History of mathematical notation

[3] The history includes Hindu–Arabic numerals, letters from the Roman, Greek, Hebrew, and German alphabets, and a variety of symbols invented by mathematicians over the past several centuries.

Many areas of mathematics began with the study of real world problems, before the underlying rules and concepts were identified and defined as abstract structures.

For example, geometry has its origins in the calculation of distances and areas in the real world; algebra started with methods of solving problems in arithmetic.

[citation needed] The majority of Mesopotamian clay tablets date from 1800 to 1600 BC, and cover topics which include fractions, algebra, quadratic and cubic equations, and the calculation of regular numbers, reciprocals, and pairs.

Abstract or pure mathematics[17] deals with concepts like magnitude and quantity without regard to any practical application or situation, and includes arithmetic and geometry.

[18][19] He used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave a remarkably accurate approximation of pi.

As in other early societies, the purpose of astronomy was to perfect the agricultural calendar and other practical tasks, not to establish a formal system; thus, the duties of the Chinese Board of Mathematics were confined to the annual preparation of the dates and predictions of the almanac.

Early Chinese mathematical inventions include a place value system known as counting rods[29][30] (which emerged during the Warring States period), certain geometrical theorems (such as the ratio of sides), and the suanpan (abacus) for performing arithmetic calculations.

[32] Shen's work on arc lengths provided the basis for spherical trigonometry developed in the 13th century by the mathematician and astronomer Guo Shoujing.

[33] As the historians L. Gauchet and Joseph Needham state, Guo Shoujing used spherical trigonometry in his calculations to improve the calendar system and Chinese astronomy.

The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, likely evolved over the course of the first millennium AD in India and was transmitted to the west via Islamic mathematics.

[51] In Summa de arithmetica, geometria, proportioni e proportionalità,[52] Luca Pacioli used plus and minus symbols and algebra, though much of the work originated from Piero della Francesca whom he appropriated and purloined.

Rafael Bombelli published his L'Algebra (1572) in which he showed how to deal with the imaginary quantities that could appear in Cardano's formula for solving cubic equations.

Simon Stevin's book De Thiende ("The Art of Tenths"), published in Dutch in 1585, contained a systematic treatment of decimal notation, which influenced all later work on the real number system.

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.

But in our opinion truths of this kind should be drawn from notions rather than from notations.At the turn of the 19th century, Carl Friedrich Gauss developed the identity sign for congruence relation and, in quadratic reciprocity, the integral part.

In 1829, Carl Gustav Jacob Jacobi published Fundamenta nova theoriae functionum ellipticarum with his elliptic theta functions.

Matrix notation would be more fully developed by Arthur Cayley in his three papers, on subjects which had been suggested by reading the Mécanique analytique[66] of Lagrange and some of the works of Laplace.

[79] In 1895 Giuseppe Peano issued his Formulario mathematico,[80] an effort to digest mathematics into terse text based on special symbols.

Bertrand Russell[94] once said, "Ordinary language is totally unsuited for expressing what physics really asserts, since the words of everyday life are not sufficiently abstract.

The first formulation of a quantum theory describing radiation and matter interaction is due to Paul Adrien Maurice Dirac, who, during 1920, was first able to compute the coefficient of spontaneous emission of an atom.

Dirac described the quantification of the electromagnetic field as an ensemble of harmonic oscillators with the introduction of the concept of creation and annihilation operators of particles.

In the following years, with contributions from Wolfgang Pauli, Eugene Wigner, Pascual Jordan, and Werner Heisenberg, and an elegant formulation of quantum electrodynamics due to Enrico Fermi,[100] physicists came to believe that, in principle, it would be possible to perform any computation for any physical process involving photons and charged particles.

Studies by Felix Bloch with Arnold Nordsieck,[101] and Victor Weisskopf,[102] in 1937 and 1939, revealed that such computations were reliable only at a first order of perturbation theory, a problem already pointed out by Robert Oppenheimer.

[103] Infinities emerged at higher orders in the series, making such computations meaningless and casting serious doubts on the internal consistency of the theory itself.

The notation establishes an encoded abstract representation-independence, producing a versatile specific representation (e.g., x, or p, or eigenfunction base) without much ado, or excessive reliance on, the nature of the linear spaces involved.

Later, multi-index notation eliminates conventional notions used in multivariable calculus, partial differential equations, and the theory of distributions, by abstracting the concept of an integer index to an ordered tuple of indices.

The orbifold notation system, invented by William Thurston, has been developed for representing types of symmetry groups in two-dimensional spaces of constant curvature.

Combinatorial LCF notation, devised by Joshua Lederberg and extended by Harold Scott MacDonald Coxeter and Robert Frucht, was developed for the representation of cubic graphs that are Hamiltonian.