Number

• 1. Introduction

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  A number is a mathematical object used to count, measure, and label. That sentence sounds simple, yet behind it sits one of humanity's longest arguments. Consider the word "eleven" and the numeral "11". One is a number word, the other is a symbol that stands for it, and in everyday speech almost no one keeps them apart. Now consider a notched bone, dated to roughly 43,000 years ago, scratched by someone counting something we will never know. Between that bone and the symbol for zero lies a story of ideas that people resisted, feared, and sometimes refused outright to accept. Who first treated nothing as a value? Why did a man supposedly drown for proving that the square root of 2 cannot be written as a fraction? How did debts, impossible pyramids, and a circle's edge force whole new kinds of numbers into existence? This is the story of how the idea of number kept being stretched, century after century, until it held things that the people who started it would never have called numbers at all.

• 2. Marks On Bone

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  Bones and other artifacts survive with marks cut into them that many researchers read as tally marks. The Lebombo bone, dated about 43,000 years ago, and the Ishango bone, dated about 22,000 to 30,000 years ago, are sometimes called the oldest arithmetic artifacts, though that reading is disputed. These notches may have tracked elapsed time, such as days or lunar cycles, or kept records of quantities like animals. A tallying system has no concept of place value, which limits how large a number it can express. Even so, tallying counts as the first kind of abstract numeral system. The earliest unambiguous numbers in the archaeological record come from Mesopotamia, a base 60 system dated to about 3400 BC, with place value emerging in the 3rd millennium BCE. The earliest known base 10 system dates to 3100 BC in Egypt. A Babylonian clay tablet dated to 1900 estimates the ratio of a circle's circumference to its diameter as 3.125, possibly the oldest approximation of pi. The instinct to count may run deeper than any of these marks, since a perceptual sense for quantity is shared with other species and likely predates language itself.

• 3. The Argument Over Nothing

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  Viewing zero as a number required a fundamental shift in philosophy, identifying nothingness with a value. The Ancient Greeks seemed unsure of its status, asking themselves how nothing could be something, and the paradoxes of Zeno of Elea depend in part on that uncertainty. The first known recorded use of zero as an integer dates to AD 628, in the Brahmasphutasiddhanta, the main work of the Indian mathematician Brahmagupta. He is usually considered the first to formulate the mathematical concept of zero, treating 0 as a number and discussing operations with it, including division by zero. He set down rules such as zero plus a positive number is a positive number, and a negative number plus zero is the negative number. Before Brahmagupta, zero appeared mainly as a placeholder. Babylonian and Egyptian texts used it, and Egyptians used the word nfr to mark a zero balance in double entry accounting. Indian texts used the Sanskrit word shunya for the concept of void, and Panini, in the 5th century BC, used a null operator in the Ashtadhyayi, an early algebraic grammar of Sanskrit. The idea arose independently across the world. The late Olmec people of south-central Mexico used a shell glyph as a placeholder by 38 BC, and it was the Maya who developed zero as a cardinal number, in a base 20 system built from dots and bars. In the Old World, Ptolemy was using a small circle with a long overbar by 130 AD, the first documented use of a true zero there. The concept began reaching Europe through Islamic sources around the year 1000.

• 4. Numbers People Refused

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  The abstract concept of negative numbers was recognized as early as 100 to 50 BC in China, where the Nine Chapters on the Mathematical Art used red rods for positive coefficients and black rods for negative ones. The first reference in a Western work came in the 3rd century AD in Greece, when Diophantus called an equation with a negative solution absurd in his Arithmetica. During the 600s, negative numbers were in use in India to represent debts, and Brahmagupta used them to produce the general quadratic formula still in use today. Resistance ran deep even among those who used them. In 12th-century India, Bhaskara gave negative roots for quadratic equations but wrote that the negative value is in this case not to be taken, for it is inadequate, since people do not approve of negative roots. European mathematicians mostly resisted the idea until the 17th century. Fibonacci allowed negative solutions in financial problems where they meant debts, in chapter 13 of the Liber Abaci of 1202, and later as losses in Flos. Rene Descartes called them false roots, while Nicolas Chuquet in the 15th century used them as exponents and called them absurd numbers. As recently as the 18th century, it was common practice to ignore any negative result an equation returned, on the assumption that it meant nothing at all.

• 5. Fractions And A Drowning

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  The concept of fractional numbers likely dates to prehistoric times. The Ancient Egyptians wrote rational numbers in their own fraction notation, in texts such as the Rhind Mathematical Papyrus and the Kahun Papyrus, and the Rhind Papyrus derives a circle's area from its diameter, yielding an estimate of pi near 3.16049. Euclid's Elements, from roughly 300 BC, studied rational numbers as part of number theory, as did the Indian Sthananga Sutra. Trouble entered with numbers that no fraction could capture. The Babylonians, as early as 1800 BCE, approximated irrational quantities such as the square root of 2 on clay tablets, with an accuracy like six decimal places, as on the tablet YBC 7289. The first existence proof of irrational numbers is usually attributed to the Pythagorean Hippasus, who produced a likely geometrical proof that the square root of 2 is irrational. According to legend, Pythagoras believed in the absoluteness of numbers and could neither disprove nor accept irrational ones. So, the story goes, he sentenced Hippasus to death by drowning, to stop the spread of this unsettling news. The 16th century brought final European acceptance of negative integers and fractional numbers, and by the 17th century mathematicians generally used decimal fractions in modern notation. Rene Descartes introduced the concept of real numbers in the 17th century. In 1683, while studying compound interest, Jacob Bernoulli found that ever shorter compounding intervals drove growth toward a base of 2.71828, later named Euler's number.

• 6. Beyond The Algebraic

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  A transcendental number is a value that is not the root of any polynomial with integer coefficients, which makes it not algebraic and excludes every rational number. Leonhard Euler proved in the 18th century that the irrational numbers are those whose simple continued fractions are not finite, and that Euler's number is irrational. Johann Lambert proved the irrationality of pi in 1761. The existence of transcendental numbers was first established by Liouville in 1844 and 1851. Hermite proved in 1873 that e is transcendental, and Lindemann proved in 1882 that pi is transcendental. Then came a stranger result about infinity itself. Georg Cantor showed that the set of all real numbers is uncountably infinite while the set of all algebraic numbers is only countably infinite, which means there is an uncountably infinite number of transcendental numbers. In mathematics, infinity is treated as a concept rather than a number, the property of having no end. The earliest known conception appears in the Yajurveda, an ancient Indian script, which states that if the whole is subtracted from the whole, the leftover will still be the whole. Jain mathematicians around 400 BC distinguished five types of infinity. John Wallis first introduced the familiar infinity symbol in a mathematical context in 1655. In the 1960s, Abraham Robinson showed how infinitely large and infinitesimal numbers could be rigorously defined, founding nonstandard analysis and the system of hyperreal numbers.

• 7. Roots Of Negative One

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  The earliest fleeting reference to square roots of negative numbers appears in the work of Heron of Alexandria in the 1st century AD, when he considered the volume of an impossible frustum of a pyramid. They grew prominent in the 16th century, when Italian mathematicians such as Niccolo Fontana Tartaglia and Gerolamo Cardano found closed formulas for the roots of third and fourth degree polynomials. Those formulas sometimes demanded the manipulation of square roots of negative numbers even when only real solutions were wanted, which was doubly unsettling given that negative numbers themselves were not on firm ground. Rene Descartes is sometimes credited with coining the term imaginary for these quantities in 1637, intending it as derogatory. The confusion ran deep enough to bedevil Euler, who adopted the special symbol i to guard against a recurring mistake. De Moivre's formula arrived in 1730, and Euler's formula of complex analysis in 1748, a special case of which yields Euler's identity, showing a connection among the most fundamental numbers in mathematics. The existence of complex numbers was not completely accepted until Caspar Wessel described their geometrical interpretation in 1799. Carl Friedrich Gauss rediscovered and popularized it soon after, and in the same year gave the first generally accepted proof of the fundamental theorem of algebra. Gauss studied numbers of the form a plus bi with integer parts, now called Gaussian integers, while his student Gotthold Eisenstein studied a related type now called Eisenstein integers. Ernst Kummer generalized this work and invented ideal numbers, which Felix Klein expressed as geometrical entities in 1893.

• 8. The Significance Of Numbers

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  Numbers have carried cultural, symbolic, and religious meaning throughout history and across many cultures. In Ancient Greece, number symbolism shaped the development of Greek mathematics, prompting investigations in number theory that still interest researchers. According to Plato, the Pythagoreans assigned specific characteristics and meaning to particular numbers, believing that things themselves are numbers. Folktales show clear preferences. Three and seven hold special significance in European culture, while four and five appear more prominently in Chinese folktales. Numbers also carry luck. In Western society the number 13 is considered unlucky, while in Chinese culture the number eight is considered auspicious. The most heavily studied class of all may be the prime numbers, the natural numbers that are not a product of two smaller natural numbers, beginning 2, 3, 5, 7, and 11. Euclid devoted one book of the Elements to them, proving the infinitude of the primes and the fundamental theorem of arithmetic. In 240 BC, Eratosthenes used his sieve to isolate primes quickly. Around 1000 AD, Ibn al-Haytham discovered Wilson's theorem, and Fibonacci first described trial division in 1202. Adrien-Marie Legendre conjectured the prime number theorem in 1796, finally proved by Jacques Hadamard and Charles de la Vallee-Poussin in 1896. The Goldbach conjecture and the Riemann hypothesis, the latter formulated by Bernhard Riemann in 1859, remain unproven and unrefuted to this day.