by DK
Chinese rod system
The earliest ideas of negative quantities seem to have arisen in commercial accounting: the seller received money for what had been sold (a positive quantity), and the buyer spent the same amount, resulting in a deficit (a negative quantity). For their commercial arithmetic, the ancient Chinese used small bamboo rods, laid out on a large board. Positive and negative quantities were represented by rods of different colors and could be added together. The Chinese military strategist Sun Tzu, who lived around 500 BCE, used such rods to make calculations before battles.
By 150 BCE, the rod system had developed into alternating horizontal and vertical rods in sets of up to five. Later, during the Sui dynasty (581–618 CE), the Chinese also used triangular rods for positive quantities and rectangular rods for negative quantities. The system was employed for trading and tax calculations: amounts received were represented by red rods, and debts by black rods. When rods of different colors were added together, they canceled each other out—like income erasing a debt. The polarized nature of positive numbers (red rods) and negative numbers (black rods) was also in tune with the Chinese concept that opposing but complementary forces—yin and yang—governed the Universe.
In the Chinese rod numeral system, red indicates positive numbers, while black indicates negative numbers. To make the number being represented as clear as possible, horizontal and vertical symbols are used alternately—for example, the number 752 would use a vertical 7, then a horizontal 5, followed by a vertical 2. Blank spaces represent zero.
Fluctuating fortunes
Over a period of several centuries, starting around 200 BCE, the ancient Chinese produced a book of collected scholarship called The Nine Chapters on the Mathematical Art. This work, which encapsulated the essence of their mathematical knowledge, included algorithms that assumed negative quantities were possible—for example, as solutions to problems on profit and loss.
In contrast, the mathematics of ancient Greece was based on geometry and geometrical magnitudes, or their ratios. As these quantities—actual lengths, areas, and volumes—can only be positive, the idea of a negative number did not make sense to Greek mathematicians.
By the time of Diophantus, around 250 CE, linear and quadratic equations were used to solve problems, but any unknown quantity was still represented geometrically—by a length. So the idea of negative numbers as solutions to these equations was still seen as an absurdity.
An important advance in the arithmetical use of negative numbers came around 400 years later from India, in the work of the mathematician Brahmagupta (c. 598–668). He set out arithmetic rules for negative quantities, and even used a symbol to indicate negative numbers. Like the ancient Chinese, Brahmagupta looked at numbers in financial terms, as “fortunes” (positive) and “debts” (negative), and stated the following rules for multiplying with positive and negative quantities:
The product of two fortunes is a fortune. The product of two debts is a fortune. The product of a debt and a fortune is a debt. The product of a fortune and a debt is a debt.
It makes no sense to find the product of two piles of coins, as only the actual quantities can be multiplied, not the money itself (just as you cannot multiply apples by apples). Brahmagupta was therefore performing arithmetic with positive and negative quantities, while using fortunes and debts as a way to try to understand what negative numbers represented.
The Persian mathematican and poet al-Khwarizmi (c. 780–c. 850)— whose theories, particularly on algebra, influenced later European mathematicians—was familiar with the rules of Brahmagupta and understood the use of negative numbers for dealing with debts. However, he could not accept the use of negative numbers in algebra, believing them to be meaningless. Instead, al-Khwarizmi followed geometric methods to solve linear or quadratic equations.
Temperature readings on the Celsius scale display negative numbers to show when something such as an ice crystal is colder than 0°C—the point at which water freezes.
A negative multiplied by a negative makes a positive. This is why all positive numbers have two square roots (a positive and a negative) and negative numbers have no real square roots—because a positive number squared is positive, and a negative number squared is also positive.
Accepting the negative
Throughout the Middle Ages, European mathematicians remained unsure of negative quantities as numbers. This was still the case in 1545 when Italian polymath Gerolamo Cardano published his Ars Magna (The Great Art), in which he explained how to solve linear, quadratic, and cubic equations. He could not exclude negative solutions to his equations and even used a sign, “m,” to denote a negative number. He could not, however, accept the value of negative numbers, calling them “fictitious.” René Descartes (1596–1650) also accepted negative quantities as solutions to equations but referred to them as “false roots” rather than true numbers.
English mathematician John Wallis (1616–1703) gave some meaning to negative numbers by extending the number line below zero. This way of seeing numbers as points on a line finally led to the acceptance of negative numbers on equal terms with positive numbers, and by the end of the 1800s, they had been formally defined within mathematics, separate from notions of quantities. Today, negative numbers are used in many areas, ranging from banking and temperature scales to the charge on subatomic particles. Any ambiguity about their status in mathematics is long gone.
Negative numbers are evidence of inconsistency or absurdity.
Augustus De Morgan
British mathematician
Investors rush to withdraw their money from the Seamen’s Savings Bank in New York in 1857. The panic was caused by American banks loaning out many millions of dollars (a negative quantity) without the reserves (a positive quantity) to back this up.
Mathematics in ancient China
Jiuzhang suanshu, or The Nine Chapters on the Mathematical Art, reveals the mathematical methods known to the ancient Chinese. It is written as a collection of 246 practical problems and their solutions.
The first five chapters are mostly about geometry (areas, lengths, and volumes) and arithmetic (ratios, and square and cube roots). Chapter six covers taxes, and includes the ideas of direct, inverse, and compound proportions, most of which did not appear in Europe until around the 1500s. Chapters seven and eight deal with solutions to linear equations, including the rule of “double false position,” whereby two test (or “false”) values for the solution to a linear equation are used in repeated steps to yield the actual solution. The final chapter concerns applications of the “Gougu” (equivalent to Pythagoras’s theorem), and the solving of quadratic equations.
See also: Positional numbers • Diophantine equations • Zero • Algebra • Imaginary and complex numbers
IN CONTEXT
KEY FIGURE
Diophantus (c. 200–c. 284 CE)
FIELD
Algebra
BEFORE
c. 800 BCE The Indian scholar Baudhayana finds solutions to some “Diophantine” equations.
AFTER
c. 1600 François Viète lays the foundations for solutions of Diophantine equations.
1657 Pierre de Fermat writes his last theorem (about a Diophantine equation) in his copy of Arithmetica.
1900 The 10th problem on David Hilbert’s list of unsolved research problems is the quest to find an algorithm to solve all Diophantine equations.
1970 Mathematicians in Russia show that there is no algorithm that can solve all Diophantine equations.
In the 3rd century CE, the Greek mathematician Diophantus, a pioneer of number theory and arithmetic, created a prodigious work called Arithmetica. In 13 volumes, only six of which have survived, he explored 130 problems involving equations and was the first person to use a symbol for an unknown quantity—a cornerstone of algebra. It is only in the past 100 years that mathematicians have fully explored what are now known as Diophantine equations. Today, the equations are considered to be one of the mos
t interesting areas of number theory.
Diophantine equations are a type of polynomial—an equation in which the powers of the variables (unknown quantities) are integers, such as x3 + y4 = z5. The aim of Diophantine equations is to find all the variables, but solutions must be integers or rational numbers (those that can be written as one integer divided by another, such as 8⁄3). In Diophantine equations, the coefficients—integers such as the 4 in 4x, that multiply a variable—are also rational numbers. Diophantus only used positive numbers, but mathematicians now look for negative solutions as well.
The symbolism that Diophantus introduced for the first time… provided a short and readily comprehensible means of expressing an equation.
Kurt Vogel
German mathematical historian
The quest for solutions
Many of the problems now called Diophantine equations were known well before Diophantus’s time. In India, mathematicians explored some of them from around 800 BCE, as the ancient Shulba Sutras texts reveal. In the 6th century BCE, Pythagoras created his quadratic equation for calculating the sides of a right-angled triangle; its x2 + y2 = z2 form is a Diophantine equation.
Diophantine equations of the kind xn + yn = zn may look simple to calculate, but only those with squares are solvable. If the power (n in the equation) is greater than 2, the equation has no integer solutions for x, y, and z—as Fermat asserted in a marginal note in 1657 and British mathematician Andrew Wiles finally proved in 1994.
The Arithmetica of Diophantus strongly influenced 17th-century mathematicians as the study of modern algebra developed. This volume of the book was published in Latin in 1621.
A source of fascination
Diophantine equations are vast in number and form, and mostly very difficult to solve. In 1900, David Hilbert suggested that the question of whether or not they could all be solved was one of the greatest challenges facing mathematicians.
The equations are now grouped in three classes: those with no solution, those with a finite number of solutions, and those with an infinite number of solutions. Rather than finding solutions, however, mathematicians are often more interested in discovering whether solutions exist at all. In 1970, Russian mathematician Yuri Matiyasevich settled Hilbert’s query, which he and three others had studied for years, concluding that no general algorithm to solve a Diophantine equation exists. Yet studies continue, as the fascination of these equations is largely theoretical. Mathematicians, who are driven by curiosity, believe there is still more to discover.
DIOPHANTUS
Little is known about the life of the Greek mathematician and philosopher Diophantus, but he was probably born in Alexandria, Egypt, in C. 200 CE. His 13-volume Arithmetica was well-received—the Alexandrian mathematician Hypatia wrote about the first six volumes—but fell into relative obscurity until the 1500s, when interest in his ideas was revived.
The Greek Anthology, a compilation of mathematical games and verses published around 500 CE, contains one number problem purporting to be an epitaph to Diophantus that appeared on his tombstone. Written as a puzzle, it suggests he married at the age of 35, and five years later had a son, who died at the age of 40 when he was half his father’s age. Diophantus is then said to have lived a further four years, dying at the age of 84.
Key work
C. 250 CE Arithmetica
See also: The Rhind papyrus • Pythagoras • Hypatia • The equals sign and other symbology • 23 problems for the 20th century • The Turing machine • Proving Fermat’s last theorem
IN CONTEXT
KEY FIGURE
Hypatia of Alexandria (c. 355–415 CE)
FIELDS
Arithmetic, geometry
BEFORE
6th century BCE Pythagoras’s wife Theano and other women actively participate in the Pythagorean community.
c. 100 BCE Mathematician and astronomer Aglaonike of Thessaly wins renown for her ability to predict lunar eclipses.
AFTER
1748 Italian mathematician Maria Agnesi writes the first textbook to explain differential and integral calculus.
1874 Russian mathematician Sofia Kovalevskaya is the first woman to be awarded a doctorate in mathematics.
2014 Iranian mathematician Maryam Mirzakhani is the first woman to win the Fields Medal.
History mentions only a few pioneering female mathematicians in the ancient world, among them Hypatia of Alexandria. An inspirational teacher, she was appointed head of the city’s Platonist school in 400 CE.
Hypatia is not known to have contributed any original research, but she is credited with editing and writing commentaries on several classic mathematical, astronomical, and philosophical texts. It is likely that she helped her father, Theon, a respected Alexandrian scholar, to produce his definitive edition of Euclid’s Elements, and his Almagest and Handy Tables of Ptolemy. She also continued his project of preserving and expanding the classic texts, in particular providing commentaries on Diophantus’s 13-volume Arithmetica, and Apollonius’s work on conic sections. Hypatia may have intended these editions to serve as textbooks for students, as she offered commentaries providing clarification, and developed some of the concepts further.
Hypatia won great renown for her teaching, scientific knowledge, and wisdom, but in 415 she was killed by Christian zealots for her “pagan” philosophy. As attitudes toward women in academia became less tolerant, mathematics and astronomy would be almost exclusively male preserves until the Enlightenment opened up new opportunities for women in the 1700s.
The Alexandrian scholar Hypatia, depicted here in an 1889 painting by Julius Kronberg, was revered as a heroic martyr after her murder. She later became a symbol for feminists.
See also: Euclid’s Elements • Conic sections • Diophantine equations • Emmy Noether and abstract algebra
IN CONTEXT
KEY FIGURE
Zu Chongzhi (429–501 CE)
FIELD
Geometry
BEFORE
c. 1650 BCE The area of a circle is calculated using π as (16⁄9)2 ≈ 3.1605 in the Rhind papyrus.
c. 250 BCE Archimedes finds an approximate value for π using a polygon algorithm method.
AFTER
c. 1500 Indian astronomer Nilakantha Somayaji uses an infinite series (the sum of terms of an infinite sequence, such as 1⁄2 + 1⁄4 + 1⁄8 + 1⁄16) to compute π.
1665–66 Isaac Newton calculates π to 15 digits.
1975–76 Iterative algorithms allow computer calculations of π to millions of digits.
Like their counterparts in Greece, mathematicians in ancient China realized the importance of π (pi)—the ratio of a circle’s circumference to its diameter—in geometric and other calculations. Various values for π were suggested from the 1st century CE onward. Some were sufficiently accurate for practical purposes, but several Chinese mathematicians sought more precise methods for determining π. In the 3rd century, Liu Hui approached the task using the same method as Archimedes—drawing regular polygons with increasing numbers of sides inside and outside a circle. He found that a 96-sided polygon allowed a calculation of π as 3.14, but by repeatedly doubling the number of sides up to 3,072, he reached a value of 3.1416.
More precision
In the 5th century, astronomer and mathematician Zu Chongzhi, who was renowned for his meticulous calculations, set about obtaining an even more accurate value for π. Using a 12,288-sided polygon, he calculated that π is between 3.1415926 and 3.1415927, and suggested two fractions to express the ratio: the Yuelü, or approximate ratio, of 22⁄7, which had been in use for some time; and his own calculation, the Milü, or close ratio, of 355⁄113. This later became known as “Zu’s ratio.” Zu’s calculations of π were not bettered until European mathematicians set about the task during the Renaissance, almost a millennium later.
I cannot help thinking that Zu Chongzhi was a genius of Antiquity.
Takebe Katahiro
Japanese mathematician
S
ee also: The Rhind papyrus • Irrational numbers • Calculating pi • Euler’s identity • Buffon’s needle experiment
INTRODUCTION
As the Roman Empire collapsed and Europe entered the Middle Ages, the center of scientific and mathematical scholarship shifted from the eastern Mediterranean to China and India. From about the 5th century CE, India began a “Golden Age” of mathematics, building on its own long tradition of scholarship, but also on ideas brought in by the Greeks. Indian mathematicians made significant advances in the fields of geometry and trigonometry, which had practical applications in astronomy, navigation, and engineering, but the most far-reaching innovation was the development of a character to represent the number zero.
The use of a specific symbol— a simple circle, rather than a blank space or placeholder—to denote zero is attributed to the brilliant mathematician Brahmagupta, who described the rules of its use in calculation. In fact, the character may already have been in use for some time. It would have fitted well with India’s numeral system, which is the prototype of our modern Hindu–Arabic numerals. Yet it is thanks to Islam that these and other ideas from India’s Golden Age (which continued until the 12th century) went on to influence the history of mathematics.
Persian powerhouse