History of mathematical notation
The history of mathematical notation covers the introduction, development, and cultural diffusion of mathematical symbols and the conflicts between notational methods that arise during a notation's move to popularity or obsolescence. Mathematical notation comprises the symbols used to write mathematical equations and formulas. Notation generally implies a set of well-defined representations of quantities and symbols operators. 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.
The historical development of mathematical notation can be divided into three stages:
- Rhetorical stage—where calculations are performed by words and tallies, and no symbols are used.
- Syncopated stage—where frequently used operations and quantities are represented by symbolic syntactical abbreviations, such as letters or numerals. During antiquity and the medieval periods, bursts of mathematical creativity were often followed by centuries of stagnation. As the early modern age opened and the worldwide spread of knowledge began, written examples of mathematical developments came to light.
- Symbolic stage—where comprehensive systems of notation supersede rhetoric. The increasing pace of new mathematical developments, interacting with new scientific discoveries, led to a robust and complete usage of symbols. This began with mathematicians of medieval India and mid-16th century Europe, and continues through the present day.
Rhetorical stage
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. The earliest mathematical notations emerged from these problems.There can be no doubt that most early peoples who left records knew something of numeration and mechanics and that a few were also acquainted with the elements of land-surveying. In particular, the ancient Egyptians paid attention to geometry and numbers, and the ancient Phoenicians performed practical arithmetic, book-keeping, navigation, and land-surveying. The results attained by these people seem to have been accessible to travelers, facilitating dispersal of the methods. It is probable that the knowledge of the Egyptians and Phoenicians was largely the result of observation and measurement, and represented the accumulated experience of many ages. Subsequent studies of mathematics by the Greeks were largely indebted to these previous investigations.
Beginning of notation
Written mathematics began with numbers expressed as tally marks, with each tally representing a single unit. Numerical symbols consisted probably of strokes or notches cut in wood or stone, which were intelligible across cultures. For example, one notch in a bone represented one animal, person, or object. Numerical notation's distinctive feature—symbols having both local and intrinsic values—implies a state of civilization at the period of its invention.The earliest evidence of written mathematics dates back to the ancient Sumerians and the system of metrology from 3000 BC. From around 2500 BC onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises and division problems. The earliest traces of Babylonian numerals also date back to this period. Babylonian mathematics has been reconstructed from more than 400 clay tablets unearthed since the 1850s. Written in cuneiform, these tablets were inscribed whilst the clay was soft and then baked hard in an oven or by the heat of the sun. Some of these appear to be graded homework.
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. The tablets also include multiplication tables and methods for solving linear and quadratic equations. The Babylonian tablet YBC 7289 gives an approximation of that is accurate to an equivalent of six decimal places.
Babylonian mathematics were written using a sexagesimal numeral system. From this derives the modern-day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 degrees in a circle, as well as the use of minutes and seconds of arc to denote fractions of a degree. Babylonian advances in mathematics were facilitated by the fact that 60 has many divisors: the reciprocal of any integer which is a multiple of divisors of 60 has a finite expansion in base 60. Also, unlike the Egyptians, Greeks, and Romans, the Babylonians had a true place-value system, where digits written in the left column represented larger values, much as in the decimal system. They lacked, however, an equivalent of the decimal point, and so the place value of a symbol often had to be inferred from the context.
Initially, the Mesopotamians had symbols for each power of ten. Later, they wrote numbers in almost exactly the same way as in modern times. Instead of using unique symbols for each power of ten, they wrote only the coefficients of each power of ten, with each digit separated by only a space. By the time of Alexander the Great, they had created a symbol that represented zero and was a placeholder.
Rhetorical algebra was first developed by the ancient Babylonians and remained dominant up to the 16th century. In this system, equations are written in full sentences. For example, the rhetorical form of is "The thing plus one equals two" or possibly "The thing plus 1 equals 2".
The ancient Egyptians numerated by hieroglyphics. Egyptian mathematics had symbols 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 Hindu–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 the number 'four' were replaced by a single horizontal line. This is found in the Rhind Mathematical Papyrus and the Moscow 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.
The peoples with whom the Greeks of Asia Minor were likely to have come into frequent contact were those inhabiting the eastern littoral of the Mediterranean; Greek tradition uniformly assigned the special development of geometry to the Egyptians, and the science of numbers to either the Egyptians or the Phoenicians.
Syncopated stage
The history of mathematics cannot with certainty be traced back to any school or period before that of the Ionian Greeks. Still, the subsequent history may be divided into periods, the distinctions between which are tolerably well-marked. Greek mathematics, which originated with the study of geometry, tended to be deductive and scientific from its commencement. Since the fourth century AD, Pythagoras has commonly been given credit for discovering the Pythagorean theorem, a theorem in geometry that states that in a right-angled triangle the area of the square on the hypotenuse is equal to the sum of the areas of the squares of the other two sides. However, this geometric relationship appears in a few earlier ancient mathematical texts, notably Plimpton 322, a Babylonian tablet of mathematics from around 1900 BC. The study of mathematics as a subject in its own right began in the 6th century BC with the Pythagoreans, who coined the term "mathematics" from the ancient Greek mathema, meaning "subject of instruction".Plato's influence was especially strong in mathematics and the sciences. He helped to distinguish between pure and applied mathematics by widening the gap between "arithmetic" and "logistic". Greek mathematics greatly refined the methods and expanded the subject matter of mathematics. Aristotle is credited with what later would be called the law of excluded middle.
Abstract or pure mathematics deals with concepts like magnitude and quantity without regard to any practical application or situation, and includes arithmetic and geometry. In contrast, in mixed or applied mathematics, mathematical properties and relationships are applied to real-world objects to model laws of physics, for example in hydrostatics, optics, and navigation.
Archimedes is generally considered to be the greatest mathematician of antiquity and one of the greatest of all time. 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. He also defined the spiral bearing his name, formulae for the volumes of surfaces of revolution, and an ingenious system for expressing very large numbers.
The ancient Greeks made steps in the abstraction of geometry. Euclid's Elements is the earliest extant documentation of the axioms of plane geometry—though Proclus tells of an earlier axiomatisation by Hippocrates of Chios —and is one of the oldest extant Greek mathematical treatises. Consisting of thirteen books, it collects theorems proven by other mathematicians, supplemented by some original work. The document is a successful collection of definitions, postulates, propositions, and mathematical proofs of the propositions, and covers topics such as Euclidean geometry, geometric algebra, elementary number theory, and the ancient Greek version of algebraic systems. The first theorem given in the text, Euclid's lemma, captures a fundamental property of prime numbers. The text was ubiquitous in the quadrivium and was instrumental in the development of logic, mathematics, and science. Autolycus' On the Moving Sphere is another ancient mathematical manuscript of the time.
The next phase of notation for algebra was syncopated algebra, in which some symbolism is used, but which does not contain all of the characteristics of symbolic algebra. For instance, there may be a restriction that subtraction may be used only once within one side of an equation, which is not the case with symbolic algebra. Syncopated algebraic expression first appeared in a series of books called Arithmetica, by Diophantus of Alexandria, followed by Brahmagupta's Brahma Sphuta Siddhanta.