by DK
I have had my results for a long time, but I do not yet know how to arrive at them.
Carl Gauss
Legacy of the theorem
The proofs by Gauss and Argand established the validity of complex numbers as roots of polynomials. The FTA stated that anyone faced with solving an equation built from real numbers could be sure that they would find their solution within the realm of complex numbers. These groundbreaking ideas formed the foundations of complex analysis.
Mathematicians since Argand have continued to work on proving the FTA using new methods. In 1891, for example, German Karl Weierstrass created a method—now known as the Durand–Kerner method, due to its rediscovery by mathematicians in the 1960s—for simultaneously finding all of the roots of a polynomial.
Applications of the FTA
An Einstein ring, first discovered in 1998, is the deformation of light from a source into a ring through gravitational lensing.
Research on the fundamental theorem of algebra has led to breakthroughs in other fields. In the 1990s, British mathematicians Terrence Sheil-Small and Alan Wilmshurst extended the FTA to harmonic polynomials. These may have an infinite number of roots, but in some cases, there are a finite number. In 2006, American mathematicians Dmitry Khavinson and Genevra Neumann proved that there was an upper limit to the number of roots of a certain class of harmonic polynomials. After publishing their results, they were told that their proof settled a conjecture by South Korean astrophysicist Sun Hong Rhie. Her conjecture concerned images of distant astronomical light sources. Massive objects in the Universe bend passing light rays from distant sources in a phenomenon called gravitational lensing, creating multiple images seen through a telescope. Rhie posited that there would be a maximum number of images produced; this turned out to be exactly the upper bound found by Khavinson and Neumann.
See also: Quadratic equations • Negative numbers • Algebra • Cubic equations • Imaginary and complex numbers • The algebraic resolution of equations • The complex plane
INTRODUCTION
Progress in mathematics accelerated through the 1800s, with science and mathematics now becoming respected academic studies. As the Industrial Revolution spread and 1848’s “Year of Revolution” saw nationalism surge across old empires, there was a renewed drive to understand the workings of the Universe in scientific terms, rather than through religion or philosophy. Pierre-Simon Laplace, for example, applied the theories of calculus to celestial mechanics. He proposed a form of scientific determinism, saying that with the relevant knowledge of moving particles, the behavior of everything in the Universe could be predicted.
Another characteristic of 19th-century mathematics was its increasing tendency toward the theoretical. This trend was fostered by the influential work of Carl Friedrich Gauss, regarded by many in the field as the greatest of all mathematicians. He dominated the study of mathematics for much of the first half of the century, making contributions to the fields of algebra, geometry, and number theory, and giving his name to such concepts as Gaussian distribution, Gaussian function, Gaussian curvature, and Gaussian error curve.
New fields
Gauss was also a pioneer of non-Euclidean geometries, which epitomized the revolutionary spirit of 19th-century mathematics. The subject was taken up by Nicolai Lobachevsky and János Bolyai, who independently developed theories of hyperbolic geometry and curved spaces, resolving the problem of Euclid’s parallel postulate. This opened up a completely new approach to geometry, paving the way for the nascent field of topology (the study of space and surfaces) which was also influenced by the possibility of more than three dimensions offered by William Hamilton’s discovery of quaternions.
Perhaps the best known of the pioneers of topology is August Möbius, inventor of the Möbius strip, which had the unusual property of being a two-dimensional surface with only one side. Non-Euclidean geometries were further developed by Bernhard Riemann, who identified and defined different types of geometry in multiple dimensions.
Riemann did not confine his studies to geometry, however. As well as his work on calculus, he made important contributions to number theory, following in the footsteps of Gauss. The Riemann hypothesis, derived from the Riemann zeta function concerning complex numbers, is as yet unsolved. Other notable discoveries in number theory at this time include the creation of set theory and the description of an “infinity of infinities” of Georg Cantor, Eugène Catalan’s conjecture about the powers of natural numbers, and the application of elliptic functions to number theory proposed by Carl Gustav Jacob Jacobi.
Jacobi was, like Riemann, multi-talented, often linking different fields of mathematics in new ways. His primary interest was in algebra, another area of mathematics that was becoming increasingly abstract during the 1800s. The groundwork for the growing field of abstract algebra was laid by Évariste Galois, who, although he died young, also developed group theory while determining a general algebraic method for solving polynomial equations.
New technologies
Not all mathematics in this period was purely theoretical—and even some of the abstract concepts soon found more practical applications. Siméon Poisson, for example, used his knowledge of pure mathematics to develop ideas such as the Poisson distribution, a key concept in the field of probability theory. Charles Babbage, on the other hand, responded to practical demand for a means of accurate and quick calculation with his mechanical calculating device, the “Difference Engine.” In so doing, he laid the groundwork for the invention of computers. Babbage’s work in turn inspired Ada Lovelace to devise the forerunner of modern computer algorithms.
Meanwhile, there were other developments in mathematics that were to have far-reaching implications for later technological progress. Using algebra as his starting point, George Boole devised a form of logic based on a binary system, and using the operators AND, OR, and NOT. These became the foundation of modern mathematical logic, but just as importantly paved the way for the language of computers almost a century later.
IN CONTEXT
KEY FIGURE
Jean-Robert Argand (1768–1822)
FIELD
Number theory
BEFORE
1545 Italian scholar Gerolamo uses negative square roots to solve cubic equations in his book Ars Magna.
1637 French philosopher and mathematician René Descartes develops a way to plot algebraic expressions as coordinates on a grid.
AFTER
1843 Irish mathematician William Hamilton extends the complex plane by adding two more imaginary units to create quaternions—expressions that are plotted in a 4-D space.
1859 By merging two complex planes, Bernhard Riemann develops a 4-D surface to help him analyze complex functions.
After centuries of suspicion, mathematicians finally embraced the concept of negative numbers in the 1700s. They did so by using imaginary numbers in algebra. In 1806, the key contribution of Swiss-born mathematician Jean-Robert Argand was to plot complex numbers (made up of a real and imaginary component) as coordinates on a plane created by two axes—x for real numbers and y for imaginary numbers. This complex plane provided the first geometrical interpretation of the distinctive properties of complex numbers.
There can be very little… science and technology that is not dependent on complex numbers.
Keith Devlin
British mathematician
Algebraic roots
Imaginary numbers had emerged in the 1500s when Italian mathematicians such as Gerolamo Cardano and Niccolò Fontana Tartaglia found that solving cubic equations required a square root of a negative number. The square of a real number cannot be negative—any real number multiplied by itself results in a positive—so they decided to treat as a new unit that operated separately from the real numbers. Leonhard Euler first used i to denote the imaginary unit () in his attempts to prove the fundamental theorem of algebra (FTA). This theorem states that all polynomial equations of degree n have n roots. This means that if x2 is t
he highest power in an algebraic expression made up of a single variable (such as x) and real coefficients (numbers multiplying the variable), the expression has a degree of two, and also two roots; roots are values of x that make the polynomial equal to zero. Many seemingly simple polynomials, however, such as x2 + 1, do not equal zero if x is a real number. Plotting x2 + 1 on a graph with an x and y axis creates a neat curve that never passes through the origin, or (0,0) point. To make the FTA work for x2 + 1, Gauss and others used real numbers combined with imaginary numbers to create complex numbers. All numbers are in essence complex. For example, the real number 1 is the complex number 1 + 0i, while the number i is 0 + i. The equation x2 + 1 can equal zero when x is i or -i.
An Argand diagram uses the x and y axes to represent real numbers and imaginary numbers, combining them to plot complex numbers. This diagram shows two numbers: 3 + 5i and 7 + 2i.
Argand’s discovery
As Argand began to plot complex numbers, he discovered that the imaginary number i does not get bigger if raised to higher powers. Instead, it follows a four-step pattern that repeats infinitely: i1 = i; i2 = –1; i3 = –i, i4 = 1; i5 = i, and so on. This can be visualized on the complex plane. Multiplying real numbers by imaginary numbers produces 90° rotations through the complex plane. So 1 × i = i, which does not appear on the real number x axis at all, but on the imaginary y axis. Continuing to multiply by i results in more 90° rotations, which is why every four multiplications arrive back at the start point.
Plots of complex numbers— or Argand diagrams—make complicated polynomials easier to solve. The complex plane is now a powerful tool that works far beyond the interests of number theory.
JEAN-ROBERT ARGAND
Little is known of Jean-Robert Argand’s early life. He was born in Geneva in 1768, but appears to have had no formal education in mathematics. In 1806, he moved to Paris to manage a bookshop, and self-published the work containing the geometrical interpretation of complex numbers for which he is known. (Norwegian cartographer Casper Wessel is now known to have used similar constructions in 1799.) Argand’s essay was republished in a mathematics journal in 1813, and in the next year, he used the complex plane to produce the first rigorous proof of the fundamental theorem of algebra. Argand published eight more articles before his death in Paris in 1822.
Key work
1806 Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriques (Essay on a method of representing imaginary quantities geometrically)
See also: Quadratic equations • Cubic equations • Imaginary and complex numbers • Coordinates • The fundamental theorem of algebra
IN CONTEXT
KEY FIGURE
Joseph Fourier (1768–1830)
FIELD
Applied mathematics
BEFORE
1701 In France, Joseph Sauveur suggests that vibrating strings oscillate with many waves of different lengths at the same time.
1753 Swiss mathematician Daniel Bernoulli shows that a vibrating string consists of an infinite number of harmonic oscillations.
AFTER
1965 In the US, James Cooley and John Tukey develop the Fast Fourier Transform (FFT), an algorithm that is able to speed up Fourier analysis.
2000s Fourier analysis is used to create a number of speech recognition programs for computers and smartphones.
The sound created by vibrating strings has been a topic of research for more than 2,500 years. In about 550 BCE, Pythagoras discovered that if you take two taut strings of the same material and the same tension, but one is twice the length of the other, the short string will vibrate with twice the frequency of the longer string and the resulting notes will be an octave apart.
Two centuries later, Aristotle suggested that sound traveled through the air in waves, although he incorrectly thought that higher-pitched sounds traveled faster than lower-pitched ones. In the 1600s, Galileo recognized that sounds are produced by vibrations: the higher the frequency of the vibrations, the higher the pitch of the sound we perceive.
Sounds are made of a complex series of tones. Fourier analysis can separate out pure tones, which can be represented as sine waves on a graph, from each other. Tones have frequency, which determines pitch, and amplitude, which determines volume.
Heat and harmony
By the end of the 1600s, physicists including Joseph Sauveur were making great strides in studying the relationships between the waves in stretched strings and the pitch and frequency of sounds that they produced. In the course of their research, mathematicians showed that any string will support a potentially infinite series of vibrations, starting from the fundamental (the string’s lowest natural frequency) and including its harmonics (integer multiples of the fundamental). The pure tone of a single pitch is produced by a smooth repetitive oscillation called a sine wave (see graph). The sound quality of a musical instrument results principally from the number and relative intensities of the harmonics present in the sound, or its harmonic content. The result is a variety of waves interfering with each other.
Joseph Fourier was attempting to solve the problem of how heat diffused through a solid object. He developed an approach that would allow him to calculate the temperature at any location within an object, at any time after a source of heat had been applied to one of its edges.
Fourier’s studies of heat distribution showed that no matter how complex a waveform, it could be broken down into its constituent sine waves, a process that is now called Fourier analysis. Since heat in the form of radiation is a wave, Fourier’s discoveries about heat distribution had applications to the study of sound. A sound wave can be understood in terms of the amplitudes of its constituent sine waves, a set of numbers that is sometimes referred to as the harmonic spectrum.
Today, Fourier analysis plays a key role in many applications including digital file compression, analyzing MRI scans, speech recognition software, musical pitch correction software, and determining the composition of planetary atmospheres.
Fourier analysis of the way materials vibrate allows engineers to construct buildings that resonate at different frequencies from a typical earthquake and thus avoid the kind of damage that occurred in Mexico City in 2017.
JOSEPH FOURIER
Jean-Baptiste Joseph Fourier was born in Auxerre, France, in 1768. A tailor’s son, he went to military school, where his keen interest in mathematics led him to become a successful teacher of the subject.
Fourier’s career was disrupted by two arrests—one for criticizing the French Revolution, the other for supporting it—but in 1798, he accompanied Napoleon’s forces into Egypt as a diplomat. Napoleon later made him a baron, and then a count. After Napoleon’s fall in 1815, Fourier moved to Paris to become director of the Statistical Bureau of the Seine, where he pursued his studies in mathematical physics, including work on the Fourier series (a series of sine waves that characterize sounds). In 1822, Fourier was made the secretary of the French Academy of Sciences, a post he held until his death in 1830. Fourier is one of 72 scientists whose names are inscribed on the Eiffel Tower.
Key work
1822 Théorie analytique de la chaleur (The Analytical Theory of Heat)
See also: Pythagoras • Trigonometry • Bessel functions • Elliptic functions • Topology • The Langlands Program
IN CONTEXT
KEY FIGURE
Pierre-Simon Laplace (1749–1827)
FIELD
Mathematical philosophy
BEFORE
1665 Calculus is developed by Isaac Newton to analyze and describe the motion of falling bodies and other complex mechanical systems.
AFTER
1872 Ludwig Boltzmann uses statistical mechanics to show how the thermodynamics of a system always results in an increase in entropy.
1963 Edward Lorenz describes the Lorenz attractor, a model that produces chaotic results with every tiny change to the initial parameters.
1872 American mathematician D
avid Wolpert disproves Laplace’s demon by treating the “intellect” as a computer.
In 1814, Pierre-Simon Laplace, a French mathematician who combined mathematics and science with philosophy and politics, presented a thought experiment now known as Laplace’s demon. Laplace never used the word “demon” himself; it was introduced in later retellings, evoking a supernatural being made godlike by mathematics.
Laplace imagined an intellect that could analyze movements of all atoms in the Universe in order to accurately predict their future paths. His experiment was an exploration of determinism, a philosophical concept that says that all future events are determined by causes in the past.
The orrery, a “clockwork universe” showing the movement of the celestial bodies in the Solar System, became a popular device after the publication of Newton’s universal theory of gravity.
Mechanical analysis
Laplace was inspired by classical mechanics—a field of mathematics describing the behavior of moving bodies, based on Isaac Newton’s laws of motion. In a Newtonian universe, atoms (and even light particles) follow the laws of motion, and bounce around in a jumble of trajectories. Laplace’s “intellect” would be capable of capturing and analyzing all of their movements; it would create a single formula that uses present movements to ascertain past and predict future ones.