by DK
Polynomial equations involving numbers and a single unknown quantity (x, and powers of x such as x2 and x3) are a powerful tool for solving real-world problems. An example of a polynomial equation is x2 + x + 41 = 0. While such equations can be solved approximately by repeated numerical calculations, solving them exactly (algebraically) was not achieved until the 1700s. The quest led to many mathematical innovations, including new types of numbers—such as negative and complex numbers—as well as modern algebraic notation and group theory.
Searching for solutions
The Babylonians and ancient Greeks used geometrical methods to solve problems that are now usually expressed by quadratic equations. In medieval times, more abstract algorithmic approaches were established, and by the 1500s, mathematicians knew certain relations between the coefficients of a polynomial equation and its roots, and had devised formulas to solve cubic (highest power 3) and quartic equations (highest power 4). In the 1600s, a general theory of polynomial equations, now called the fundamental theorem of algebra, took shape. This stated that an equation of degree n (where the highest power of x is xn) has exactly n roots or solutions, which may be real or complex numbers.
An algebraic equation is made up of variables and coefficients. The highest power of the equation determines how many solutions it has: in this case, there are three solutions.
Roots and permutations
In his Reflections on the algebraic resolution of equations (1771), French-Italian mathematician Joseph-Louis Lagrange introduced a general approach for solving polynomial equations. His work was theoretical—he investigated the structure of polynomial equations to find the circumstances under which a formula could be found for solving them. Lagrange combined the technique of using a related, lower-degree polynomial equation whose coefficients were related to the coefficients of the original equation with a striking innovation—he considered the possible permutations (reorderings) of the roots. Lagrange’s insight into the symmetries that arose from these permutations showed why the cubic and quartic equations could be solved by formulas, and showed (due to the different permutations of symmetries and roots) why a formula for the quintic equation needed a different approach.
Within 20 years of Lagrange’s work, Italian mathematician Paolo Ruffini began to prove that there was no general formula for the quintic equation. Lagrange’s investigation into permutations (and symmetries) formed the basis of the even more abstract and general group theory advanced by French mathematician Évariste Galois, who used it to prove why it is impossible to resolve equations of degree 5 or higher algebraically—that is, why there is no general formula for solving such equations.
JOSEPH-LOUIS LAGRANGE
Born Giuseppe Lodovico Lagrangia in Turin in 1736, Lagrange embraced his family’s French heritage and went by the French version of his name. As a young mathematician, self-taught, he worked on the tautochrone problem and developed a new formal method to find the function that solved such problems. At the age of 19, he wrote to Leonhard Euler, who recognized his talent. Lagrange applied his method, which Euler named the calculus of variations, to study a wide range of physical phenomena, including the vibration of strings. In 1766, at Euler’s recommendation, he was made Director of Mathematics at the Berlin Academy, and in 1787 he moved to the Académie des Sciences in Paris. Despite being an academic and a foreigner, Lagrange survived the French Revolution and Reign of Terror, and died in Paris in 1813.
Key works
1771 Reflections on the algebraic resolution of equations
1788 Analytical Mechanics
1797 Theory of analytic functions
See also: Quadratic equations • Algebra • The binomial theorem • Cubic equations • Huygens’s tautochrone curve • The fundamental theorem of algebra • Group theory
IN CONTEXT
KEY FIGURE
Georges Louis Leclerc, Comte de Buffon (1707–88)
FIELD
Probability
BEFORE
1666 Liber de ludo aleae (On Games of Chance) by Italian mathematician Gerolamo Cardano is published.
1718 Abraham de Moivre publishes The Doctrine of Chances, the first textbook on probability.
AFTER
1942–46 The Manhattan Project, a US-led body for developing nuclear weapons, makes extensive use of Monte Carlo methods (computational processes that model risk by generating random variables).
Late 1900s Quantum Monte Carlo methods are used to explore particle interactions in microscopic systems.
In 1733, the mathematician and naturalist George Leclerc, the Comte de Buffon, raised—and answered—a fascinating question. If a needle is dropped onto a series of parallel lines, all the same width apart, what is the likelihood that the needle will cross one of the lines? Now known as Buffon’s needle experiment, it was one of the earliest probability calculations.
An elegant illustration
Buffon originally used the needle experiment to estimate π (pi)—the ratio of a circle’s circumference to its diameter. He did this by dropping a needle of length l many times onto a series of parallel lines distance d apart, where d is greater than the needle’s length (d > l). Buffon then counted the number of times the needle crossed the line as a proportion of total attempts (p) and came up with the formula that π is approximately equal to twice the length of the needle l, divided by the distance (d) multiplied by the proportion of needles crossing the line: π ≈ 2l ÷ dp. The probability of the needle crossing a line can be calculated by multiplying each side of the formula by p, then dividing each side by π to get p ≈ 2l ÷ πd.
The relationship with π can be used in a number of probability problems. One example involves a quarter circle, inscribed in a square, which curves from the top left corner of the square to the bottom right. The bottom horizontal edge of the square is the x axis and the left vertical edge is the y axis, with a value of 0 in the lower left corner and 1 in the corners at each end of the curve. When two numbers between 0 and 1 are chosen at random as the x and y coordinates, whether the point will lie inside the quarter circle (success) or outside it (failure) can be deduced by examining , where a is the x coordinate and b is the y coordinate. The result is > 1 for points outside the curve and < 1 for points within it. The point is chosen at random, so could be anywhere in the square. Points on the line of the quarter circle can be counted as a success. The chance of “success” is πr2 (the area of a circle) ÷ 4. If the radius is 1, r2 = 1, so the area is just π; for a quarter circle, divide π by 4 to get approximately 0.78. The whole area is the area of the square, which is 1 × 1 = 1, so the probability of landing in the shaded area is approximately 0.78 ÷ 1 = 0.78.
Using pi, the probability of a randomly chosen point falling within the quarter circle in this square can be calculated as roughly 78 percent.
The Monte Carlo method
This problem is an example of a wider class of experiments that employ a statistical approach called the Monte Carlo method, a code name coined by Polish-American scientist Stanislaw Ulam and his colleagues for the random sampling used during secret work on nuclear weapons in World War II. Monte Carlo methods went on to be useful in modern applications, especially once computers made it far less time-consuming to repeat a probability experiment over and over again.
Buffon’s needle experiment demonstrated how probability can be connected to pi. Buffon classed needles as “successful” (pink) if they crossed a line when dropped, or “unsuccessful” (blue) if they didn’t, then calculated the probability of “success.”
In wind energy yield analysis, the predicted energy output of a wind farm during its lifetime is calculated, giving different levels of uncertainty, by using Monte Carlo probability methods.
GEORGES-LOUIS LECLERC, COMTE DE BUFFON
Born in Montbard, France, in 1707, Georges-Louis Leclerc was urged by his parents to pursue a career in law, but was more interested in botany, medicine, and mathematics, which he studied at the University of Angers, France. At the age
of 20, he explored the binomial theorem.
Independently wealthy, Buffon was able to write and study tirelessly, corresponding with many of the scientific elite of his day. His interests were wide-ranging, and his output was immense—on topics ranging from ship-building to natural history and astronomy. The comte also translated a number of scientific works.
Appointed keeper of the Jardin du Roi, the royal botanical gardens in Paris, in 1739, Buffon enriched its collections and doubled its size. He held the post until his death in Paris in 1788.
Key works
1749–1786 Histoire naturelle (Natural History)
1778 Les époques de la nature
(The Epochs of Nature
See also: Calculating pi • Probability • The law of large numbers • Bayes’ theorem • The birth of modern statistics
IN CONTEXT
KEY FIGURE
Carl Gauss (1777–1855)
FIELD
Algebra
BEFORE
1629 Albert Girard states that a polynomial of degree n will have n roots.
1746 The first attempt at a proof of the fundamental theorem of algebra (FTA) is made by Jean d’Alembert.
AFTER
1806 Robert Argand publishes the first rigorous proof of the FTA that allows polynomials with complex coefficients.
1920 Alexander Ostrowski proves the remaining assumptions in Gauss’s proof of the FTA.
1940 Hellmuth Kneser gives the first constructive variant of the Argand FTA proof that allows for the roots to be found.
This method of solving problems by honest confession of one’s ignorance is called Algebra.
Mary Everest Boole
British mathematician
An equation asserts that one quantity is equal to another, and provides a means of determining an unknown number. Since Babylonian times, scholars have sought solutions to equations, periodically encountering seemingly insoluble examples. In the 5th century BCE, Hippasus’s attempts to resolve x2 = 2 and his realization that was irrational (neither a whole number nor a fraction) are said to have led to his death for betraying Pythagorean beliefs. Some 800 years later, Diophantus had no knowledge of negative numbers, so could not accept an equation where x is negative, such as 4 = 4x + 20, where x is -4.
Polynomials and roots
In the 1700s, one of the most studied areas of mathematics involved polynomial equations. These are often used to solve problems in mechanics, physics, astronomy, and engineering, and involve powers of an unknown value, such as x2. The “root” of a polynomial equation is a specific numerical value that will replace the unknown value to make the polynomial equal 0. In 1629, French mathematician Albert Girard showed that a polynomial of degree n will have n roots. The quadratic equation x2 + 4x - 12 = 0, for example, has two roots, x = 2 and x = -6, both of which produce the answer 0. It has two roots because of the x2 term – 2 is the equation’s highest power. If any quadratic equation is drawn on a graph, these roots can be easily found: they are where the line touches the x axis. Although his theorem was useful, Girard’s work was hindered by the fact that he had no concept of complex numbers. These would be key to producing a fundamental theorem of algebra (FTA) for solving all possible polynomials.
Gerolamo Cardano encountered negative roots while working on cubic equations in the 1500s. His acceptance of these as valid solutions was an important step in algebra.
Complex numbers
The collection of all positive and negative, rational and irrational numbers together make up the real numbers. Some polynomials, however, do not have real-number roots. This was a problem faced by Italian mathematician Gerolamo Cardano and his peers in the 1500s; while working on cubic equations, they found that some of their solutions involved square roots of negative numbers. This seemed impossible, because a negative number multiplied by itself produces a positive result.
The problem was solved in 1572 when another Italian, Rafael Bombelli, set out the rules for an extended number system that included numbers such as alongside the real numbers. In 1751, Leonhard Euler investigated the imaginary roots of polynomials, and called the “imaginary unit,” or i. All imaginary numbers are multiples of i. Combining the real and the imaginary, such as a + bi (where a and b are any real numbers, and i = ), creates what is called a complex number.
Once mathematicians had accepted the necessity of negative and complex numbers for solving certain equations, the question remained as to whether finding roots of higher-degree polynomials would require the introduction of yet new types of number. Euler and other mathematicians, notably Carl Gauss in Germany, would seek to address this question, eventually concluding that the roots of any polynomial are either real or complex numbers—no further types of number are needed.
Imaginary numbers are a fine and wonderful refuge of the divine spirit.
Gottfried Leibniz
CARL GAUSS
Born in Brunswick, Germany, in 1777, Carl Gauss showed his mathematical talents early: aged only three, he corrected an error in his father’s payroll calculations, and by the age of five he was taking care of his father’s accounts. In 1795, he entered Göttingen University and in 1798, he constructed a regular heptadecagon (a polygon with 17 sides) using only a ruler and compasses—the biggest advance in polygon construction since Euclid’s geometry some 2,000 years earlier. Gauss’s Arithmetical Investigations, written at the age of 21 and published in 1801, was key to defining number theory. Gauss also made advances in astronomy (such as the rediscovery of the astroid Ceres), cartography, the study of electromagnetism, and the design of optical instruments. However, he kept many of his ideas to himself; a great number were discovered in his unpublished papers after his death in 1855.
Key work
1801 Disquisitiones Arithmeticae (Arithmetical Investigations)
Early research
The FTA can be stated in a number of ways, but its most common formulation is that every polynomial with complex coefficients will have at least one complex root. It can also be stated that all polynomials of degree n containing complex coefficients have n complex roots.
The first significant attempt at proving the FTA was made in 1746 by French mathematician Jean le Rond d’Alembert in his “Recherches sur le calcul intégral” (“Research on integral calculus”). D’Alembert’s proof argued that if a polynomial P(x) with real coefficients has a complex root, x = a + ib, then it also has a complex root, x = a - ib. To prove this theorem, he used a complicated idea now known as “d’Alembert’s lemma.” In mathematics, a lemma is an intermediary proposition used to solve a bigger theorem. However, d’Alembert did not prove his lemma satisfactorily; his proof was correct, but contained too many holes to satisfy his fellow mathematicians.
Later attempts to prove the FTA included those of Leonhard Euler and Joseph-Louis Lagrange. While useful to later mathematicians, these were also unsatisfactory. In 1795, Pierre-Simon Laplace tried an FTA proof using the polynomial’s “discriminant,” a parameter determined from its coefficients which indicates the nature of its roots, such as real or imaginary. His proof contained an unproved assumption that d’Alembert had avoided—that a polynomial will always have roots.
There are only two kinds of certain knowledge: awareness of our own existence and the truths of mathematics.
Jean d’Alembert
Jean d’Alembert was the first to attempt to prove the FTA. In France, it is called the d’Alembert–Gauss theorem, acknowledging the influence of d’Alembert on Gauss.
Gauss’s proof
In 1799, at the age of 21, Carl Friedrich Gauss published his doctoral thesis. It began with a summary and criticism of d’Alembert’s proof, among others. Gauss pointed out that each of these earlier proofs had assumed part of what they were trying to prove. One such assumption was that polynomials of odd degree (such as cubics and quintics) always have a real root. This is true, but Gauss argued that the point needed to be proved. His first proof was based on assumptions about algebraic curves
. Although these were plausible, they were not rigorously proved in Gauss’s work. It was not until 1920, when Ukrainian mathematician Alexander Ostrowski published his proof, that Gauss’s assumptions could all be justified. Arguably, Gauss’s first, geometric proof suffered for being premature—in 1799, the concepts of continuity and of the complex plane, which would have helped him explain his ideas, had not yet been developed.
Argand’s additions
Gauss published an improved proof of the FTA in 1816 and a further refinement at his golden jubilee lecture (celebrating 50 years since his doctorate) in 1849. Unlike his first geometric approach, his second and third proofs were more algebraic and technical in nature. Gauss published four proofs of the FTA, but did not fully resolve the problem. He failed to address the obvious next step: although he had established that every real number equation would have a complex number solution, he had not considered equations built from complex numbers such as x2 = i.
In 1806, Swiss mathematician Jean-Robert Argand found a particularly elegant solution. Any complex number, z, can be written in the form a + bi, where a is the real part of z and bi is the imaginary part. Argand’s work allowed complex numbers to be represented geometrically. If the real numbers are drawn along the x axis and the imaginary numbers are drawn along the y axis, then the whole plane between them becomes the realm of the complex numbers. Argand proved that the solution for every equation built from complex numbers could be found among the complex numbers on his diagram and that there was therefore no need to extend the number system. Argand’s was the first truly rigorous proof of the FTA.