User:Mpatel/sandbox/History of mathematical notation

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.

Indices and roots

Abstract algebra

Vectors and matrices

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,^[1] 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

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

Logic and set theory

Propositional calculus

Sets and classes

Proofs

Notes

^ "Earliest Uses of Symbols of Calculus". 2004-12-01. {{cite web}}: Cite has empty unknown parameters: |month= and |coauthors= (help); Unknown parameter |accessmonthday= ignored (help); Unknown parameter |accessyear= ignored (|access-date= suggested) (help)

References

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

External links

[1] "Earliest Uses of Symbols of Calculus". 2004-12-01. {{cite web}}: Cite has empty unknown parameters: |month= and |coauthors= (help); Unknown parameter |accessmonthday= ignored (help); Unknown parameter |accessyear= ignored (|access-date= suggested) (help)

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