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.

Introduction

Numerals

Algebra

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.^[1]

Indices and roots

Abstract algebra

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,^[2] 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).

Analysis

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.^[3] 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:

The direct, scalar, or dot product of two vectors
The skew, vector, or cross product of two vectors

Several kinds of triple products and products of more than three vectors were also examined. The above mentioned triple product expansion was also included.

Special numbers

Zero

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).

Geometry and topology

Differential geometry and tensor calculus

Logic and set theory

Propositional calculus

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.

Category theory

Probability and Statistics

Other

Notes

^ Florian Cajori (1919). A History of Mathematics. Macmillan.
^ "Earliest Uses of Symbols of Calculus". 2004-12-01. Retrieved October 22, 2008. {{cite web}}: Cite has empty unknown parameters: |month= and |coauthors= (help)
^ Lagrange, JL (1773). "Solutions analytiques de quelques problèmes sur les pyramides triangulaires". Oeuvres. Vol. vol 3. {{cite book}}: |volume= has extra text (help)

References

Florian Cajori (1929) A History of Mathematical Notations, 2 vols. Dover reprint in 1 vol., 1993. ISBN 0486677664.

External links

Category:History of mathematics *

[1] Florian Cajori (1919). A History of Mathematics. Macmillan.

[2] "Earliest Uses of Symbols of Calculus". 2004-12-01. Retrieved October 22, 2008. {{cite web}}: Cite has empty unknown parameters: |month= and |coauthors= (help)

[3] Lagrange, JL (1773). "Solutions analytiques de quelques problèmes sur les pyramides triangulaires". Oeuvres. Vol. vol 3. {{cite book}}: |volume= has extra text (help)

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