Number

In mathematics, the notion of a number has been extended over the centuries to include zero (0), negative numbers such as negative one (−1), rational numbers such as one half ${\displaystyle \left(\frac{1}{2}\right)}$, real numbers such as the square root of 2 ${\displaystyle \left(\sqrt{2}\right)}$, and pi (π), and complex numbers which extend the real numbers with a square root of −1 (i), and its combinations with real numbers by adding or subtracting its multiples. Calculations with numbers are done with arithmetical operations, the most familiar being addition, subtraction, multiplication, division, and exponentiation. Their study or usage is called arithmetic, a term which may also refer to number theory, the study of the properties of numbers.

Viewing the concept of zero as a number required a fundamental shift in philosophy, identifying nothingness with a value. During the 19th century, mathematicians began to develop the various systems now called algebraic structures, which share certain properties of numbers, and may be seen as extending the concept. Some algebraic structures are explicitly referred to as numbers (such as the p-adic numbers and hypercomplex numbers) while others are not, but this is more a matter of convention than a mathematical distinction.

History

First use of numbers

Bones and other artifacts have been discovered with marks cut into them that many believe are tally marks. Some historians suggest that the Lebombo bone (dated about 43,000 years ago) and the Ishango bone (dated about 22,000 to 30,000 years ago) are the oldest arithmetic artifacts but this interpretation is disputed. These tally marks may have been used for counting elapsed time, such as numbers of days, lunar cycles or keeping records of quantities, such as of animals. A perceptual system for quantity thought to underlie numeracy, is shared with other species, a phylogenetic distribution suggesting it would have existed before the emergence of language.

A tallying system has no concept of place value (as in modern decimal notation), which limits its representation of large numbers. Nevertheless, tallying systems are considered the first kind of abstract numeral system.

The earliest unambiguous numbers in the archaeological record are the Mesopotamian base 60 (sexagesimal) system (c. 3400 BC); place value emerged in the 3rd millennium BCE. The earliest known base 10 system dates to 3100 BC in Egypt. A Babylonian clay tablet dated to 1900–1600 BC provides an estimate of the circumference of a circle to its diameter of $3\frac{1}{8}$ = 3.125, possibly the oldest approximation of π.

Numerals

Numbers should be distinguished from numerals, the symbols used to represent numbers. The Egyptians invented the first ciphered numeral system, and the Greeks followed by mapping their counting numbers onto Ionian and Doric alphabets. (However, in 300 BC, Archimedes first demonstrated the use of a positional numeral system to display extremely large numbers in The Sand Reckoner.) Roman numerals, a system that used combinations of letters from the Roman alphabet, remained dominant in Europe until the spread of the Hindu–Arabic numeral system around the late 14th century, and the Hindu–Arabic numeral system remains the most common system for representing numbers in the world today. The key to the effectiveness of the system was the symbol for zero, which was developed by ancient Indian mathematicians around 500 AD.

Zero

The first known recorded use of zero as an integer dates to AD 628, and appeared in the Brāhmasphuṭasiddhānta, the main work of the Indian mathematician Brahmagupta. He is usually considered the first to formulate the mathematical concept of zero. Brahmagupta treated 0 as a number and discussed operations involving it, including division by zero. He gave rules of using zero with negative and positive numbers, such as "zero plus a positive number is a positive number, and a negative number plus zero is the negative number". By this time (the 7th century), the concept had clearly reached Cambodia in the form of Khmer numerals, and documentation shows the idea later spreading to China and the Islamic world. The concept began reaching Europe through Islamic sources around the year 1000.

There are other uses of zero before Brahmagupta, though the documentation is not as complete as it is in the Brāhmasphuṭasiddhānta. The earliest uses of zero was as simply a placeholder numeral in place-value systems, representing another number as was done by the Babylonians. Many ancient texts used 0, including Babylonian and Egyptian texts. Egyptians used the word nfr to denote zero balance in double entry accounting. Indian texts used a Sanskrit word Shunye or shunya to refer to the concept of void. In mathematics texts this word often refers to the number zero. In a similar vein, Pāṇini (5th century BC) used the null (zero) operator in the Ashtadhyayi, an early example of an algebraic grammar for the Sanskrit language (also see Pingala).

Records show that the Ancient Greeks seemed unsure about the status of 0 as a number: they asked themselves "How can 'nothing' be something?" leading to interesting philosophical and, by the Medieval period, religious arguments about the nature and existence of 0 and the vacuum. The paradoxes of Zeno of Elea depend in part on the uncertain interpretation of 0. (The ancient Greeks even questioned whether 1 was a number.)

The late Olmec people of south-central Mexico began to use a placeholder symbol for zero, a shell glyph, in the New World, by 38 BC. It would be the Maya who developed zero as a cardinal number, employing it in their numeral system and in the Maya calendar. Maya used a base 20 numerical system by combining a number of dots (base 5) with a number of bars (base 4). George I. Sánchez in 1961 reported a base 4, base 5 "finger" abacus.

By 130 AD, Ptolemy, influenced by Hipparchus and the Babylonians, was using a symbol for 0 (a small circle with a long overbar) within a sexagesimal numeral system otherwise using alphabetic Greek numerals. Because it was used alone, not as just a placeholder, this Hellenistic zero was the first documented use of a true zero in the Old World. In later Byzantine manuscripts of his Syntaxis Mathematica (Almagest), the Hellenistic zero had morphed into the Greek letter Omicron (otherwise meaning 70 in isopsephy).

A true zero was used in tables alongside Roman numerals by 525 (first known use by Dionysius Exiguus), but as a word, nulla meaning nothing, not as a symbol. When division produced 0 as a remainder, nihil, also meaning nothing, was used. These medieval zeros were used by all future medieval computists (calculators of Easter). An isolated use of their initial, N, was used in a table of Roman numerals by Bede or a colleague about 725, a true zero symbol.

Negative numbers

The abstract concept of negative numbers was recognized as early as 100–50 BC in China. The Nine Chapters on the Mathematical Art contains methods for finding the areas of figures; red rods were used to denote positive coefficients, black for negative. The first reference in a Western work was in the 3rd century AD in Greece. Diophantus referred to the equation equivalent to 4x + 20 = 0 (the solution is negative) in Arithmetica, saying that the equation gave an absurd result.

During the 600s, negative numbers were in use in India to represent debts. Diophantus' previous reference was discussed more explicitly by Indian mathematician Brahmagupta, in Brāhmasphuṭasiddhānta in 628, who used negative numbers to produce the general form quadratic formula that remains in use today. However, in the 12th century in India, Bhaskara gives negative roots for quadratic equations but says the negative value "is in this case not to be taken, for it is inadequate; people do not approve of negative roots".

European mathematicians, for the most part, resisted the concept of negative numbers until the 17th century, although Fibonacci allowed negative solutions in financial problems where they could be interpreted as debts (chapter 13 of Liber Abaci, 1202) and later as losses (in Flos). René Descartes called them false roots as they cropped up in algebraic polynomials yet he found a way to swap true roots and false roots as well. At the same time, the Chinese were indicating negative numbers by drawing a diagonal stroke through the right-most non-zero digit of the corresponding positive number's numeral. An early European experimenter with negative numbers was Nicolas Chuquet during the 15th century. He used them as exponents, but referred to them as "absurd numbers".

As recently as the 18th century, it was common practice to ignore any negative results returned by equations on the assumption that they were meaningless.

Rational numbers

It is likely that the concept of fractional numbers dates to prehistoric times. The Ancient Egyptians used their Egyptian fraction notation for rational numbers in mathematical texts such as the Rhind Mathematical Papyrus and the Kahun Papyrus. The Rhind Papyrus includes an example of deriving the area of a circle from its diameter, which yields an estimate of π as $(\frac{16}{9}{)}^2$ ≈ 3.16049.... Classical Greek and Indian mathematicians made studies of the theory of rational numbers, as part of the general study of number theory. A particularly influential example of these is Euclid's Elements, dating to roughly 300 BC. Of the Indian texts, the most relevant is the Sthananga Sutra, which also covers number theory as part of a general study of mathematics.

The concept of decimal fractions is closely linked with decimal place-value notation; the two seem to have developed in tandem. For example, it is common for the Jain math sutra to include calculations of decimal-fraction approximations to pi or the square root of 2. Similarly, Babylonian math texts used sexagesimal (base 60) fractions.

Real numbers and irrational numbers

The Babylonians, as early as 1800 BCE, demonstrated numerical approximations of irrational quantities such as √2 on clay tablets, with an accuracy analogous to six decimal places, as in the tablet YBC 7289. These values were primarily used for practical calculations in geometry and land measurement. There were practical approximations of irrational numbers in the Indian Shulba Sutras composed between 800 and 500 BC.

The first existence proofs of irrational numbers is usually attributed to Pythagoras, more specifically to the Pythagorean Hippasus, who produced a (most likely geometrical) proof of the irrationality of the square root of 2. The story goes that Hippasus discovered irrational numbers when trying to represent the square root of 2 as a fraction. However, Pythagoras believed in the absoluteness of numbers. He could not disprove the existence of irrational numbers, or accept them, so according to legend, he sentenced Hippasus to death by drowning, to impede the spread of this unsettling news.

The 16th century brought final European acceptance of negative integers and fractional numbers. By the 17th century, mathematicians generally used decimal fractions with modern notation. The concept of real numbers was introduced in the 17th century by René Descartes. While studying compound interest, in 1683 Jacob Bernoulli found that as the compounding intervals grew ever shorter, the rate of exponential growth converged to a base of 2.71828...; this key mathematical constant would later be named Euler's number (e). Irrational numbers began to be studied systematically in the 18th century, with Leonhard Euler who proved that the irrational numbers are those numbers whose simple continued fractions is not finite and that Euler's number (e) is irrational. The irrationality of π was proved in 1761 by Johann Lambert.

It is in the second half of the 19th century that real numbers, and thus irrational numbers, were rigorously defined, with the work of Augustin-Louis Cauchy, Charles Méray (1869), Karl Weierstrass (1872), Eduard Heine (1872), Georg Cantor (1883), and Richard Dedekind (1872).

Transcendental numbers and reals

A transcendental number is a numerical value that is not the root of a polynomial with integer coefficients. This means it is not algebraic and thus excludes all rational numbers. The existence of transcendental numbers was first established by Liouville (1844, 1851). Hermite proved in 1873 that e is transcendental and Lindemann proved in 1882 that π is transcendental. Finally, Cantor showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite, so there is an uncountably infinite number of transcendental numbers.

Infinity and infinitesimals

In mathematics, infinity is considered an abstract concept rather than a number; instead of being "greater than any number", infinite is the property of having no end. The earliest known conception of mathematical infinity appears in the Yajurveda, an ancient Indian script, which at one point states, "If [the whole] was subtract from [the whole], the leftover will still be [the whole]". Infinity was a popular topic of philosophical study among the Jain mathematicians c. 400 BC. They distinguished between five types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually.

Aristotle defined the traditional Western notion of mathematical infinity. He distinguished between actual infinity and potential infinity—the general consensus being that only the latter had true value. Galileo Galilei's Two New Sciences discussed the idea of one-to-one correspondences between infinite sets, known as Galileo's paradox. The next major advance in the theory was made by Georg Cantor; in 1895 he published a book about his new set theory, introducing, among other things, transfinite numbers and formulating the continuum hypothesis. The symbol ${\displaystyle \text{∞}}$, often used to represent an infinite quantity, was first introduced in a mathematical context by John Wallis in 1655.

In the 1960s, Abraham Robinson showed how infinitely large and infinitesimal numbers can be rigorously defined and used to develop the field of nonstandard analysis. The system of hyperreal numbers represents a rigorous method of treating the ideas about infinite and infinitesimal numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of infinitesimal calculus by Newton and Leibniz.

A modern geometrical version of infinity is given by projective geometry, which introduces "ideal points at infinity", one for each spatial direction. Each family of parallel lines in a given direction is postulated to converge to the corresponding ideal point. This is closely related to the idea of vanishing points in perspective drawing.

Complex numbers

The earliest fleeting reference to square roots of negative numbers occurred in the work of the mathematician and inventor Heron of Alexandria in the 1st century AD, when he considered the volume of an impossible frustum of a pyramid. They became more prominent when in the 16th century closed formulas for the roots of third and fourth degree polynomials were discovered by Italian mathematicians such as Niccolò Fontana Tartaglia and Gerolamo Cardano. It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers.

This was doubly unsettling since they did not even consider negative numbers to be on firm ground at the time. René Descartes is sometimes credited with coining the term "imaginary" for these quantities in 1637, intending it as derogatory. (See imaginary number for a discussion of the "reality" of complex numbers.) A further source of confusion was that the equation $${\displaystyle {\left(\sqrt{-1}\right)}^2=\sqrt{-1}\sqrt{-1}=-1}$$ seemed capriciously inconsistent with the algebraic identity $${\displaystyle \sqrt{a}\sqrt{b}=\sqrt{ab},}$$ which is valid for positive real numbers a and b, and was also used in complex number calculations with one of a, b positive and the other negative. The incorrect use of this identity, and the related identity $${\displaystyle \frac{1}{\sqrt{a}}=\sqrt{\frac{1}{a}}}$$ in the case when both a and b are negative even bedeviled Euler. This difficulty eventually led him to the convention of using the special symbol i in place of ${\displaystyle \sqrt{-1}}$ to guard against this mistake.

The 18th century saw the work of Abraham de Moivre and Leonhard Euler. De Moivre's formula (1730) states: $${\displaystyle (\cos \theta +i\sin \theta {)}^n=\cos n\theta +i\sin n\theta }$$ while Euler's formula of complex analysis (1748) gave us: $${\displaystyle \cos \theta +i\sin \theta =e^{i\theta }.}$$ A special case of this formula yields Euler's identity: $${\displaystyle e^{i\pi }+1=0}$$ showing a profound connection between the most fundamental numbers in mathematics.

The existence of complex numbers was not completely accepted until Caspar Wessel described the geometrical interpretation in 1799. Carl Friedrich Gauss rediscovered and popularized it several years later, and as a result the theory of complex numbers received a notable expansion. However, the idea of the graphic representation of complex numbers had appeared as early as 1685, in Wallis's De algebra tractatus.

In the same year, Gauss provided the first generally accepted proof of the fundamental theorem of algebra, showing that every polynomial over the complex numbers has a full set of solutions in that realm. Gauss studied complex numbers of the form a + bi, where a and b are integers (now called Gaussian integers) or rational numbers. His student, Gotthold Eisenstein, studied the type a + bω, where ω is a complex root of x³ − 1 = 0 (now called Eisenstein integers). Other such classes (called cyclotomic fields) of complex numbers derive from the roots of unity x^k − 1 = 0 for higher values of k. This generalization is largely due to Ernst Kummer, who also invented ideal numbers, which were expressed as geometrical entities by Felix Klein in 1893.

In 1850 Victor Alexandre Puiseux took the key step of distinguishing between poles and branch points, and introduced the concept of essential singular points. This eventually led to the concept of the extended complex plane.

Prime numbers

Prime numbers may have been studied throughout recorded history. They are natural numbers that are not a product of two smaller natural numbers. It has been suggested that the Ishango bone includes a list of the prime numbers between 10 and 20. The Rhind papyrus display different forms for prime numbers. But the formal study of prime numbers is first documented by the ancient Greek. Euclid devoted one book of the Elements to the theory of primes; in it he proved the infinitude of the primes and the fundamental theorem of arithmetic, and presented the Euclidean algorithm for finding the greatest common divisor of two numbers.

In 240 BC, Eratosthenes used the Sieve of Eratosthenes to quickly isolate prime numbers. But most further development of the theory of primes in Europe dates to the Renaissance and later eras. At around 1000 AD, Ibn al-Haytham discovered Wilson's theorem. Ibn al-Banna' al-Marrakushi found a way to speed up the Sieve of Eratosthenes by only testing up to the square root of the number. Fibonacci communicated Islamic mathematical contributions to Europe, and in 1202 was the first to describe the method of trial division.

In 1796, Adrien-Marie Legendre conjectured the prime number theorem, describing the asymptotic distribution of primes. Other results concerning the distribution of the primes include Euler's proof that the sum of the reciprocals of the primes diverges, and the Goldbach conjecture, which claims that any sufficiently large even number is the sum of two primes. Yet another conjecture related to the distribution of prime numbers is the Riemann hypothesis, formulated by Bernhard Riemann in 1859. The prime number theorem was finally proved by Jacques Hadamard and Charles de la Vallée-Poussin in 1896. Goldbach and Riemann's conjectures remain unproven and unrefuted.

Cultural and symbolic significance

Numbers have held cultural, symbolic and religious significance throughout history and in many cultures. In Ancient Greece, number symbolism heavily influenced the development of Greek mathematics, stimulating the investigation of many problems in number theory which are still of interest today. According to Plato, Pythagoreans attributed specific characteristics and meaning to particular numbers, and believed that "things themselves are numbers".

Folktales in different cultures exhibit preferences for particular numbers, with three and seven holding special significance in European culture, while four and five are more prominent in Chinese folktales. Numbers are sometimes associated with luck: in Western society, the number 13 is considered unlucky while in Chinese culture the number eight is considered auspicious.