by DK
Mersenne’s influence
Regius’s work on primes was continued by others who proposed new hypotheses with 2n - 1. The most significant was that of French monk Marin Mersenne in 1644). He stated that 2n-1 was valid when n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257. Mersenne’s work rekindled interest in the topic, and primes generated by 2n - 1 are now known as Mersenne primes (Mn).
The use of computers has made it possible to find more Mersenne primes. Two of Mersenne’s n values (67 and 257) were proved incorrect, but in 1947, three new primes were found: n = 61, 89, and 107 (M61, M89, M107), and in 2018, the Great Internet Mersenne Prime Search uncovered the 51st known Mersenne prime.
The beauty of number theory [is] related to the contradiction between the simplicity of the integers and the complicated structure of the primes.
Andreas Knauf
German mathematician
See also: Euclid’s Elements • Eratosthenes’ sieve • The Riemann hypothesis • The prime number theorem
IN CONTEXT
KEY FIGURE
Pedro Nunes (1502–78)
FIELD
Graph theory
BEFORE
150 CE The Greco-Roman mathematician Ptolemy establishes the concepts of latitude and longitude.
c. 1200 The magnetic compass is used by navigators in China, Europe, and the Arab world.
1522 Portuguese navigator Ferdinand Magellan’s ship completes the first voyage around the world.
AFTER
1569 Flemish mapmaker Gerardus Mercator’s map projection allows navigators to plot rhumb-line courses as straight lines on the map.
1617 A spiral rhumb line is named a “loxodrome” by Dutch mathematician Willebrord Snell.
From around 1500, as ships began to cross the world’s oceans, navigators met a problem—plotting a course across the world that took account of the Earth’s curved surface. The problem was solved by the introduction of the rhumb line by Portuguese mathematician Pedro Nunes in his Treatise on the Sphere (1537).
The rhumb spiral
A rhumb line cuts across every meridian (line of longitude) at the same angle. Because meridians get closer toward the poles, rhumb lines bend around into a spiral. Such spirals were called loxodromes by Dutch mathematician Willebrord Snell in 1617; they became a key concept in the geometry of space.
The rhumb line helps navigators because it gives a single compass bearing for a voyage. In 1569, Mercator maps—on which lines of longitude are drawn parallel, so that all rhumb lines are straight—were introduced. This further enabled people to plot a course just by drawing a straight line on the map. The shortest distance across the globe is not a rhumb, however, but a “great circle”—any circle that centers on the center of the Earth. It only became practical to follow a great circle course with the invention of GPS.
A loxodrome starts at the North or South Pole, and spirals around the globe, crossing each meridian at the same angle. A rhumb line is all or part of this spiral.
See also: Coordinates • Huygens’s tautochrone curve • Graph theory • Non-Euclidean geometries
IN CONTEXT
KEY FIGURE
Robert Recorde (c. 1510–58)
FIELD
Number systems
BEFORE
250 CE Greek mathematician Diophantus uses symbols to represent variables (unknown quantities) in Arithmetica.
1478 The Treviso Arithmetic explains in simple language how to perform addition, subtraction, multiplication, and division calculations.
AFTER
1665 In England, Isaac Newton develops infinitesimal calculus, which introduces ideas such as limits, functions, and derivatives. These processes require new symbols for abbreviation.
1801 Carl Friedrich Gauss introduces the symbol for congruence—equal size and shape.
In the 16th century, when Welsh doctor and mathematician Robert Recorde began his work, there was little consensus on the notation used in arithmetic. Hindu–Arabic numerals, including zero, were already established, but there was little to represent calculations.
In 1543, Recorde’s The Grounde of Artes introduced the symbols for addition (+) and subtraction (˗) to mathematics in England. These signs had first appeared in print in Mercantile Arithmetic (1489), by German mathematician Johannes Widman, but were probably already used by German merchants before Widman’s book was published. These symbols slowly replaced the letters “p” for plus and “m” for minus as they were taken up by scholars, first in Italy, then in England.
In 1557, Recorde went on to recommend a new symbol of his own. In The Whetstone of Witte, he used a pair of identical parallel lines (=) to represent “equals,” claiming that “no two things can be more equal” than these. Recorde suggested that symbols would save mathematicians from having to write out calculations in words. The equals sign was widely adopted, and the 17th century also saw the creation of many of the other symbols used today, such as those for multiplication (×) and division (÷).
Robert Recorde tested the equals sign (=) in his own calculations, as seen here in one of his exercise books. Recorde’s sign was noticeably longer than the modern form.
Notating algebra
While the earliest algebraic techniques date back more than two millennia to the Babylonians, most calculations before the 16th century were recorded in words—sometimes abbreviated, but not in a uniform way. English mathematician Thomas Harriot and French mathematician François Viète, who each made important contributions to developments in algebra, used letters to produce consistent symbolic notation. In their system, the most noticeable difference from today’s notation is the use of a repeated letter to indicate a power. For example, a3 was aaa and x4 was xxxx.
A modern system
French mathematician Nicholas Chuquet used superscripts in 1484 to represent exponents (“to the power of”), but did not record them as such; for example, 6x2 was 6.2. It took more than 150 years for superscripts to become common; René Descartes used recognizable examples in 1637 when writing 3x + 5x3, yet continued to write x2 as xx. Only in the early 1800s, when the influential German mathematician Carl Gauss favored using x2, did superscript notation begin to stick. Descartes also made a contribution with his use of x, y, and z for the unknowns in equations, and a, b, and c for known figures.
Algebraic notation may have taken a long time to catch on, but when a symbol made sense and helped mathematicians work through problems, it became the norm. Improved contact between mathematicians in different parts of the world in the 1600s also led to such notations being adopted much more swiftly.
To avoid the tedious repetition of these words, is equal to, I will set, as I do often in work use, a pair of parallels.
Robert Recorde
ROBERT RECORDE
Born in Tenby, Wales, around 1510, Recorde grew up to study medicine first at Oxford University, then at Cambridge, where he qualified as a physician in 1545. He taught mathematics at both universities and wrote the first English book on algebra in 1543. In 1549, after a period practicing medicine in London, Recorde was made controller of the Bristol mint. However, after he refused to issue funds to William Herbert, the future Earl of Pembroke, for his army, the mint was closed.
In 1551, Recorde was given charge of the Dublin mint, which included silver mines in Germany. When he failed to show a profit, the mines were also closed. Recorde later tried to sue Pembroke for misconduct, but was instead countersued for libel. Sent to a London prison in 1557 for failure to pay the fine, Recorde died there in 1558.
Key works
1543 Arithmetic: or the Grounde of Artes
1551 The Pathway to Knowledge
1557 The Whetstone of Witte
See also: Positional numbers • Negative numbers • Algebra • Decimals • Logarithms • Calculus
IN CONTEXT
KEY FIGURE
Rafael Bombelli (1526–72)
FIELD
Algebra
BEFORE
; 1500s In Italy, Scipione del Ferro, Tartaglia, Antonio Fior, and Ludovico Ferrari compete publicly to solve cubic equations.
1545 Gerolamo Cardano’s Ars Magna, a book of algebra, includes the first published calculation involving complex numbers.
AFTER
1777 Leonhard Euler introduces the notation i for .
1806 Jean-Robert Argand publishes a geometrical interpretation of complex numbers, leading to the Argand diagram.
In the late 1500s, Italian mathematician Rafael Bombelli broke new ground when he laid down the rules for using imaginary and complex numbers in his book Algebra. An imaginary number, when squared, produces a negative result, defying the usual rules that any number (positive or negative) results in a positive number when squared. A complex number is the sum of any real number (on the number line) and an imaginary number. Complex numbers take the form a + bi, where a and b are real and i = .
Over the centuries, scholars have needed to extend the concept of the number in order to solve different problems. Imaginary and complex numbers were new tools in this endeavor, and Bombelli’s Algebra advanced understanding of how these and other numbers work. To solve the simplest equations, such as x + 1 = 2, only natural numbers (positive integers) are needed. To solve x + 2 = 1, however, x must be a negative integer, while solving x2 + 2 = 1 requires the square root of a negative number. This did not exist with the numbers at Bombelli’s disposal, so had to be invented—leading to the concept of the imaginary unit (). Negative numbers were still mistrusted in the 1500s; imaginary and complex numbers were not widely accepted for many decades.
Some people believe in imaginary friends. I believe in imaginary numbers.
R. M. ArceJaeger
American author
Fierce rivalry
The idea of complex numbers first emerged early in Bombelli’s lifetime as Italian mathematicians sought to find solutions to cubic equations as efficiently as possible, without relying on the geometrical methods devised by Persian polymath Omar Khayyam in the 12th century. As most quadratic equations could be solved with an algebraic formula, the search was on for a similar formula that worked for cubic equations. Scipione del Ferro, a mathematics professor at Bologna University, took a major step forward when he discovered an algebraic method for solving some cubic equations, but the quest for a comprehensive formula continued.
Italian mathematicians of this era would publicly challenge one another to solve cubic equations and other problems in the least possible time. Achieving fame in such contests became essential for any scholar who wanted to gain a post as a mathematics professor at a prestigious university. As a result, many mathematicians kept their methods secret rather than sharing them for the common good. Del Ferro tackled equations of the form x3 + cx = d. He passed his technique on to only two people, Antonio Fior and Annibale della Nave, swearing them to secrecy. Del Ferro soon had competition from Niccolò Fontana (known as Tartaglia, or “the stutterer”). An itinerant teacher of considerable mathematical ability, but with few financial resources, Tartaglia discovered a general method for solving cubic equations independently of del Ferro. When del Ferro died in 1526, Fior decided the time had come for him to unleash del Ferro’s formula upon the world. He challenged Tartaglia to a cubic duel, but was beaten by Tartaglia’s superior methods. Gerolamo Cardano heard of this and persuaded Tartaglia to share his methods with him. As with del Ferro, the condition was that the method should never be published.
I shall call [the imaginary unit] ‘plus of minus’ when added and when subtracted, ‘minus of minus.’
Rafael Bombelli
Beyond positive numbers
At this time all equations were solved using positive numbers. Working with Tartaglia’s method, Cardano had to grapple with the notion that using the square roots of negative numbers might help solve cubic equations. He was evidently prepared to experiment with the method, but appears not to have been convinced. He called such negative solutions “fictitious” and “false” and described the intellectual effort involved in finding them as “mental torture.” His Ars Magna shows his use of the negative square root. He wrote: “Multiply 5 + by 5 , making 25 ˗(˗15), which is + 15. Hence this product is 40.” This is the first recorded calculation involving complex numbers, but the significance of this breakthrough escaped Cardano; he branded his work “subtle” and “useless.”
Rafael Bombelli set out the rules for operations on complex numbers. He used the term “plus of minus” to describe a positive imaginary unit and “minus of minus” to describe a negative imaginary unit. Multiplying a positive imaginary unit by a negative imaginary unit, for example, equals a positive integer; while multiplying a negative imaginary unit by a negative imaginary unit equals a negative integer.
Explaining the numbers
Rafael Bombelli assimilated the tussles between the various mathematicians solving cubic equations. He read Cardano’s Ars Magna with great admiration. His own work, Algebra, was a more accessible version, and was a thorough and innovative survey of the subject. It investigated the arithmetic of negative numbers, and included some economical notation that represented a major advance on what had gone before.
The work outlines the basic rules for calculating with positive and negative quantities, such as: “Plus times plus makes plus; Minus times minus makes plus.” It then sets out new rules for adding, subtracting, and multiplying imaginary numbers in terminology that differs from that used by mathematicians today. For example, he stated that “Plus of minus multiplied by plus of minus makes minus”—meaning a positive imaginary number multiplied by a positive imaginary number equals a negative number: × = ˗n. Bombelli also gave practical examples of how to apply his rules for complex numbers to cubic equations, where solutions require finding the square root of some negative number. Although Bombelli’s notation was advanced for his time, the use of algebraic symbols was still in its infancy. Two centuries later, Swiss mathematician Leonhard Euler introduced the symbol i to denote the imaginary unit.
The shortest route between two truths in the real domain passes through the complex domain.
Jacques Hadamard
French mathematician
Applying complex numbers
Imaginary and complex numbers joined the ranks of other sets, such as natural numbers, real numbers, rational numbers, and irrational numbers, that were used to solve equations and perform a range of other increasingly sophisticated mathematical tasks.
Over the decades, sets of such numbers acquired their own universal symbols that could be used in formulae. For instance, the bold capital N is used for natural numbers from the set {0, 1, 2, 3, 4…}, enclosed in curly brackets to denote a set. In 1939, American mathematician Nathan Jacobson established the bold capital C to signify the set of complex numbers, {a + bi}, where a and b are real and i = .
Complex numbers enable all polynomial equations to be solved completely, but have also proved immensely useful in many other branches of mathematics—even in number theory (the study of integers, especially positive numbers). By treating the integers as complex numbers (the sum of a real value and an imaginary value), number theorists can use powerful techniques of complex analysis (a study of functions with complex numbers) to investigate the integers. The Riemann zeta function, for example, is a function of complex numbers that provides information about primes. In other practical areas, physicists use complex numbers in the study of electromagnetism, fluid dynamics, and quantum mechanics, while engineers need them for designing electronic circuits, and for studying audio signals.
There is an ancient and innate sense in people that numbers ought not to misbehave.
Douglas Hofstadter
Cognitive scientist
A series of cups shows blue food dye being dripped over an ice cube (left). As the ice cube melts, the heavier blue dye sinks. Complex numbers are used to model the velocity (speed and direction) of such fluids.
RAFAEL BOMBELLI
Born in Bologna, Italy, in 1526
, Rafael Bombelli was the eldest of six children; his father was a wool merchant. Although Bombelli did not receive a college education, he was taught by an engineer–architect and became an engineer himself, specializing in hydraulics. He also developed an interest in mathematics, studying the work of ancient and contemporary mathematicians. While waiting for a drainage project to recommence, he embarked on his major work, Algebra, which laid out a primitive but thorough arithmetic of complex numbers for the first time.
Greatly impressed by a copy of Diophantus’s Arithmetica found in the Vatican library, Bombelli helped to translate it into Italian – work that led him to revise Algebra. Three volumes were published in 1572, the year he died; the last two incomplete volumes were published in 1929.
Key work
1572 Algebra
See also: Quadratic equations • Irrational numbers • Negative numbers • Cubic equations • The algebraic resolution of equations • The fundamental theorem of algebra • The complex plane
IN CONTEXT
KEY FIGURE
Simon Stevin (1548–1620)
FIELD
Number systems
BEFORE
830 CE Al-Kindi’s four-volume On the use of Indian numerals spreads the place value system based on the Hindu numerals throughout the Arab world.
