was that of the Egyptians
Egyptians
Egyptians are nation an ethnic group made up of Mediterranean North Africans, the indigenous people of Egypt.Egyptian identity is closely tied to geography. The population of Egypt is concentrated in the lower Nile Valley, the small strip of cultivable land stretching from the First Cataract to...
. They 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
Hieratic
Hieratic refers to a cursive writing system that was used in the provenance of the pharaohs in Egypt and Nubia that developed alongside the hieroglyphic system, to which it is intimately related...
instead of hieroglyphic
Egyptian hieroglyphs
Egyptian hieroglyphs were a formal writing system used by the ancient Egyptians that combined logographic and alphabetic elements. Egyptians used cursive hieroglyphs for religious literature on papyrus and wood...
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 first found in the Rhind Mathematical Papyrus
Rhind Mathematical Papyrus
The Rhind Mathematical Papyrus , is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum. It dates to around 1650 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.
Like the Egyptians, 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
Coefficient
In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...
of that number. Each digit was at first 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.
Greek notation
The GreeksAncient Greece
Ancient Greece is a civilization belonging to a period of Greek history that lasted from the Archaic period of the 8th to 6th centuries BC to the end of antiquity. Immediately following this period was the beginning of the Early Middle Ages and the Byzantine era. Included in Ancient Greece is the...
at first employed Attic numeration, which was based on the system of the Egyptians and was later adapted and used by the Romans
Ancient Rome
Ancient Rome was a thriving civilization that grew on the Italian Peninsula as early as the 8th century BC. Located along the Mediterranean Sea and centered on the city of Rome, it expanded to one of the largest empires in the ancient world....
. Numbers one through four were vertical lines, like in the hieroglyphics. The symbol for five was the Greek letter pi, which is the first letter of the Greek word for five, pente. (This is not to be confused with the modern constant also denoted by, and referred to as "pi," which is a constant equal to the ratio of the diameter of a circle to its circumference. Greek mathematicians did not have a formal name for what we now know as "pi," and certainly did not associate their letter pi, equivalent to "P" in English, with it.) Numbers six through nine were pente with vertical lines next to it. Ten was represented by the first letter of the word for ten, deka, one-hundred by the first letter from the word for one-hundred, etc.
The Ionian numeration used the entire alphabet and three archaic letters.
|Α (α)
|Β (β)
|Г (γ)
|Δ (δ)
|Ε (ε)
|Ϝ (ϝ)
|Z
Z
Z is the twenty-sixth and final letter of the basic modern Latin alphabet.-Name and pronunciation:In most dialects of English, the letter's name is zed , reflecting its derivation from the Greek zeta but in American English, its name is zee , deriving from a late 17th century English dialectal...
(ζ)
|H
H
H .) is the eighth letter in the basic modern Latin alphabet.-History:The Semitic letter ⟨ח⟩ most likely represented the voiceless pharyngeal fricative . The form of the letter probably stood for a fence or posts....
(η)
|θ (θ)
|I
I
I is the ninth letter and a vowel in the basic modern Latin alphabet.-History:In Semitic, the letter may have originated in a hieroglyph for an arm that represented a voiced pharyngeal fricative in Egyptian, but was reassigned to by Semites, because their word for "arm" began with that sound...
(ι)
|K
K
K is the eleventh letter of the English and basic modern Latin alphabet.-History and usage:In English, the letter K usually represents the voiceless velar plosive; this sound is also transcribed by in the International Phonetic Alphabet and X-SAMPA....
(κ)
|Λ (λ)
|Μ (μ)
|Ν (ν)
|Ξ (ξ)
|Ο (ο)
|Π (π)
|Ϟ (ϟ)
|Ρ (ρ)
|Σ (σ)
|Τ (τ)
|Υ (υ)
|Ф (φ)
|Χ (χ)
|Ψ (ψ)
|Ω (ω)
|Ϡ (ϡ)
|-
|1
|2
|3
|4
|5
|6
|7
|8
|9
|10
|20
|30
|40
|50
|60
|70
|80
|90
|100
|200
|300
|400
|500
|600
|700
|800
|900>
This system appeared in the third century BC, before the letters digamma (Ϝ), koppa (Ϟ), and sampi (Ϡ) became archaic. When lowercase letters appeared, these replaced the uppercase ones as the symbols for notation. Multiples of one-thousand were written as the first nine numbers with a stroke in front of them; thus one-thousand was, α, two-thousand was, β, etc. M was used to multiply numbers by ten-thousand. The number 88,888,888 would be written as M,ηωπη*ηωπη
Greek mathematical reasoning was almost entirely geometric
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
(albeit often used to reason about non-geometric subjects such as number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
), and hence the Greeks had no interest in algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
ic symbols. The great exception was Diophantus
Diophantus
Diophantus of Alexandria , sometimes called "the father of algebra", was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica. These texts deal with solving algebraic equations, many of which are now lost...
of Alexandria
Alexandria
Alexandria is the second-largest city of Egypt, with a population of 4.1 million, extending about along the coast of the Mediterranean Sea in the north central part of the country; it is also the largest city lying directly on the Mediterranean coast. It is Egypt's largest seaport, serving...
the first great algebraists. His Arithmetica
Arithmetica
Arithmetica is an ancient Greek text on mathematics written by the mathematician Diophantus in the 3rd century AD. It is a collection of 130 algebraic problems giving numerical solutions of determinate equations and indeterminate equations.Equations in the book are called Diophantine equations...
was one of the first texts to use symbols in equations. It was not completely symbolic, but was much more than previous books. An unknown number was called s. The square of s was
; the cube was ; the fourth power was ; and the fifth power was . The expression would be written as SS2 C3 x5 M S4 u6.Chinese notation
China
Chinese civilization may refer to:* China for more general discussion of the country.* Chinese culture* Greater China, the transnational community of ethnic Chinese.* History of China* Sinosphere, the area historically affected by Chinese culture...
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.
Indian notation
The algebraic notation of the Indian mathematicianIndian mathematics
Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics , important contributions were made by scholars like Aryabhata, Brahmagupta, and Bhaskara II. The decimal number system in use today was first...
, Brahmagupta
Brahmagupta
Brahmagupta was an Indian mathematician and astronomer who wrote many important works on mathematics and astronomy. His best known work is the Brāhmasphuṭasiddhānta , written in 628 in Bhinmal...
, was syncopated. Addition was indicated by placing the numbers side by side, subtraction by placing a dot over the subtrahend, 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.
Beginning of Hindu-Arabic numerals
Despite their name, Arabic numerals actually started in India. The reason for this misnomerMisnomer
A misnomer is a term which suggests an interpretation that is known to be untrue. Such incorrect terms sometimes derive their names because of the form, action, or origin of the subject becoming named popularly or widely referenced—long before their true natures were known.- Sources of misnomers...
is Europeans first saw the numerals used in an Arabic book, Concerning the Hindu Art of Reckoning, by Mohommed ibn-Musa al-Khwarizmi. 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.
One of the first European books that advocated using the numerals was Liber Abaci
Liber Abaci
Liber Abaci is a historic book on arithmetic by Leonardo of Pisa, known later by his nickname Fibonacci...
, by Leonardo of Pisa, better known as Fibonacci
Fibonacci
Leonardo Pisano Bigollo also known as Leonardo of Pisa, Leonardo Pisano, Leonardo Bonacci, Leonardo Fibonacci, or, most commonly, simply Fibonacci, was an Italian mathematician, considered by some "the most talented western mathematician of the Middle Ages."Fibonacci is best known to the modern...
. 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.
North African notation
Abū al-Hasan ibn Alī al-QalasādīAbu al-Hasan ibn Ali al-Qalasadi
Abū al-Ḥasan ibn ʿAlī ibn Muḥammad ibn ʿAlī al-Qalaṣādī was a Muslim mathematician from Al-Andalus specializing in Islamic inheritance jurisprudence...
(1412–1482) was the last major medieval Arab algebraist, who improved on the algebraic notation
Mathematical notation
Mathematical notation is a system of symbolic representations of mathematical objects and ideas. Mathematical notations are used in mathematics, the physical sciences, engineering, and economics...
earlier used in the Maghreb
Maghreb
The Maghreb is the region of Northwest Africa, west of Egypt. It includes five countries: Morocco, Algeria, Tunisia, Libya, and Mauritania and the disputed territory of Western Sahara...
by Ibn al-Banna
Ibn al-Banna
Ibn al‐Bannāʾ al‐Marrākushī al-Azdi also known as Abu'l-Abbas Ahmad ibn Muhammad ibn Uthman al-Azdi. [29) December 1256 – c. 1321) was a Moroccan mathematician, astronomer, Islamic scholar, Sufi, and a one time astrologer....
in the 13th century and by Ibn al-Yāsamīn in the 12th century. In contrast to the syncopated notations of their predecessors, Diophantus
Diophantus
Diophantus of Alexandria , sometimes called "the father of algebra", was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica. These texts deal with solving algebraic equations, many of which are now lost...
and Brahmagupta
Brahmagupta
Brahmagupta was an Indian mathematician and astronomer who wrote many important works on mathematics and astronomy. His best known work is the Brāhmasphuṭasiddhānta , written in 628 in Bhinmal...
, which lacked symbols for mathematical operations
Operation (mathematics)
The general operation as explained on this page should not be confused with the more specific operators on vector spaces. For a notion in elementary mathematics, see arithmetic operation....
, 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
Table of mathematical symbols
This is a listing of common symbols found within all branches of mathematics. Each symbol is listed in both HTML, which depends on appropriate fonts being installed, and in , as an image.-Symbols:-Variations:...
using characters from the Arabic alphabet
Arabic alphabet
The Arabic alphabet or Arabic abjad is the Arabic script as it is codified for writing the Arabic language. It is written from right to left, in a cursive style, and includes 28 letters. Because letters usually stand for consonants, it is classified as an abjad.-Consonants:The Arabic alphabet has...
.
Pre-calculus
Two of the most widely used mathematical symbols are addition and subtraction, + and −. The plus sign was first used by Nicole Oresme in Algorismus proportionum, possibly an abbreviation for "et", which is "and" in Latin (in much the same way the ampersandAmpersand
An ampersand is a logogram representing the conjunction word "and". The symbol is a ligature of the letters in et, Latin for "and".-Etymology:...
began as "et"). The minus sign was first used by Johannes Widmann
Johannes Widmann
Johannes Widmann was a German mathematician who invented the addition and the subtraction signs.Born at in Eger, Bohemia, Widmann attended the University of Leipzig in the 1480s, and published Behende und hubsche Rechenung auff allen Kauffmanschafft Johannes Widmann (c. 1460 - after 1498) was a...
in Mercantile Arithmetic. Widmann used the minus symbol with the plus symbol, to indicate deficit and surplus, respectively. The radical
Nth root
In mathematics, the nth root of a number x is a number r which, when raised to the power of n, equals xr^n = x,where n is the degree of the root...
symbol
for square root was introduced by Christoph RudolffChristoph Rudolff
Christoff Rudolff was the author of the first German textbook on algebra.Rudolff was from 1517 to 1521 a student of Henricus Grammateus at the University of Vienna and was the author of a book computing, under the title: Behend und hübsch Rechnung durch die kunstreichen regeln Algebre.He...
because it resembled a lowercase "r" (for "radix").
In 1557 Robert Recorde
Robert Recorde
Robert Recorde was a Welsh physician and mathematician. He introduced the "equals" sign in 1557.-Biography:A member of a respectable family of Tenby, Wales, he entered the University of Oxford about 1525, and was elected a fellow of All Souls College in 1531...
published The Whetstone of Witte
The Whetstone of Witte
The Whetstone of Witte is the shortened title of Robert Recorde's mathematics book published in 1557. The full title being, The Whetstone of Witte, whiche is the seconde parte of Arithmeteke: containing the extraction of rootes; the cossike practise, with the rule of equation; and the workes of...
which used the equal sign (=) as well as plus and minus signs for the English reader. The New algebra
New algebra
The new algebra or symbolic analysis is a formalization of algebra promoted by François Viète in 1591 and by his successors...
(1591) of François Viète
François Viète
François Viète , Seigneur de la Bigotière, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to its innovative use of letters as parameters in equations...
introduced the modern notational manipulation of algebraic expressions. In 1631 William Oughtred
William Oughtred
William Oughtred was an English mathematician.After John Napier invented logarithms, and Edmund Gunter created the logarithmic scales upon which slide rules are based, it was Oughtred who first used two such scales sliding by one another to perform direct multiplication and division; and he is...
introduced the multiplication sign (×) and abbreviations sin and cos for the trigonometric functions.
William Jones used π in Synopsis palmariorum mathesios in 1706 because it is the first letter of the Greek word perimetron (περιμετρον), which means perimeter
Perimeter
A perimeter is a path that surrounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length - it can be thought of as the length of the outline of a shape. The perimeter of a circular area is called circumference.- Practical uses :Calculating...
in Greek. This usage was popularized by Euler in 1737.
Calculus
Calculus had two main systems of notation, each created by one of the creators: that developed by Isaac NewtonIsaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
and the notation developed by Gottfried Leibniz
Gottfried Leibniz
Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....
. Leibniz's is the notation used most often today. Newton's was simply a dot or dash placed above the function. For example, the derivative of the function x would be written as
. The second derivative of x would be written as , etc. In modern usage, this notation generally denotes derivatives of physical quantities with respect to time, and is used frequently in the science of mechanicsMechanics
Mechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment....
.
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. For example, the derivative of the function x with respect to the variable t in Leibniz's notation would be written as
. 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 SLong s
The long, medial or descending s is a form of the minuscule letter s formerly used where s occurred in the middle or at the beginning of a word, for example "ſinfulneſs" . The modern letterform was called the terminal, round, or short s.-History:The long s is derived from the old Roman cursive...
, representing the Latin
Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...
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.
Euler
Leonhard EulerLeonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
was one of the most prolific mathematicians in history, and perhaps was also the most prolific inventor of canonical notation. His contributions include his use of e
E (mathematical constant)
The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...
to represent the base of natural logarithm
Natural logarithm
The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...
s. It is not known exactly why
was chosen, but it was probably because the first four letters of the alphabet were already commonly used to represent variables and other constants. Euler was also one of the first to use to represent piPi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
consistently. The use of
was first suggested by William Jones, who used it as shorthand for perimeterPerimeter
A perimeter is a path that surrounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length - it can be thought of as the length of the outline of a shape. The perimeter of a circular area is called circumference.- Practical uses :Calculating...
. Euler was also the first to use
to represent the square root of negative one, , although he earlier used it as an infinite number. (Nowadays the symbol created by John Wallis, , is used for infinity.) For summationSummation
Summation is the operation of adding a sequence of numbers; the result is their sum or total. If numbers are added sequentially from left to right, any intermediate result is a partial sum, prefix sum, or running total of the summation. The numbers to be summed may be integers, rational numbers,...
, Euler was the first to use sigma
Sigma
Sigma is the eighteenth letter of the Greek alphabet, and carries the 'S' sound. In the system of Greek numerals it has a value of 200. When used at the end of a word, and the word is not all upper case, the final form is used, e.g...
, Σ, as in
. For functionsFunction (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
, Euler was the first to use the notation
to represent a function of .Peano
In 1895 Giuseppe PeanoGiuseppe Peano
Giuseppe Peano was an Italian mathematician, whose work was of philosophical value. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The standard axiomatization of the natural numbers is named the Peano axioms in...
issued the first edition of his Formulario mathematico
Formulario mathematico
Formulario Mathematico is a book by Giuseppe Peano which expresses fundamental theorems of mathematics in a symbolic language developed by Peano...
, an effort to digest mathematics into terse text based on special symbols. The copy he passed to Bertrand Russell
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
in 1900 at a Paris conference so impressed Russell that he too was taken with the drive to render mathematics more concisely. The result was Principia Mathematica
Principia Mathematica
The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913...
written with Alfred North Whitehead
Alfred North Whitehead
Alfred North Whitehead, OM FRS was an English mathematician who became a philosopher. He wrote on algebra, logic, foundations of mathematics, philosophy of science, physics, metaphysics, and education...
. This treatise marks a watershed in modern literature where symbol became dominant. Peano's Formulario Mathematico, though less popular than Russell's work, continued through five editions. The fifth appeared in 1908 and included 4200 formulas and theorems.
Logic
Once logic was recognized as an important part of mathematics, it received its own notation. Some of the first was the set of symbols used in Boolean algebra, created by George BooleGeorge Boole
George Boole was an English mathematician and philosopher.As the inventor of Boolean logic—the basis of modern digital computer logic—Boole is regarded in hindsight as a founder of the field of computer science. Boole said,...
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
(AND), (OR), and (NOT). With these symbols, and letters to represent different truth values, one can make logical statements such as , 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. Most of these symbols can be found in propositional calculusPropositional calculus
In mathematical logic, a propositional calculus or logic is a formal system in which formulas of a formal language may be interpreted as representing propositions. A system of inference rules and axioms allows certain formulas to be derived, called theorems; which may be interpreted as true...
, a formal system
Formal system
In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive an expression from one or more other premises that are antecedently supposed or derived . The axioms and rules may be called a deductive apparatus...
described as
. is the set of elements, such as the a in the example with Boolean algebra above. is the set that contains the subsets that contain operations, such as or . contains the inference rules, which are the rules dictating how inferences may be logically made, and contains the axioms. (See also: Basic and Derived Argument Forms). With these symbols, proofsMathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
can be made that are completely artificial.
While proving his incompleteness theorems, Kurt Gödel
Kurt Gödel
Kurt Friedrich Gödel was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the...
created an alternative to the symbols normally used in logic. He used Gödel number
Gödel number
In mathematical logic, a Gödel numbering is a function that assigns to each symbol and well-formed formula of some formal language a unique natural number, called its Gödel number. The concept was famously used by Kurt Gödel for the proof of his incompleteness theorems...
s, which were numbers that represented operations with set numbers, and variables with the first 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 first 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. For example, take the statement "There exists a number x such that it is not y". Using the symbols of propositional calculus, this would become
. If the Gödel numbers replace the symbols, it becomes {8, 4, 11, 9, 8, 11, 5, 1, 13, 9}. There are ten numbers, so the first ten prime numbers are found and these are {2, 3, 5, 7, 11, 13, 17, 19, 23, 29}. Then, the Gödel numbers are made the powers of the respective primes and multiplied, giving . The resulting number is approximately .