User:Mpatel/sandbox/History of mathematical notation

The history of mathematical notation is an extensive topic describing the inception and development of symbols used in mathematics throughout recorded history. The contributions of many cultures to mathematics has led to a rich collection of mathematical notation, much of which is still used.

<!-- Mathematical notation comprises the symbols used to write mathematical equations and formulas. It includes Arabic numerals, letters from the Roman, Greek, Hebrew, and German alphabets, and a host of symbols invented by mathematicians over the past several centuries.

(See table of mathematical symbols for a list of symbols.)

Beginning of notation
Written mathematics began with numbers expressed as tally marks, 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 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 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.

Greek notation
The Greeks 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 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 Ionian numeration used the entire alphabet and three archaic letters. This system appeared in the third century B.C., 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 nongeometric subjecs such as number theory), and hence the Greeks had no interest in algebraic 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 $$\Delta^y$$; the cube was $$K^y$$; the fourth power was $$\Delta^y\Delta$$; and the fifth power was $$\Delta K^y$$. The expression $$2x^4+3x^3-4x^2+5x-6$$ would be written as SS2 C3 x5 M S4 u6.

Chinese 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.

Indian notation
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, 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 Arabic numerals
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.

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.

North African notation
Abū al-Hasan ibn Alī al-Qalasādī (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.

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 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. The radical symbol $$\sqrt{\,\,}$$ for square root was introduced by Christoph Rudolff. The symbol for the constant pi, π, was also first used during this time.

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 Francois Viete introduced the modern notational manipulation of algebraic expressions. In 1631 William Oughtred introduced the multiplication sign (x) and abbreviations sin and cos for the trignometric 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. 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, others during Euler's time and almost all after it used Jones's notation.

Calculus
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. For example, the derivative of the function x would be written as $$\dot{x}$$. The second derivative of x would be written as $$\ddot{x}$$, 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 $${ dx \over dt }$$. This notation makes explicit the variable with respect to which the derivative of the function is taken.

Leibniz also created the integral symbol, $$\int_{-N}^{N} f(x)\, dx $$. 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.

Euler
Leonhard Euler 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 logarithms. It is not known exactly why e 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 $$\pi\,\!$$ to represent pi consistently. The use of $$\pi\,\!$$ was first suggested by William Jones, who used it as shorthand for perimeter. Euler was also the first to use i to represent the square root of negative one, $$\sqrt{-1}$$, although he earlier used it as an infinite number. (Nowadays the symbol created by John Wallis, $$\infty$$, is used for infinity.)  For summation, Euler was the first to use sigma, Σ, as in $$\sum_{n=1}^\infty\frac{1} {n^2}$$. For functions, Euler was the first to use the notation $$f(x)$$ to represent a function of x.

Peano
In 1895 Giuseppe Peano 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.

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 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 $$\land$$ (AND), $$\lor$$ (OR), and $$\lnot$$ (NOT). With these symbols, and letters to represent different truth values, one can make logical statements such as $$a\lor\lnot a=1$$, 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. Most of these symbols can be found in propositional calculus, a formal system described as $$\mathcal{L} = \mathcal{L}\ (\Alpha,\ \Omega,\ \Zeta,\ \Iota)$$. $$\Alpha$$ is the set of elements, such as the a in the example with Boolean algebra above. $$\Omega$$ is the set that contains the subsets that contain operations, such as $$\lor$$ or $$\land$$. $$\Zeta$$ contains the inference rules, which are the rules dictating how inferences may be logically made, and $$\Iota$$ 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 numbers, 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 $$(\exists x)(x=\lnot y)$$. 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 $$2^8\times3^4\times5^{11}\times7^9\times11^8\times13^{11}\times17^5\times19^1\times23^{13}\times29^9$$. The resulting number is approximately $$3.096262735\times10^{78}$$. -->

Basic operators
The earliest known use of the equals sign (=) was by Robert Recorde 1557 in The Whetstone of Witte. The equality symbol was slightly longer than that in present use.

The obelus symbol to denote division was first used by Johann Rahn in 1659 in Teutsche Algebra.

The × symbol for multiplication was introduced by William Oughtred in 1631.

Vectors, matrices and tensors
The notation for the scalar and vector products was introduced in Vector Analysis by Josiah Willard Gibbs.

Calculus and analysis
The independent discovery of the calculus by Isaac Newton and Gottfried Wilhelm Leibniz led to dual notations, especially for the derivative. Other calculus notations have developed, giving rise to many that are still used today.

Differentials and derivatives
Leibniz 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 $${ dx \over dt }$$. This notation makes explicit the variable with respect to which the derivative of the function is taken.

Newton used a dot placed above the function. For example, the derivative of the function x would be written as $$\dot{x}$$. The second derivative of x would be written as $$\ddot{x}$$, etc. In modern usage, Newton's notation generally denotes derivatives of physical quantities with respect to time, and is used frequently in mechanics.

Other notations for the derivative include the dash notation used by Joseph Louis Lagrange and the differential operator notation (sometimes called "Euler's notation") introduced by Louis François Antoine Arbogast in De Calcul des dérivations et ses usages dans la théorie des suites et dans le calcul différentiel (1800) and used by Leonhard Euler.

All four notations for derivatives are used today, but Leibniz notation is the most common.

Integrals
Leibniz also created the integral symbol, $$\int_{-N}^{N} e^x\, dx $$. 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 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.

Limits
The symbol $$\lim$$ to denote a limit was used by Karl Weierstrass in 1841. However, the same symbol with a period was first used by Simon L'Huilier in his 1786 essay Exposition élémentaire des principes des calculs superieurs. The notation $$\lim_{x \to x_0}$$ was introduced by G. H. Hardy in A Course of Pure Mathematics (1908).

Vector calculus
In 1773, Joseph Louis Lagrange introduced the component form of both the dot and cross products in order to study the tetrahedron in three dimensions. In 1843 the Irish mathematical physicist Sir William Rowan Hamilton introduced the quaternion product, and with it the terms "vector" and "scalar". Given two quaternions [0, u] and [0, v], where u and v are vectors in R3, their quaternion product can be summarized as [−u·v, u×v]. James Clerk Maxwell used Hamilton's quaternion tools to develop his famous electromagnetism equations, and for this and other reasons quaternions for a time were an essential part of physics education.

However, Oliver Heaviside in England and Josiah Willard Gibbs in Connecticut felt that quaternion methods were too cumbersome, often requiring the scalar or vector part of a result to be extracted. Thus, about forty years after the quaternion product, the dot product and cross product were introduced — to heated opposition. Pivotal to (eventual) acceptance was the efficiency of the new approach, allowing Heaviside to reduce the equations of electromagnetism from Maxwell's original 20 to the four commonly seen today.

Largely independent of this development, and largely unappreciated at the time, Hermann Grassmann created a geometric algebra not tied to dimension two or three, with the exterior product playing a central role. William Kingdon Clifford combined the algebras of Hamilton and Grassmann to produce Clifford algebra, where in the case of three-dimensional vectors the bivector produced from two vectors dualizes to a vector, thus reproducing the cross product.

The cross notation, which began with Gibbs, inspired the name "cross product". Originally appearing in privately published notes for his students in 1881 as Elements of Vector Analysis, Gibbs’s notation — and the name — later reached a wider audience through Vector Analysis (Gibbs/Wilson), a textbook by a former student. Edwin Bidwell Wilson rearranged material from Gibbs's lectures, together with material from publications by Heaviside, Föpps, and Hamilton. He divided vector analysis into three parts:
 * "First, that which concerns addition and the scalar and vector products of vectors. Second, that which concerns the differential and integral calculus in its relations to scalar and vector functions. Third, that which contains the theory of the linear vector function."

Two main kinds of vector multiplications were defined, and they were called as follows: Several kinds of triple products and products of more than three vectors were also examined. The above mentioned triple product expansion was also included.
 * The direct, scalar, or dot product of two vectors
 * The skew, vector, or cross product of two vectors

e, $$\pi$$ and i
The symbol b for the base of natural logarithms was used by Leibniz. However, the symbol e was first used by Euler 1727, the first published use being in Euler's Mechanica (1736).

Sets and classes
A common way of defining sets is through the use of set-builder notation.

Proofs
The latin phrase Q.E.D. was used by Euclid and Archimedes to indicate the end of a proof. More recently, various incarnations of the Halmos symbol are used for the same purpose.