Mathematic notation comprises the symbols used in expressing mathematical expressions, equations, and formulas. It is normally made up of Arabic numerals, Roman alphabet letters, and letters from the Greek alphabet. Numerals sans-serif Arabic numerals, known formally as Hindu-Arabic numerals, and also known as Indian numerals, Hindu numerals, European numerals, and Western numerals, are the most common symbolic representation of numbers around the world. ... The Latin alphabet, also called the Roman alphabet, is the most widely used alphabetic writing system in the world today. ... Due to technical limitations, some web browsers may not display some special characters in this article. ...

Beginning of notation

When mathematics was first recorded in writing, numbers were normally expressed as tally marks. Each tally represented a single unit, making it easy to record. One notch in a bone represented one animal, or person, or anything else. The first notation that made use of several symbols 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 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 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. A sign warning hikers on the trail to Hanakapiai Beach. ... Development of hieratic script from hieroglyphs; after Champollion. ... Hieroglyphs are a system of writing used by the Ancient Egyptians, using a combination of logographic, syllabic, and alphabetic elements. ... The Moscow and Rhind Mathematical Papyri are two of the oldest mathematical texts discovered. ...

Like the Egyptians, the Mesopotamians had symbols for each power of ten. Later, they wrote their numbers in almost the exact 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. This is an article about the ancient middle eastern region. ... In mathematics, a coefficient is a multiplicative factor of a certain object such as a variable (for example, the coefficients of a polynomial), a basis vector, a basis function and so on. ... Alexander the Great (Greek: [1], Megas Alexandros; July 356 BC — June 11, 323 BC), also known as Alexander III, king of Macedon (336–323 BC), was one of the most successful military commanders in history, conquering most of the known world before his death; he is frequently included in a... The sexagesimal (base-sixty) is a numeral system with sixty as the base. ...

See also: Moscow and Rhind Mathematical Papyri The Moscow and Rhind Mathematical Papyri are two of the oldest mathematical texts discovered. ...

Greek notation

The Greeks changed the notation they used. The first was the 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 pente, which was the first letter of the word for five. 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 Roman Forum was the central area around which ancient Rome developed. ...

The Ionian numeration used the entire alphabet and three archaic letters.

This system appeared in the third century B.C., before the letters vau (F), koppa, and sampi 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^[1] Look up A, a in Wiktionary, the free dictionary. ... The letter B is the second letter of the modern Latin alphabet. ... Ge or He (Г, г) is a letter of the Cyrillic alphabet, pronounced differently in different languages. ... Delta (upper case Δ, lower case δ) is the fourth letter of the Greek alphabet. ... The letter E is the fifth letter in the Latin alphabet. ... The letter F is the sixth letter in the Latin alphabet. ... Note: This page contains IPA phonetic symbols in Unicode. ... H is also a multi a-side single by Japanese singer Ayumi Hamasaki. ... Note: A theta probe is a device for measuring soil moisture. ... The lowercase i redirects here. ... The letter K is the eleventh letter in the Latin alphabet. ... Lambda (upper case Λ, lower case λ) is the 11th letter of the Greek alphabet. ... The letter M is the thirteenth letter in the Latin alphabet. ... This article is about the letter N. For the Flash game, see N (game). ... Xi (upper case Ξ, lower case ξ) is the 14th letter of the Greek alphabet. ... This is for the letter O. For Oxygen, see here. ... Lower-case pi The mathematical constant Ï€ is a real number which may be defined as the ratio of a circles circumference (Greek περιφέρεια, periphery) to its diameter in Euclidean geometry, and which is in common use in mathematics, physics, and engineering. ... The letter koppa in the Early Cyrillic alphabet Koppa (?, ?) is an archaic letter of the Cyrillic alphabet, originally derived from the Greek letter Qoppa. ... The letter P is the sixteenth letter in the Latin alphabet. ... For other uses, see Sigma (disambiguation). ... For other uses, see T (disambiguation). ... Upsilon (upper case , lower case ) is the 20th letter of the Greek alphabet. ... Ef (Ф, Ñ„) is the twenty-first letter of the Cyrillic alphabet. ... The letter X is the twenty-fourth letter in the Latin alphabet. ... For other uses, see Psi. ... Omega (Ω ω) is the 24th and last letter of the Greek alphabet. ... Sampi (Upper case Ϡ, lower case ϡ) is an obsolete letter of the Greek alphabet and has a numeric value of 900. ...

The Greeks were more interested in geometry and had little need for algebraic symbols. Diophantus of Alexandria was one of the first and most famous of Greek 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 Δ^y; the cube was K^y; the fourth power was Δ^yΔ; and the fifth power was ΔK^y. The expression 2x⁴ + 3x³ − 4x² + 5x − 6 would be written as SS2 C3 x5 M S4 u6. Table of Geometry, from the 1728 Cyclopaedia. ... Algebra is a branch of mathematics concerning the study of structure, relation and quantity. ... Cover of the 1621 edition of Diophantus Arithmetica, translated into Latin by Claude Gaspard Bachet de Méziriac. ... Alexandria Modern Alexandria, from Qaitbays Citadel Alexandria, sphinx made of pink granite, Ptolemaic. ... Arithmetica, an ancient text on mathematics written by classical period Greek mathematician Diophantus in the second century AD is a collection of 130 algebra problems giving numerical solutions of determinate equations (those with a unique solution), and indeterminate equations. ...

Chinese notation

Main article: Chinese_numbers#Suzhou (蘇州) or huāmǎ (花碼) numerals

The numbers 0-9 in Chinese huāmǎ (花碼) numerals

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. Today, speakers of Chinese use three numeral systems: There is the ubiquitous system of Arabic digits and two ancient Chinese numeral systems. ... The system of Roman numerals is a numeral system originating in ancient Rome, and was adapted from Etruscan numerals. ...

Beginning of Arabic numerals

Main article: History of the Hindu-Arabic numeral system

Despite their name, arabic numerals actually started in India. The reason for this misnomer 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. To meet Wikipedias quality standards, this article or section may require cleanup. ... Look up Misnomer in Wiktionary, the free dictionary. ... Soviet postage stamp commemorating the 1200th anniversary of Muhammad al‑Khwarizmi in 1983. ...

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. Liber Abaci (1202) is an historic book on arithmetic by Leonardo of Pisa, known later by his nickname Fibonacci. ... Leonardo of Pisa (c. ... In mathematics, the Fibonacci numbers form a sequence defined recursively by: In words: you start with 0 and 1, and then produce the next Fibonacci number by adding the two previous Fibonacci numbers. ...

Pre-calculus

Two of the most widely used mathematical symbols are addition and subtraction, + and −. The plus sign was first used by Nicole d'Oresme in Algorismus proportionum, possibly an abbriviation for "et", which is "and" in Latin (in much the same way the ampersand 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.^[2] The symbol for the constant pi, π, was also first used during this time. 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. Ironically, the mathematician Leonhard Euler (who would begin much of his own notation that is used today) did not use π but its equivalent in the Roman alphabet, p. However, other during Euler's time and many after it used Jones's notation. The plus (+) and minus (−) signs are used universally to represent the operations of addition and subtraction, and have been extended to many other meanings, more or less analogous. ... Portrait of Nicole Oresme: Miniature of Nicole Oresmes Traité de l’espere, Bibliothèque Nationale, Paris, France, fonds français 565, fol. ... The roman ampersand at left is stylised, but the italic one at right reveals its origin in the Latin word An ampersand (&, &, &), also commonly called an and sign, is a logogram representing the conjunction and. The symbol is a ligature of the letters in et, which is Latin for and... The plus (+) and minus (−) signs are used universally to represent the operations of addition and subtraction, and have been extended to many other meanings, more or less analogous. ... Johannes Widmann, a German mathematician, invented the addition (+) and the subtraction (-) signs in 1489. ... William Jones is a common name, especially in Wales, and there have been several well-known individuals of this name, including: // Academics and authors William Jones (historian) (1860–1932) Sir William Jones (mathematician) (~1675–1749), father of Sir William Jones (philologist) Sir William Jones (philologist) (1746–1794) son of Sir... Events March 27 - Concluding that Emperor Iyasus I of Ethiopia had abdicated by retiring to a monastery, a council of high officials appoint Tekle Haymanot I Emperor of Ethiopia May 23 - Battle of Ramillies September 7 - The Battle of Turin in the War of Spanish Succession - forces of Austria and... The perimeter is the distance around a given two-dimensional object. ...

Calculus

Calculus had two main systems of notation, each created by one of the creators: that developed by Issac 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. 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. Sir Isaac Newton in Knellers portrait of 1689. ... This article is 82 kilobytes or more in size. ... Mechanics (Greek ) is the branch of physics concerned with the behaviour of physical bodies when subjected to forces or displacements, and the subsequent effect 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 dervatives 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, likely to represent sum. When finding areas under curves, integration is often illustrated by dividing the area into tall, thin rectangles. Infintesimally thin rectangles, when added, yield the area. The process of add up the infintesmal areas in integration, hence the S for sum.

See also: Newton's notation, Leibniz's notation In mathematics, the derivative is defined as the instantaneous rate of change of a function. ... In mathematics, the derivative is defined as the instantaneous rate of change of a function. ...

Euler

Leonhard Euler was one of the most prolific mathematicians in history, and contributed to notation more than any else. His contributions include his use of e to indicate the base of natural logarithms. It is not known exactly why e was chosen, but was probably because the first four letters were already commonly used to represent variables and other constants. He was also one of the first to use π to represent pi consistently. The use of π was first suggested by William Jones, who used it as shorthand for perimeter. He was also the first to use i 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 . And Euler was the first to use the notation that is probably used more than anything else, f(x), to represent the function of x. 1937 portrait of Leonhard Euler by Johann Georg Brucker. ... The letter E is the fifth letter in the Latin alphabet. ... The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is equal to 2. ... William Jones is a common name, especially in Wales, and there have been several well-known individuals of this name, including: // Academics and authors William Jones (historian) (1860–1932) Sir William Jones (mathematician) (~1675–1749), father of Sir William Jones (philologist) Sir William Jones (philologist) (1746–1794) son of Sir... The perimeter is the distance around a given two-dimensional object. ... John Wallis John Wallis (November 22, 1616 - October 28, 1703) was an English mathematician who is given partial credit for the development of modern calculus. ... For other uses, see Sigma (disambiguation). ... Look up Function in Wiktionary, the free dictionary. ...

Logic

When 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 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 (AND), (OR), and (NOT). With these symbols, and letters to represent different elements, one can make logical statements such as , that is "The existence of element a OR the existence of element NOT a is 1", meaning it is true either a exists or it doesn't. 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 . Α is the set of elements, such as the a in the example with Boolean algebra above. Ω is the set that contains the subsets that contain operations, such as or . Ζ contains the inference rules, which are the rules dictating how inferences may be logically made, and Ι contains the axioms. (See also: Basic and Derived Argument Forms). With these symbols, proofs can be made that are completely artificial. In abstract algebra, a Boolean algebra is an algebraic structure (a collection of elements and operations on them obeying defining axioms) that captures essential properties of both set operations and logic operations. ... George Boole [], (November 2, 1815 – December 8, 1864) was a mathematician and philosopher. ... 1854 (MDCCCLIV) was a common year starting on Sunday (see link for calendar). ... In mathematical logic the propositional calculus or sentential calculus is a formal deduction system whose atomic formulas are propositional variables. ... In logic, mathematics, and computer science, a formal system is a formal grammar used for modelling purposes. ... ... For the algebra software named Axiom, see Axiom computer algebra system. ... In mathematical logic the propositional calculus or sentential calculus is a formal deduction system whose atomic formulas are propositional variables. ... The word proof can mean: originally, a test assessing the validity or quality of something. ...

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 first prime numbers greater than 10. (See a table of the numbers here). 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.^[3] For example, take the statement "There is exists a number x so 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 . In mathematical logic, Gödels incompleteness theorems are two celebrated theorems proven by Kurt Gödel in 1931. ... Kurt Gödel Kurt Gödel (IPA: ) (April 28, 1906 Brno, then Austria-Hungary, now Czech Republic – January 14, 1978 Princeton, New Jersey) was a logician, mathematician, and philosopher of mathematics. ... In formal number theory a Gödel numbering is a function which assigns to each symbol and formula of some formal language a unique natural number called a Gödel number (GN). ... In formal number theory a Gödel numbering is a function which assigns to each symbol and formula of some formal language a unique natural number called a Gödel number (GN). ...

References

1. ^ Boyer, Carl B. A History of Mathematics, 2nd edition, John Wiley & Sons, Inc., 1991.
2. ^ Miller, Jeff. "Earliest Uses of Various Mathematical Symbols." 04 June 2006. Gulf High School. 24 Sep 2006 <http://members.aol.com/jeff570/mathsym.html>.
3. ^ Casti, John L. 5 Golden Rules. New York: MJF Books , 1996.

External links

• Mathematical Notation: Past and Future
• History of Mathematical Notation
• Earliest Uses of Mathematical Notation