I.

Introduction

Mathematics, a way of describing relationships between numbers and other measurable quantities. Mathematics can express simple equations as well as interactions among the smallest particles and the farthest objects in the known universe. Mathematics allows scientists to communicate ideas using universally accepted terminology. It is truly the language of science.

We benefit from the results of mathematical research every day. The fiber-optic network carrying our telephone conversations was designed with the help of mathematics. Our computers are the result of millions of hours of mathematical analysis. Weather prediction, the design of fuel-efficient automobiles and airplanes, traffic control, and medical imaging all depend upon mathematical analysis.

For the most part, mathematics remains behind the scenes. We use the end results without really thinking about the complexity underlying the technology in our lives. But the phenomenal advances in technology over the last 100 years parallel the rise of mathematics as an independent scientific discipline.

Until the 17th century, arithmetic, algebra, and geometry were the only mathematical disciplines, and mathematics was virtually indistinguishable from science and philosophy. Developed by the ancient Greeks, these systems for investigating the world were preserved by Islamic scholars and passed on by Christian monks during the Middle Ages. Mathematics finally became a field in its own right with the development of calculus by English mathematician Isaac Newton and German philosopher and mathematician Gottfried Wilhelm Leibniz during the 17th century and the creation of rigorous mathematical analysis during the 18th century by French mathematician Augustin Louis Cauchy and his contemporaries. Until the late 19th century, however, mathematics was used mainly by physicists, chemists, and engineers.

At the end of the 1800s, scientific researchers began probing the limits of observation, investigating the parts of the atom and the nature of light. Scientists discovered the electron in 1897. They had learned that light consisted of electromagnetic waves in the 1860s, but physicist Albert Einstein showed in 1905 that light could also behave as particles. These discoveries, along with inquiries into the wavelike nature of matter, led in turn to the rise of theoretical physics and to the creation of complex mathematical models that demonstrated physical laws. Einstein mathematically demonstrated the equivalence of mass and energy, summarized by the famous equation E=mc², in his special theory of relativity in 1905. Later, Einstein’s general theory of relativity (1915) extended special relativity to accelerated systems and showed gravity to be an effect of acceleration. These mathematical models marked the creation of modern physics. Their success in predicting new physical phenomena, such as black holes and antimatter, led to an explosion of mathematical analysis. Areas in pure mathematics—that is, theory as opposed to applied, or practical, mathematics—became particularly active.

A similar explosion of activity began in applied mathematics after the invention of the electronic computer, the ENIAC (Electronic Numerical Integrator and Calculator), in 1946. Initially built to calculate the trajectory of artillery shells, ENIAC was later used for nuclear weapons research, weather prediction, and wind-tunnel design. Computers aided the development of efficient numerical methods for solving complex mathematical systems.

Without mathematics to describe physical phenomena, we might be living in a world with beautiful art, literature, and philosophy, but no technology. Even the medical advances of the last 50 years might not have occurred. Science and technology, in their turn, have provided many of the problems that motivated progress in mathematics. Such problems include the behavior of weather systems, the motion of subatomic particles, and the creation of speedier and smaller computers that can perform multiple tasks simultaneously.

II.

Mathematics: The Language of Science

Experimental scientists observe phenomena and conduct experiments to obtain data about the way the universe behaves. Theoretical scientists generalize and draw conclusions from these results to form models of how the universe works. Mathematical scientists then study these models to understand their underlying principles and try to deduce what the models predict about unknown behavior or phenomena. Computational scientists use numerical simulations to study these models on computers. The cycle repeats as experimental scientists try to verify the predictions of mathematical and computational scientists through experiments. Social scientists also use mathematical techniques, primarily probability and statistics, to help resolve uncertainty about questions such as how various factors affect human behavior, how these variable factors are related, and how groups differ in their responses.

Mathematics attempts to capture the complexity of a problem using mathematical notation (signs and symbols) and concepts (theorems and proofs). Mathematical notation is a powerful tool, especially for representing entities, processes, or relationships that are impossible to visualize. For example, in modern geometry, mathematicians may work with more than three dimensions of space, even with infinite dimensions. Although these spaces are difficult to imagine, objects in these spaces can be studied through mathematics. Einstein’s discovery of relativity depended on studying objects in four dimensions, with time as the fourth dimension. Mathematicians develop simple corresponding models in two or three dimensions, then use the symbols and logic of mathematics to extend their intuition to infinite dimensions.

A.

Symbols, Equations, and Theories

Mathematics studies relationships using symbols (numbers or letters), logic, and formal proof. Equality is one of the most fundamental relationships that two objects can have. If two things are equal, and we know something about one object, we can then deduce the same thing about the other object. Expressions of equality, called equations, are one of the main subjects of mathematical analysis. We often express equality in terms of variable quantities, such as x and y. A main tool of mathematics involves transforming one form of equality to another by changing variables. In a very simple example, if we know that x = y and y = z, then we also know that x = z.

Mathematicians strive for simplicity and generality, which lie at the core of what they call elegance. Simplicity means the use of a minimal number of assumptions or hypotheses in a proof or theory. Generality is the ability to apply the mathematical theory to different situations. A 14th-century Franciscan friar, William of Ockham, expressed this principle when he said, “Entities should not be multiplied unnecessarily.” The principle of economy in logic is sometimes known as Ockham’s razor. To put it another way: If you have two competing theories that both explain the observed results, choose the one that is the simplest, until additional evidence comes along.

Scientists continue to search for one of the ultimate expressions of mathematical elegance: a unified field theory. Such a theory would describe the behavior of all things in the universe in a consistent set of equations and unify the four known interactions—the strong, weak, electromagnetic, and gravitational forces. Einstein hoped that a unified field theory could be found, and he worked on this project from 1928 until his death in 1955.

How do we reconcile these grand mathematical ideas that seek a fixed order with what we know about real life, where things are unpredictable, random events occur, and order and structure often disappear and are replaced by chaos? Mathematicians study real-world change, such as the behavior of weather systems, by means of chaos theory. They have determined that any nonlinear system (system that cannot be predicted on the basis of past behavior) that has sufficient variables (unknown quantities) can behave in a chaotic manner. Systems besides weather known to be chaotic include heart rhythms, the rise and fall of animal populations, and chemical reactions. In some cases chaotic behavior may barely be observable. Scientists long thought that if they could eliminate randomness from chaotic systems, the systems would then follow predictable rules. They know now that this is not the case.

B.

Pure Mathematics and Applied Mathematics

Mathematics, the language of science, has two dialects: pure mathematics and applied mathematics. Both kinds of mathematics are used to solve problems. Pure mathematics is the study of abstract relationships, whereas applied mathematics applies mathematical analysis to real-world problems, such as the rate of global warming. The relationship between pure and applied mathematics is a complex one, and the boundary between the two is constantly shifting.

B.1.

Pure Mathematics

Pure mathematics is more abstract than applied mathematics. It emphasizes rigorous proof, manipulates symbols rather than numbers, and seeks to obtain the most general results possible with the fewest possible assumptions. British mathematician G. H. Hardy, one of the foremost spokesmen for pure mathematics, represents this approach in his classic book A Mathematician's Apology (1941).

Pure mathematics began to come into its own during the 1800s when rigorous proof and detailed analysis became more common. The beauty of the mathematical proof—that is, its simplicity and its brevity—became just as important as the result, more important even than the specific application that inspired it. British mathematician and logician Bertrand Russell wrote in 1910, “Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.” German mathematician Hermann Weyl remarked in the early 1900s, only partly in jest, “My work has always tried to unite the true with the beautiful and when I had to choose one or the other, I usually chose the beautiful.”

B.2.

Applied Mathematics

Applied mathematics, while just as concerned with rigorous mathematical methods, emphasizes applications. Applied mathematics has had close ties with the sciences and engineering throughout its history. Applied mathematicians believe that new mathematical ideas and areas of study can come from using mathematics to solve problems in physics, chemistry, biology, medicine, engineering, and technology. Much of the current research in applied mathematics takes place outside traditional mathematics departments.

Subject areas in applied mathematics often overlap areas associated with other fields, including economics, physics, mechanics, and information theory. For example, mathematicians who study the structure of matter and the behavior of subatomic particles overlap in their area of research with physicists. Some areas of applied mathematics depend heavily on pure mathematics. Numerical analysis, which studies computational methods for solving mathematical problems, relies on the pure mathematical areas of partial differential equations and variational methods. Other areas, such as computer science, are as broad as the entire field of mathematics.

Applied mathematics is older than pure mathematics because it was used in areas that formed the core of early physics research, such as mechanics, optics, and fluid dynamics. As mathematical tools became more powerful, these areas of physics became more mathematically based.

B.3.

Historical Differentiation

One force that tends to differentiate mathematical disciplines is research funding. In the United States, pure mathematics has been funded by the National Science Foundation (NSF) and carried out in universities as well as at a few industrial research and development facilities, such as IBM Research Labs and AT&T Bell Laboratories. Within the government, the National Security Agency (NSA) has been a big supporter of pure mathematics areas related to cryptography, such as algebra, graph theory, and number theory.

Since the end of World War II in 1945, government laboratories have employed thousands of pure and applied mathematicians. These facilities include Los Alamos National Laboratory, Oak Ridge National Laboratory, Brookhaven National Laboratory, and Lawrence Berkeley National Laboratory and Lawrence Livermore National Laboratory. The military, through the Office of Naval Research, the Air Force Office of Scientific Research, and the Army Research Office, has also supported both pure and applied mathematics in the United States. The applied mathematics community has received large amounts of research funding from the energy, computing, and communications industries.

The distinction between pure and applied mathematics is not rigid, however. Many mathematicians receive their training in pure mathematics, then become interested in applying their expertise to other areas. Other mathematicians seek to build links between different areas of mathematics. Interdisciplinary mathematics, in which methods from more than one area of mathematics are used, is one of the fastest growing areas of mathematics.

Trends also influence the direction of mathematics. A research area can become popular, as catastrophe theory did in the 1970s and 1980s, then virtually disappear. Mathematical research has traditionally been tied to individuals, schools, and even countries. In the 1800s new mathematical areas were started by individuals in the great schools of Europe and the United States. These universities included Göttingen (Germany), Moscow (Russia), Paris (France), Cambridge (Britain), and Princeton University and the University of Chicago in the United States. As pioneering researchers trained students in mathematics, their universities became associated with particular areas of mathematics. Because many of these universities were the leading institutions in their countries, the countries themselves became associated with different mathematical interests.

III.

Branches of Mathematics

The American Mathematical Society, a professional organization of mathematicians, classifies subject areas of mathematics as either pure mathematics, applied mathematics, or borderline areas. Out of the many subject areas of mathematics, this article describes ten significant areas.

A.

Arithmetic

Arithmetic, one of the oldest branches of mathematics, arises from the most fundamental of mathematical operations: counting. The arithmetic operations—addition, subtraction, multiplication, division, and placeholding—form the basis of the mathematics that we use regularly. In many countries arithmetic is the primary area of mathematical study during the first six years of school.

Although arithmetic itself is not an area of mathematics research, research on how best to teach arithmetic is crucial to the field of mathematics education. Models of learning and mastering the basics of arithmetic are often used in cognitive science—the study of the processes of acquiring, storing, and using knowledge. Cognitive sciences encompass a range of activities, including the design of computer-aided instructional systems and the study of artificial intelligence. Arithmetic and logic also form the basis for all computer software—the instructions that tell computers what to do.

B.

Algebra

Algebra is the branch of mathematics that uses symbols to represent arithmetic operations. One of the earliest mathematical concepts was to represent a number by a symbol and to represent rules for manipulating numbers in symbolic form as equations. For example, we can represent the numbers 2 and 3 by the symbols x and y. From observation we know that it does not matter in which order we add the numbers (2 + 3 = 3 + 2), and we can represent this equivalence as the equation x + y = y + x. The equation is valid no matter what numbers x and y represent. Because algebra uses symbols rather than numbers, it can produce general rules that apply to all numbers. What most people commonly think of as algebra involves the manipulation of equations and the solving of equations.

An area of mathematics research is also called algebra, or modern algebra. It developed after the discovery that laws such as the commutative law (x + y = y + x) held true not only for the addition of real numbers (rational and irrational numbers) but could extend to more complex operations and objects. Interest eventually focused on the concepts themselves and the conclusions that could be drawn about sets of objects with certain properties. Among the objects studied by modern algebra are groups, rings, and fields. Algebra also can be combined with other areas of pure mathematics such as geometry and a branch of geometry called topology.

C.

Geometry

Geometry is the branch of mathematics that deals with the properties of space. Students in high school study plane geometry—the geometry of flat surfaces—and may move on to solid geometry, the geometry of three-dimensional solids. But geometry has many more fields, including the study of spaces with four or more dimensions.

Geometry was systematized by the ancient Greeks, especially Pythagoras and Euclid. It has been admired from ancient times onward for its simplicity and elegance. Early Greek philosophers believed that conic sections (ellipses, circles, and hyperbolas) were the foundations of the universe. Newton wrote his Principia Mathematica (1687), one of the great mathematical treatises, almost entirely using geometry and trigonometry, rather than the calculus he had just invented. He could not yet use calculus because no one else would have understood the treatise.

Astronomy was one of the earliest sciences to implement the ideas of geometry. Astronomers built mechanical devices consisting of gears and fixed spheres that described the orbits of celestial bodies with astonishing accuracy. German mathematician Johannes Kepler used geometry in the late 16th and early 17th centuries to argue that the universe was not Earth-centered and to prove that planets revolved around the Sun in elliptical orbits.

The creation of a coordinate system (pair of intersecting lines, or axes) to describe the equations of geometry led to analytical geometry. This area, which merged geometry and algebra, was developed in the early 17th century by French philosopher and mathematician René Descartes. The discovery of analytical geometry was critical to the development of calculus later in that century. By applying calculus to geometry, mathematicians recognized that curved surfaces had their own intrinsic geometry, leading to the development of differential geometry. This idea became important in modern physics. Einstein, for example, used it to show that gravitation results from the geometric curvature of the four-dimensional space-time continuum of the universe.

D.

Trigonometry

The study of triangles in plane geometry led to trigonometry. Originally trigonometry was concerned with the measurement of angles and the determination of three parts or a triangle (sides or angles) when the remaining three parts were known. If we know two angles and the length of one side of a triangle, for example, we can compute the other angle and the length of the remaining sides. Trigonometry uses triangles because all shapes in plane geometry can be broken down into triangles.

The relationships between the sides and angles of triangles can be expressed as ratios called trigonometric functions and used in calculations. Similar triangles—triangles with the same angles—have the same trigonometric functions because the lengths of their sides are in the same ratio. Right triangles (triangles with one angle of 90 degrees) are used to define three important trigonometric functions: sine (usually abbreviated sin), cosine (cos), and tangent (tan). As mathematics progressed the properties and applications of the trigonometric functions, or ratios associated with angles, became more important. The relationships between the ratios have many applications in the fields of physics and engineering. More complex applications result from the periodic (regularly recurring) properties of trigonometric functions and apply to physical phenomena, such as light, sound, and electricity.

Most of the elementary applications of trigonometry make use of triangles in a plane. Three-dimensional trigonometry is concerned with relationships between triangles drawn on the surface of a sphere and with solid angles—that is, volumes that extend from angles on the surface on a sphere.

E.

Calculus

Calculus is the branch of mathematics concerned with the study of rates of change, slopes of curves at given points, areas and volumes bounded by curves, and similar problems. Scientists apply calculus to numerous problems in physics, astronomy, mathematics, and engineering. In recent years calculus has also been applied to problems in business, the biological sciences, and the social sciences. The development of calculus in the 17th century made possible the solution of many problems that had been insoluble by the methods of arithmetic, algebra, and geometry. These problems include the determination of Newton’s three laws of motion (see Mechanics) and the theory of electromagnetism.

Calculus consists of two main branches: differential calculus and integral calculus. Differential calculus deals with the rate at which quantities change. Integral calculus develops methods for finding the areas enclosed by curved boundaries. In both branches two concepts are central: function and limit.

Many relationships in nature and in mathematics can be expressed by functions. For example, a car moving at a speed of 50 mph travels a distance that changes constantly, depending on how long the car has traveled. Both distance and time are variables, but because the distance covered depends on the time of travel, distance can be represented as a function of time. A coal mine grows hotter as one descends, and so temperature can be expressed as a function of depth. A mathematical curve takes on new values of y as the value of x changes. When the value of y is determined by the value of x, we say that “y is a function of x” and we write y = f(x). A function is a rule, or equation, that tells us how to compute the y values given the x values (or vice versa). But unlike in algebra where the variables are static, the variables in calculus are constantly changing.

A key characteristic of calculus is that its solutions involve the idea of a limit. If you start out with a whole pie and repeatedly give away half of what is left, the sum of the amounts given away can never exceed 1 (the whole pie). At the same time, no matter how much you give away, a small amount will remain. Thus, you can never give away 1 entire pie. The sum of the series of pieces given away—y, ‚, ˆ, w, and so forth—approaches but never reaches 1, and so 1 represents the limit. If we call the sum of the series of pieces S, then S is a function of the number of pieces (n) in the series, or S = f(n), and the limit of f(n) as n approaches infinity is 1. Solutions in integral calculus involve breaking irregular areas and volumes into ever-smaller parts, where the notion of limit proves useful. Sir Isaac Newton was the first to clarify the notion of a limit and apply it to calculus.

F.

Probability and Statistics

Probability and statistics deal with events or experiments where outcomes are uncertain, and they assess the likelihood of possible outcomes. Probability began in an effort to assess outcomes in gambling. We know from experience that if we toss a coin enough times, heads will come up about half the time and tails about half the time. The more trials, the more closely the outcome approaches y—that is, as the limit of trials approaches infinity, the probability is y.

In simple situations, such as the toss of a coin, it is relatively easy to assign probabilities based on intuition. When we consider more complicated events, intuition becomes less reliable. Various methods of calculation then come into play to assign mathematical probabilities to outcomes. For example, permutations and combinations—arrangements of the outcomes involved—are used to analyze many problems in probability. Probability has become an indispensable tool in statistics, physics, biology, social science, business, and many other fields.

Statistics is the organization and analysis of data for the purpose of simplification, comparison, and prediction. Statistical methods are used throughout most branches of human knowledge. A scientist may use statistics to bolster a theory, design an experiment, or test the significance of experimental results. Someone in business uses statistics to estimate sales and to control quality. A scholar may apply statistical methods to literary works. For example, he or she may use data on the frequency of particular words in order to determine the unknown author of a poem.

One of the best-known uses of statistics is as a predictor. The data collected from a sample group are used to predict the results from a larger group. Politicians use polls to evaluate their campaigns; biologists study animal populations by banding small numbers of captured animals; manufacturers maintain quality control on production lines by examining small samples of the manufactured products. The results of statistics are often given in the form of estimates together with some probability about how good the estimate is.

The great usefulness of statistics as a predictor is possible because of the regularity exhibited by many natural processes and populations that at first glance appear to be highly irregular. If we measured the heights of North American adults, for example, and presented the results in a bar graph, certain regularities would begin to appear as the number of people being measured grew. The bar graph would become more and more regular, symmetrical, and bell-shaped. This curve has many names, including the normal distribution curve, the Gaussian distribution, and the bell-shaped curve.

Statisticians use the term random variable to describe the outcome of an event that is unpredictable in advance, such as the percentage of adults who measure 5 ft 8 in or the effect of a lifetime of smoking on health. Statisticians are concerned with the variability of their data—that is, by how much it deviates from the expected distribution found in a normal distribution curve. They ask whether most of the outcomes cluster around the middle, forming a high curve, or scatter, forming a low curve. One measure of variability is called the standard deviation. Statisticians determine whether different variables increase together, such as packs of cigarettes smoked daily and likelihood of lung cancer, or whether they lack correlation. The study of the behavior of random variables is known as statistics.

G.

Set Theory and Logic

Set theory is the branch of mathematics that seeks to establish statements that are true of sets, regardless of the kind of objects that make up the set. A set is a group of objects with a well-defined criterion for membership so that we can say definitely whether an object belongs to the set or not. The terminology and many of the results of set theory are used in symbolic logic, geometry, the theory of probability, and mathematical analysis.

Set theory and logic are closely related as we can see in diagrams in which circles represent sets and the relationships between them. The union of two sets A and B, written AÈB, is the set of all elements that belong to A or B. The intersection, AÇB, is the set of all elements that belong to A and B. The complement of a set A consists of elements not in A. The formal manipulation of expressions involving and, or, and not is called Boolean logic, and was developed in 1847 by British mathematician George Boole.

German mathematician Georg Cantor provided the first formulation of set theory. He extended the intuitive concept of a set to include the possibility of sets containing an infinite number of objects, and he showed that it is possible to conceive of infinities of different “sizes.”

Set theory can be used to provide an axiomatic, or logical, foundation for almost all of mathematics. The number of elements in a set, n(S), can be used to produce all the positive integers (whole numbers), 1, 2, 3, and so forth. Addition can be related to the operation of union. Subtraction, the inverse of addition, produces the negative integers, -1, -2, -3, and so on. Multiplication can be shown to be repeated addition, and introducing division, its inverse, produces rational numbers. Metamathematics examines the extent to which mathematics can proceed using set theory alone. The classic work in the field of metamathematics is the Principia Mathematica (1910-1913) by British mathematicians and philosophers Bertrand Russell and Albert North Whitehead.

H.

Number Theory

Number theory is the branch of mathematics that deals with the properties of numbers, primarily integers—whole numbers that may be positive, negative, or zero. One of the earliest problems studied in algebra was the division of integers: Is it possible to write an integer as the product of smaller integers? The integer 6, for example, can be written as 2 x 3. If an integer can be written in this way, it is called a composite number; if not, it is called a prime number. The first few prime numbers are 1, 2, 3, 5, 7, 11, 13, 17, 19, and 23. There are an infinite number of prime numbers.

Investigating prime numbers has generated many problems in pure mathematics. In the 19th century German mathematician Georg Friedrich Bernhard Riemann analyzed the distribution of prime numbers, and conjectured that they were related to the roots of a function now called the Riemann zeta function. The so-called Riemann conjecture has not yet been proved. Another famous problem in number theory is known as Fermat’s last theorem after 17th-century French mathematician Pierre de Fermat. Fermat stated that he had found a remarkable proof demonstrating that the equation aⁿ + bⁿ = cⁿ cannot be solved in integers when n is greater than 2. But he did not write down his proof, and it remained one of the most famous unsolved problems in mathematics until English mathematician Andrew Wiles proved the theorem in 1994.

The extreme difficulty in factoring large integers into prime numbers is the basis for modern cryptography. Breaking codes is essentially equivalent to finding the factors of large numbers—that is, the quantities that can be multiplied to yield the large number. Not surprisingly, the U.S. government ranks as one of the world’s largest employers of number theorists.

I.

Systems Analysis

The mathematical study of systems is called systems analysis. It plays a vital role in the understanding of communications networks and computing networks. In most mathematical models a functional relationship exists between two quantities—that is, we can express one quantity in terms of the other by an equation y = f(x). Not all relationships are this simple, however. One approach to complex systems is to focus on the connectivity between quantities, rather than on the quantities themselves. In understanding the Internet, for example, the network is of interest rather than the individual computers attached to it. Similarly, in studying communications networks, the network itself is of interest rather than the hardware involved or actual information that is transmitted. The mathematical study of the connectivity between objects is sometimes called network analysis.

Ecosystems—interdependent organisms and their environments—that regulate life on this planet form a rather different network. Mathematical ecology and population dynamics are new research areas that use mathematics to study ecosystems and other biological systems and their behavior over long periods of time. Systems analysis has even played a role in the study of learning. One model for learning, called a neural network, simulates the interconnection of the neurons in the brain.

J.

Chaos Theory

Despite advances made in systems analysis, many systems remain beyond the reach of current mathematics. Chaos theory, a relatively new area of mathematics, concerns the analysis of unpredictable systems that are extremely sensitive to initial conditions. One important example of a chaotic system is climate. Global climate modeling is an area of mathematical research that seeks to develop models for predicting the weather, given accurate data from weather satellites orbiting Earth. The problem in developing such models arises not from lack of data but from the difficulty of modeling such a complex system (Earth’s atmosphere) with a small number of equations. In such models even a thousand equations may be considered small. The solution of these equations is very sensitive to changes in the initial conditions. The term initial conditions refers to all the measurements at the starting time. A tiny inaccuracy in a single measurement of a chaotic system—such as a temperature variation of a fraction of a degree—can produce large errors in solutions to the model’s equations and predictions.

Meteorologist Edward Lorenz tried to model climate in a series of equations during the 1960s. In doing so, he produced a chaotic system of three related differential equations, now known as a Lorenz attractor, or strange attractor. Through his models he discovered the sensitivity of chaotic systems to initial conditions, which he phrased in the question “Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?”

The Lorenz attractor is an example of a fractal, a pattern produced by applying a function repeatedly, much like pushing a button on a calculator over and over. The sequence {x, f(x), f(f(x)), f(f(f(x))), ...}, when graphed in two dimensions, gives rise to beautiful, complex geometric images such as the Mandelbrot set pictured in this article. These fractal images are named after Benoit Mandelbrot, a Polish-born French mathematician who developed fractal geometry and coined the word fractal. The interesting relationship among fractals, chaos, and unstable phenomena such as turbulence is the subject of a field called nonlinear dynamics.

IV.

History of Mathematics

Counting was the earliest mathematical activity. Early humans needed counts to keep track of herds and for trade. Primitive counting systems almost certainly used the fingers of one or both hands, as evidenced by the predominance of the numbers 5 and 10 as the bases for most number systems today. The first advances in arithmetic were the conceptualization of numbers and the invention of the four fundamental operations: addition, subtraction, multiplication, and division. The earliest advances in geometry dealt with simple concepts such as the line and the circle. Further progress in mathematics had to await the Babylonians and Egyptians of about 2000 bc.

A.

Mathematics in Antiquity

The Babylonians of ancient Mesopotamia and the ancient Egyptians left the earliest records of organized mathematics. Arithmetic dominated their mathematics. In geometry, measurement and calculation were emphasized, with no trace of concepts such as axioms or proofs.

A.1.

Babylonia

Our knowledge of Babylonia comes from well-preserved clay tablets on which people wrote with wedge-shaped marks known as cuneiform. The earliest tablets date from about 3000 bc. Much of the mathematics on the tablets involved commerce. The Babylonians used arithmetic and simple algebra to exchange money and merchandise, compute simple and compound interest, calculate taxes, and allocate shares of a harvest to the state, temple, and farmer. The building of canals, granaries, and other public works also required using arithmetic and geometry. Calendar reckoning, used to determine the times for planting and for religious events, was another important application of mathematics.

The division of the circle into 360 parts and the division of the degree and the minute each into 60 parts originated in Babylonian astronomy. The Babylonians also divided the day into 24 hours, the hour into 60 minutes, and the minute into 60 seconds. Their number system was based on the number 60. The wedge-shaped symbol for 1 was repeated for numbers up to 9. Any number from 11 to 59 could be written as a combination of the symbol for 10 and the symbol for 1. For numbers 60 and higher the Babylonians used a symbol for position. The principle of position, or place value, was a significant advance in calculation. It enabled the same symbol to represent different numerical values depending on its position. The Babylonian system enabled them to represent fractions as well, but they lacked a symbol for zero, which led to ambiguities.

The Babylonians devised tables of reciprocals (numbers that yield 1 when multiplied, such as 3 and €), tables of squares and square roots, tables of cubes and cube roots, and tables of compound interest. They had a good approximation of Ã. Cuneiform tablets dealing with algebraic and geometric problems show that the Babylonians had, in effect, the quadratic formula for solving equations and could solve problems that involved ten unknowns in ten equations.

Beginning about 700 bc the Babylonians used mathematics to study the motion of the Moon and planets. This enabled them to predict the daily positions of the planets, knowledge as important for astrology as for astronomy.

In geometry, the Babylonians knew a few relationships such as the proportionality of corresponding sides in similar triangles. They could solve problems using the Pythagorean theorem and were aware that an angle inscribed in a semicircle is a right angle. They had rules for the areas of simple plane figures including regular polygons and for the volumes of simple solids. They used 3 as the value of the ratio known as pi (p), a mathematical constant that is equal to about 3.14.

A.2.

Egypt

Although the Egyptians carved hieroglyph numerals onto some of their monuments, the earliest texts we have for Egyptian mathematics are two papyruses composed about 1800 bc. The papyruses contain problems in arithmetic and geometry, including practical problems dealing with the amount of grain needed to make a given quantity of beer and the amount of grain of one quality needed to give the same result as grain of another quality.

We also know that the Egyptians used mathematics to determine wages, find the areas of fields and the volumes of granaries, assess taxes, and calculate the number of bricks needed for particular structures. In addition, the Egyptians used mathematics in astronomy for calendar reckoning. Their calendar helped them set the times of religious holidays and predict the annual flooding of the Nile.

The Egyptians based their number system on the number 10, using separate hieroglyph symbols for the successive powers of 10 (1, 10, 100, and so forth). They wrote the symbol for 1 five times to represent the number 5, the symbol for 10 six times to represent the number 60, and the symbol for 100 three times to represent the number 300. Together, these 14 symbols represented the number 365. Addition was done by totaling the units—10s, 100s, and so forth—separately in the numbers to be summed. Multiplication was based on successive doublings, and division was the inverse of the process. The Egyptians represented fractions by using the symbol for mouth, meaning “part,” above the number symbols.

After the Egyptians began writing on papyrus in a script called hieratic, they developed individual symbols for every number from one to 10, every tenth number up to 100 (20, 30, and so forth), every 100th number up to 1,000 (200, 300, and so forth), and every 1,000th number. Although this system meant more symbols to memorize, it enabled the Egyptians to write numbers more compactly.

In geometry the Egyptians had rules for areas of rectangles, triangles, trapezoids, and the circle, as well as formulas for certain volumes, including rectangular prisms (such as a brick), cylinders, and pyramids. To find the area of a circle (pr²), the Egyptians used a value of about 3.16 for p, which was closer than the Babylonian value of 3.

A.3.

Maya Mathematics

Most records of Maya achievements in mathematics were destroyed after the arrival of Spanish conquerors in Middle America during the 1500s, but four codices (manuscript volumes) remain. Although the dates of Maya achievements in mathematics are difficult to determine, these accomplishments merit attention. The Maya used a base-20 number system, which probably descended from early times when people counted on both fingers and toes. The Maya may have been the first people to employ a special symbol for zero. They used two types of systems for numerals. One employed hieroglyphs, while the more commonly used system employed a dot for 1, a bar for 5, and a shell-like symbol for zero.

The calendar was extremely important in Maya civilization, and the Maya developed two. One was based on the Sun and had 365 days. The other was a sacred almanac with 260 days divided into 13 months of 20 days each. The almanac was used for predicting lucky and unlucky days. Scholars have speculated on the reasons for the 260-day calendar and believe it may be related to other astronomical data compiled by the Maya. The Maya calculated the length of the lunar month and the solar year with remarkable precision. See also Maya Civilization.

A.4.

Greek Mathematics

The Greeks adopted elements of mathematics from the Babylonians and the Egyptians. The new element in Greek mathematics was the invention of an abstract mathematics founded on a logical structure of definitions, axioms (propositions accepted as self-evident), and proofs. According to later Greek accounts, this development began in the 6th century bc with Thales of Miletus and Pythagoras of Sámos. The mathematics that had existed before their time was a collection of conclusions based on observation. In the Greek system of deductive proof, on the other hand, a new statement was logically derived from accepted premises.

The Greeks’ insistence on deductive proof was an extraordinary step. No other civilization had conceived the idea of establishing conclusions exclusively by deductive reasoning based on explicitly stated axioms.

A.4.a.

Number System

The Greek number system was based on the alphabet. The Attic system, in use from 600 bc to 200 bc, used a stroke for 1 and the initial letters of the words for 5, 10, 100, 1,000, and 10,000—namely, the initials of pente, deka, hekaton, khilioi, and myrioi—to represent the respective numbers. A later system assigned number values to the 24 letters of the Greek alphabet and to 3 other letters that were no longer used. The letters could be combined to form numbers through 999. For higher numbers, a stroke preceding the initial letter (1 through 9) indicated a multiple of 1,000 (1,000 through 9,000). For 10,000 and above, the symbol M indicated that the numeral below should be multiplied by 10,000. See also Numerals.

A.4.b.

Pythagoras and the Pythagoreans

Pythagoras taught the importance of studying numbers in order to understand the world. We know of his achievements only from his disciples, the Pythagoreans, who made important discoveries about number theory and geometry. The Pythagoreans represented whole numbers by using arrangements of dots or pebbles, and classified these numbers according to the shapes produced. (The English word calculation is derived from the Greek word for stone or pebble.) The numbers 3, 6, 10, and so on were called triangular numbers because the pebbles could be arranged to form triangles. The numbers 4, 9, 16, and so on were called square numbers because the pebbles could be arranged as squares.

From these simple geometrical arrangements some properties of the whole numbers emerged. The Pythagoreans concluded that the sum of two consecutive triangular numbers is always a square number. They called a perfect number one that equaled the sum of its divisors, for example 6 (the sum of 1, 2, and 3), 28 (the sum of 1, 2, 4, 7, and 14), and 496 (the sum of 1, 2, 4, 8, 16, 31, 62, 124, and 248).

To the Pythagoreans a number represented more than a quantity. The number 2 suggested diversity and so was identified with opinion. Four represented justice because it was the first number that was the product of equals (2 x 2). The identification of the square number 4 with justice continues in the phrase square shooter, meaning someone who acts in an impartial and straightforward manner.

The Pythagoreans discovered that the sum of certain pairs of square numbers is also a square number. Thus, the sum of 9 (the square of 3) and 16 (the square of 4) is 25 (the square of 5). Similarly, the sum of 25 (the square of 5) and 144 (the square of 12) is 169 (the square of 13). The numbers whose squares add up to another square—such as 3, 4, and 5 or 5, 12, and 13—are called Pythagorean triples. These triples have a geometric interpretation; if the two smallest numbers of a triple are the lengths of the arms of a right triangle, then the third is the length of the hypotenuse. This interpretation suggested the more general fact, known today as the Pythagorean theorem, that in any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

In examining the right triangle whose two arms are each one unit long, the Pythagoreans were disturbed to discover that the length of the hypotenuse was Ã, a quantity they sought in vain to express as a ratio of whole numbers. Quantities not expressible as such a ratio they termed incommensurable; the modern term is irrational. Around 300 bc Euclid proved that à is incommensurable. The Pythagoreans dealt with irrational numbers by thinking of all quantities geometrically. When 1 and à are treated as lengths, the distinction between whole number and irrational number vanishes.

A.4.c.

The Importance of Geometry

The classical Greeks solved equations involving unknown quantities by using a series of geometrical constructions. Constructions called for adding, subtracting, multiplying, and dividing lines and taking the square root of a line; the method is referred to as geometrical algebra. We still speak of 25 as “the square of 5” and of 27 as “the cube of 3.”

The casting of problems in geometrical form had several major consequences. It separated numbers and geometry, for only geometry could handle incommensurable ratios. Geometry became the basis for almost all rigorous mathematics until at least 1600. As late as the 18th century, when algebra and calculus had already been well developed, rigorous mathematics meant geometry, and geometer meant “mathematician.”

Among great geometers of the 5th century bc were the philosopher Democritus and the mathematician Hippocrates of Chios. Democritus, who is better known for his theory that all matter is made up of atoms, discovered the correct formula for the volume of a pyramid. Hippocrates made discoveries related to the problem of squaring the circle—that is, constructing a square equal in area to a given circle. Two other problems that originated during the 5th century were those of trisecting (dividing into three equal parts) an angle and doubling a cube—that is, constructing a cube that is double in volume to a given cube. These problems were critical to the development of geometry, and the ancient Greeks solved all three in a variety of ways.

A.4.d.

Eudoxus and Euclid

In the 4th century bc Eudoxus of Cnidus introduced the notion of magnitude as a ratio rather than an exact number for dealing with entities such as line segments and angles. He also discovered a method for determining the areas and volumes of curved figures, such as the circle, by successive approximations. Through this method of exhausting all possibilities, he closely approximated the area of a circle by inscribing the circle in a polygon. As the number of sides in the polygon increased, the closer its area came to the area of the circle. With his method of exhaustion, Eudoxus took the first step in the creation of calculus.

Eudoxus also provided the first astronomical theory to account for observed planetary motions. His was a purely mathematical theory that showed how combinations of rotating spheres of varying radii and axes of rotation could explain the seemingly irregular motion of the Sun, Moon, and planets.

About 300 bc Euclid, a Greek mathematician who taught in Alexandria, Egypt, organized the work of many Greek mathematicians in a masterful work called the Elements. The work of Eudoxus had established the deductive organization of mathematics on the basis of axioms. From a few well-chosen axioms Euclid deduced some 500 theorems comprising all the important results of Greek mathematics to that time. Euclid began by defining terms such as line, angle, and circle. He then stated ten self-evident truths such as “The whole is greater than any of its parts.” From these ten axioms he was able to deduce all the theorems. Though mathematicians generally regarded the Elements as a model of rigor until well into the 19th century, it had serious defects, including the unconscious use of unstated assumptions.

A.4.e.

Greek Mathematics after Euclid

The character of Greek mathematics changed after about 300 bc during the so-called Alexandrian period, the period of Greek civilization following the conquests of Alexander the Great. Alexandrian mathematics was a fusion of classical Greek mathematics and Babylonian and Egyptian mathematics. In general, the mathematicians of the Alexandrian period were more inclined toward engineering than philosophy. The great Alexandrian mathematicians—Eratosthenes, Archimedes, Apollonius of Perga, Ptolemy, Diophantus, and Hipparchus—displayed the Greek genius for theoretical abstraction but were nevertheless willing to apply their talents to practical problems and quantitative (measurable) knowledge.

Eratosthenes found a simple method of accurately calculating the circumference of the earth in the 3rd century bc, and he also devised a calendar that called for an extra day every fourth year. Archimedes, the greatest mathematician of antiquity, produced theorems on complicated areas and volumes and proved them rigorously. He sought exact answers and found upper and lower limits for the values of irrational numbers. Thus, working with a 96-sided polygon, he determined that the value of p lay between and . Archimedes proved some theorems that contained new geometrical algebra. For example, he stated the problem of cutting a given sphere by a plane so that the volumes of the segments correspond to one another in a given ratio. He then solved the problem geometrically by finding the intersection of a parabola and a rectangular hyperbola. Apollonius, a younger colleague of Archimedes, produced an eight-book treatise on conic sections that established the names of the sections: ellipse, parabola, and hyperbola. The work also provided the basic treatment of the geometry of conic sections until the time of French philosopher and scientist René Descartes in the 17th century.

Archimedes was also the greatest mathematical physicist of ancient times. He used geometrical arguments to prove statements in mechanics. His book Floating Bodies is a foundation work in hydrostatics (the study of liquids at rest). Legend has it that while bathing Archimedes discovered the principle that a body immersed in water is buoyed up by a force equal to the weight of the water displaced, and in his elation he ran out naked into the street shouting “Eureka!” (“I have found it”).

During the Alexandrian period arithmetic and algebra were, for the first time, studied independently of geometry. The first sizable book in which arithmetic was treated independently was the Introduction to Arithmetic of Nicomachus, written about ad 100. Its importance for arithmetic was comparable to that of Euclid’s Elements for geometry. It remained the standard text for 1,000 years because it presented a systematic, clear, and comprehensive treatment of the integers and ratios of integers. While repeating many Pythagorean statements, the work went beyond them in seeing, though not proving, general relations. Diophantus, sometimes called the father of algebra, wrote an outstanding work on algebra about ad 250. He introduced symbolism into algebra and studied solutions to algebraic equations with several unknowns, a field now called Diophantine analysis.

The supreme achievement of the Alexandrian mathematicians was the creation of a quantitative astronomy. Hipparchus developed tables of trigonometric ratios in the 2nd century bc, for which he is sometimes credited as having invented trigonometry. His method rested on the geometric theorem that if two triangles are similar, the ratio of the lengths of any two sides of one triangle equals the corresponding ratio of the other triangle. With the tables and a readily measured distance on Earth, he was able to calculate the circumference of the Earth and the distance to the Moon. He found the distance to the Moon equal to between 59 and 67 times the radius of the Earth; the correct figure is 60 times. He found the radius of the Moon to be one-third of the Earth’s radius; the present figure is 27/100.

The development of Greek trigonometry and its application to astronomy culminated in the Almagest of astronomer Ptolemy in the 2nd century ad. The Almagest put trigonometry into the form it retained for more than 1,000 years and presented a theory of the motions of the celestial bodies that prevailed until it was overthrown by the Copernican system in the 16th century.

A.5.

Roman Mathematics

The Alexandrian period of Greek civilization ended in 31 bc with Rome’s conquest of Egypt, the last of Alexander’s kingdoms. Roman orator Cicero boasted that the Romans were not dreamers like the Greeks but applied their study of mathematics to the useful. Nothing mathematically significant was accomplished by the Romans. The Roman numeration system was based on Roman numerals, which were cumbersome for calculation. Despite this drawback, the use of Roman numerals continued in some European schools until about 1600 and in bookkeeping for another century.

B.

Indian and Islamic Mathematics

After the decline of Greece and Rome, mathematics flourished for hundreds of years in India and the Islamic world. Mathematics in India was largely a tool for astronomy, yet Indian mathematicians discovered a number of important concepts. Their mathematical masterpieces and those of the Greeks were translated into Arabic in centers of Islamic learning, where mathematical discoveries continued during the period known in the West as the Middle Ages. Our present numeration system, for example, is known as the Hindu-Arabic system.

B.1.

Indian Mathematics

The system of numbers that we use today, with each number having an absolute value and a place value (units, tens, hundreds, and so forth) originated in India. Mathematicians in India also were the first to recognize zero as both an integer and a placeholder. When the Indian numeration system was developed is not known, but digits similar to the Arabic numerals used today have been found in a Hindu temple built about 250 bc.

In the 5th century Hindu mathematician and astronomer Aryabhata studied many of the same problems as Diophantus but went beyond the Greek mathematician in his use of fractions as opposed to whole numbers to solve indeterminate equations (equations that have no unique solutions). Aryabhata also figured the value of p accurately to eight places, thus coming closer to its value than any other mathematician of ancient times. In astronomy, he proposed that Earth orbited the sun and correctly explained eclipses of the Sun and Moon.

The earliest known use of negative numbers in mathematics was by Hindu mathematician Brahmagupta about ad 630. He presented rules for them in terms of fortunes (positive numbers) and debts (negative numbers). Brahmagupta’s understanding of numbers exceeded that of other mathematicians of the time, and he made full use of the place system in his method of multiplication. Brahmagupta headed the leading astronomical observatory in India and wrote two works on mathematics and astronomy. The works dealt with topics such as eclipses, risings and settings, and conjunctions of the planets with each other and with fixed stars.

Writing in the 9th century, Jain mathematician Mahavira stated rules for operations with zero, although he thought that division by zero left a number unchanged. The best-known Indian mathematician of the early period was Bhaskara, who lived in the 12th century. Bhaskara supplied the correct answer for division by zero as well as rules for operating with irrational numbers. Bhaskara wrote six books on mathematics, including Lilavati (The Beautiful), which summarized mathematical knowledge in India up to his time, and Karanakutuhala, translated as “Calculation of Astronomical Wonders.”

B.2.

Islamic Mathematics

Indian mathematics reached Baghdād, a major early center of Islam, about ad 800. Supported by the ruling caliphs and wealthy individuals, translators in Baghdād produced Arabic versions of Greek and Indian mathematical works. The need for translations was stimulated by mathematical research in the Islamic world. Islamic mathematics also served religion in that it proved useful in dividing inheritances according to Islamic law; in predicting the time of the new moon, when the next month began; and in determining the direction to Mecca for the orientation of mosques and of daily prayers, which were delivered facing Mecca.

In the 9th century Arab mathematician al-Khwārizmī wrote a systematic introduction to algebra, Kitab al-jabr w’al Muqabalah (Book of Restoring and Balancing). The English word algebra comes from al-jabr in the treatise’s title. Al-Khwārizmī’s algebra was founded on Brahmagupta’s work, which he duly credited, and showed the influence of Babylonian and Greek mathematics as well. A 12th-century Latin translation of al-Khwārizmī’s treatise was crucial for the later development of algebra in Europe. Al-Khwārizmī’s name is the source of the word algorithm.

By the year 900 the acquisition of past mathematics was complete, and Muslim scholars began to build on what they had acquired. Alhazen, an outstanding Arab scientist of the late 900s and early 1000s, produced algebraic solutions of quadratic and cubic equations. Al-Karaji in the 10th and early 11th century completed the algebra of polynomials (mathematical expressions that are the sum of a number of terms) of al-Khwārizmī. He included polynomials with an infinite number of terms.

Later scholars, including 12th-century Persian mathematician Omar Khayyam, solved certain cubic equations geometrically by using conic sections. Arab astronomers contributed the tangent and cotangent to trigonometry. Geometers such as Ibrahim ibn Sinan in the 10th century continued Archimedes’s investigations of areas and volumes, and Kamal al-Din and others applied the theory of conic sections to solve problems in optics. Astronomer Nasir al-Din al-Tusi created the mathematical disciplines of plane and spherical trigonometry in the 13th century and was the first to treat trigonometry separately from astronomy. Finally, a number of Muslim mathematicians made important discoveries in the theory of numbers, while others explained a variety of numerical methods for solving equations.

Many of the ancient Greek works on mathematics were preserved during the Middle Ages through Arabic translations and commentaries. Europe acquired much of this learning during the 12th century, when Greek and Arabic works were translated into Latin, then the written language of educated Europeans. These Arabic works, together with the Greek classics, were responsible for the growth of mathematics in the West during the late Middle Ages.

C.

Medieval and Renaissance Mathematics

Few advances in mathematics took place in Europe during the early Middle Ages, before about 1100. Most learning was concentrated in monasteries and focused on questions of theology. A most important application of mathematics during the Middle Ages was in astrology; astrologers were called mathematici. Inasmuch as the practice of medicine was based largely on astrological determination of the proper treatment, physicians had to become mathematicians.

The introduction of Greek and Arabic works starting about 1100 played a major role in the rebirth of secular (worldly) learning in Europe. The translation of these works into Latin led to an upsurge in mathematical study in the West. English philosopher Adelard translated al-Khwārizmī’s astronomical tables and an Arabic version of Euclid’s Elements into Latin in the 12th century. Italian mathematicians such as Leonardo Fibonnaci and Luca Pacioli depended heavily on Arabic sources in improving business mathematics used for accounting and trade. Fibonnaci’s Liber abaci (1202, Book of the Abacus) introduced Arabic numbers, the Hindu-Arabic place-value decimal system, and Arabic algebra to Europe.

The late Middle Ages saw some fruitful mathematical considerations of infinite series by French prelate Nicole d’Oresme and others. But not until the 16th century did the first truly important mathematical discovery in Europe occur. This discovery was an algebraic formula for the solution of both cubic and quartic equations—equations with terms raised to the third or fourth powers. Italian mathematician Gerolamo Cardano published the formula in 1545 in his Ars Magna (Great Art). The discovery drew the attention of mathematicians to complex numbers and stimulated a series of solutions to equations of degrees higher than four.

During the 16th century mathematicians began to use symbols to make algebraic thinking and writing more concise. These symbols included +, -, ×, =, > (greater than), and < (less than). The most significant innovation, by French mathematician François Viète, was the systematic use of letters for variables in equations. Viète’s remarkable work on solving equations to the third and fourth degree influenced many mathematicians of the following century, including Fermat in France and Newton in England.

D.

Mathematics from 1600 to 1900

The scientific revolution of the 17th century spurred advances in mathematics as well. The founders of modern science—Nicolaus Copernicus, Johannes Kepler, Galileo, and Isaac Newton—studied the natural world as mathematicians, and they looked for its mathematical laws. Over time mathematics grew more and more abstract as mathematicians sought to establish the foundations of their fields in logic.

D.1.

17th-Century Mathematics

The 17th century saw the greatest advances in mathematics since the time of the ancient Greeks. The major invention of the century was calculus. Although two great thinkers—Sir Isaac Newton of England and Gottfried Wilhelm Leibniz of Germany—have received credit for its invention, they built on the work of others. As Newton noted, “If I have seen further, it is by standing on the shoulders of giants.” Major advances also were made in numerical calculation and geometry.

D.1.a.

Calculation

The 17th century opened with the discovery of logarithms by Scottish mathematician John Napier and Swiss mathematician Justus Byrgius. Logarithms enabled mathematicians to extract the roots of numbers and simplified many calculations by basing them on addition and subtraction rather than on multiplication and division. Napier, who was interested in simplification, studied the systems of the Indian and Islamic worlds and spent years producing the tables of logarithms that he published in 1614. Kepler’s enthusiasm for the tables ensured their rapid spread.

So useful did logarithms prove that, two centuries after their discovery, French astronomer Pierre Simon Laplace said Napier had doubled the lifetime of astronomers by halving their labors. The need for logarithm tables disappeared with the widespread use of electronic calculators and computers in the second half of the 20th century.

D.1.b.

Analytic Geometry

The most important development in geometry during the 17th century was the discovery of analytic geometry by René Descartes and Pierre de Fermat, working independently in France. Analytic geometry makes it possible to study geometric figures using algebraic equations.

Descartes wished to overcome the limitations of Euclidean geometry, and he did so by applying algebra to geometry. In his publication Discours de la méthode (1637; Discourse on Method), Descartes showed how to use developments in algebra since the Renaissance to investigate the geometry of curves. Descartes maintained that an acceptable curve is one that can be expressed by a unique algebraic equation in x and y. His introduction of x and y coordinates was a major step. This made possible the classification of equations by the shape of the curves they made when graphed, and it opened the study of curves. As a consequence, many new curves important for science, including the cycloid and catenary, were introduced in the 17th and 18th centuries.

Descartes’s discoveries in geometry led to a reversal of the historical roles of geometry and algebra. French mathematician Joseph Louis Lagrange observed in the 18th century, “As long as algebra and geometry proceeded along separate paths their advance was slow and their applications limited. But when these sciences joined company, they drew from each other fresh vitality and thenceforward marched on at a rapid pace toward perfection.”

Descartes’s book, together with short treatises published along with it, provided the impetus and basis for Newton’s mathematical work later in the century. Fermat, however, regarded his own work on what became known as analytic geometry as a reformulation of Apollonius’s treatise on conic sections. That treatise had provided the basic work on the geometry of curves from ancient times until Descartes.

D.1.c.

Projective Geometry

Other developments came in the field of projective geometry—that is, the properties of geometric figures projected onto another plane. French engineer Gérard Desargues published his discovery of projective geometry in 1639. The work was much appreciated by Descartes and Blaise Pascal, another French philosopher and mathematician of the 17th century. But eccentric terminology and the excitement of Descartes’s earlier publication on analytic geometry delayed the development of the ideas in this work until the early 19th century.

D.1.d.

Fermat and Number Theory

Fermat made his greatest contributions to mathematics in number theory, a field that seeks to make general statements about groups of numbers, such as prime numbers. He made his most important conjecture in number theory while reading the Arithmetica of Diophantus. Writing in the margin of his copy, Fermat stated that no solutions exist to the equation aⁿ + bⁿ = cⁿ for positive integers a, b, and c when n is greater than 2, but added that there was too little room in the margin for his proof. This conjecture, known as Fermat’s last theorem, stimulated much important work in algebra and number theory before it was finally proved in 1994.

D.1.e.

Probability

Pascal and Fermat, through correspondence in 1654 on a problem in gambling, began the mathematical study of probability. The problem concerned the division of stakes between two gamblers who wished to leave the gambling table before either has scored the n points needed to win. The solution involved calculating the probability for all possible outcomes and the associated amount of the winnings. Pascal solved the problem for two players, but a solution that involved three or more players had to wait. Pascal also invented a mechanical calculator to help his father in collecting taxes.

An unpublished piece by Pascal on gambling stimulated Dutch scientist Christiaan Huygens to publish a small work in 1657 on probabilities in dice games. Swiss mathematician Jakob Bernoulli reprinted this work in his Ars Conjectandi (Art of Conjecturing) published in 1713. Both Bernoulli and French-English mathematician Abraham De Moivre, in his Doctrine of Chances (1718), applied the newly discovered calculus to probability. They thus made advances in probability theory, which by then had important applications in the rapidly developing insurance industry.

D.1.f.

The Invention of Calculus

The discovery of differential and integral calculus by Sir Isaac Newton and Gottfried Wilhelm Leibniz ranks as the crowning achievement of 17th-century mathematics. Calculus allowed the solution of many problems that had been previously insoluble, including the determination of the laws of motion and the theory of electromagnetism.

The task of defining and calculating continuously changing quantities attracted almost all the mathematicians of the 17th century. In inventing differential calculus for the study of nature and change, Newton and Leibniz drew on earlier work by European scientists and mathematicians such as Johannes Kepler, Francesco Cavalieri, Fermat, John Wallis, and Isaac Barrow. The methods that Newton and Leibniz created, however, had far wider applications than the calculation of instantaneous rates of change. Newton and Leibniz also realized that the inverse of differentiation, or integration, had applications in calculating the sums of infinite series—another problem that had vexed mathematicians for some time.

Newton made his discoveries on calculus from 1664 to 1666 but did not publish them until 1687. Leibniz published his system of differential calculus in 1684 and of integral calculus in 1686. Controversy soon erupted over who had invented calculus first, with Newton accusing Leibniz of plagiarism. Later investigations show that Leibniz invented his calculus independently. As a consequence of the dispute, English mathematicians and continental European mathematicians stopped exchanging ideas for many years. English mathematicians persisted in the geometric development of calculus. Far greater advances resulted on the European continent from the algebraic approach of mathematicians such as Johann Bernoulli and Leonhard Euler of Switzerland and Lagrange of France.

D.2.

18th-Century Mathematics

During the 18th century calculus became the cornerstone of mathematical analysis on the European continent. Mathematicians applied the discovery to a variety of problems in physics, astronomy, and engineering. In the course of doing so, they also created new areas of mathematics.

In geometry French mathematician Gaspard Monge developed the field of descriptive geometry. Monge received his opportunity when as a draftsman he was asked to draw up a fortification plan that could be defended no matter what the enemy’s position. Monge used techniques of geometry he had developed on his own to determine firing lines and produce the fortress plan. His system of descriptive geometry has applications for solving engineering and construction problems.

Also in France, Joseph Louis Lagrange made substantial contributions in all fields of pure mathematics, including differential equations, the calculus of variations, probability theory, and the theory of equations. In addition, Lagrange put his mathematical skills to work in the solution of practical problems in mechanics and astronomy. The greatest achievement of his career came in 1788 with the publication of his Mécanique analytique (Analytical Mechanics). In this work Lagrange used the calculus of variations to deduce from one simple assumption the mechanics of fluids and solids.

The greatest mathematician of the 18th century, Leonhard Euler of Switzerland, was also the most prolific writer on mathematical subjects of all time. His treatises covered essentially the entire fields of pure and applied mathematics. He wrote major works on mechanics that preceded Lagrange’s work. He won a number of prizes for his work on the orbits of comets and planets, a field known as celestial mechanics. But Euler is best known for his work in pure mathematics. His Introductio in analysin infinitorum (Introduction to the Analysis of Infinites), published in 1748, approached calculus in terms of functions rather than the geometry of curves. Other works by Euler contributed to number theory and differential geometry (the application of differential calculus to the study of the properties of curves and curved spaces).

D.3.

19th-Century Mathematics

The 19th century was a period of intense mathematical activity. It began with German mathematician Carl Friedrich Gauss, considered the last complete mathematician because of his contributions to all branches of the field. The century saw a great effort to place all areas of mathematics on firm theoretical foundations. The support for these foundations was logic—the deduction of basic propositions from a limited set of assumptions and definitions. Mathematicians succeeded not only in firming the foundations of analysis, as the techniques of calculus were by then known, but also in making great strides in the field. Mathematicians also discovered the existence of additional geometries and algebras, and more than one kind of infinity.

D.3.a.

Gauss: All Fields

Carl Friedrich Gauss ranks as one of the greatest mathematicians who ever lived. Diaries from his childhood show that he had already made important discoveries in the theory of numbers. His book Disquisitiones Arithmeticae (Inquiries into Arithmetic), published in 1801, marks the beginning of the modern era in number theory.

Gauss also provided a geometric explanation of complex numbers as points on a plane, and the xy plane became known as the complex plane. (Complex numbers are of the form a + bi where a and b are real numbers and i is the imaginary number Á.) In his doctoral dissertation Gauss gave the first satisfactory proof of the fundamental theorem of algebra: that every polynomial equation has at least one root, or solution, in the complex plane. Complex numbers then formed a new field for analysis. Gauss called mathematics “the queen of the sciences” and number theory “the queen of mathematics.”

Often Gauss combined scientific and mathematical investigations. For example, his investigations of the orbit of Ceres, a newly discovered asteroid, led to his discovery of a statistical method called least squares. He used this method to predict the asteroid’s future appearances. His study of magnetism led to the creation of the potential theory in physics, which uses potential-energy functions to study gravity, electricity, and magnetism. His investigations in surveying produced important contributions to the geometry of curved surfaces.

D.3.b.

Analysis

Almost from the introduction of calculus, efforts had been made to supply a rigorous foundation for it. Calculus had introduced two new and complex concepts—the derivative and the definite integral. Newton and Leibniz were hazy on these matters, as were the mathematicians who extended calculus to analysis. Concepts such as limit, continuity, and differentiability were vague. Almost every mathematician of the 18th and early 19th centuries, including Euler and Lagrange, made some effort to produce a logical justification for calculus and failed. Although calculus clearly worked in solving problems, mathematicians lacked rigorous proof that explained why it worked.

Finally, in 1821 French mathematician Augustin Louis Cauchy established a rigorous foundation for calculus with his theory of limits, a purely arithmetic theory. Later mathematicians found Cauchy’s formulation still too vague because it did not provide a logical definition of “real number.” The necessary precision for calculus and mathematical analysis was attained in the 1850s by German mathematician Karl T. W. Weierstrass and his followers.

Another important advance in analysis came from French mathematician Jean Baptiste Fourier, who studied infinite series in which the terms are trigonometric functions. Known today as Fourier series, they are still powerful tools in pure and applied mathematics. The investigation of which functions could be equal to Fourier series led German mathematician Georg Cantor to the study of infinite sets and to an arithmetic of infinite numbers.

D.3.c.

Cantor: Set Theory

Georg Cantor began his mathematical investigations in number theory and went on to create set theory. In the course of his early studies on Fourier series, he developed a theory of irrational numbers (real numbers that cannot be defined as a ratio of two integers—Ã, for example, or p). The character of irrational numbers had withstood mathematical explanation since ancient times. Although they cannot be written as finite numbers, they can easily be handled algebraically—in the case of Ã, by squaring it. Cantor and another German mathematician, Julius W. R. Dedekind, defined the irrational numbers and established their properties. These explanations hastened the abandonment of many 19th-century mathematical principles.

Cantor introduced the theory of sets in 1874 in a paper on different kinds of infinities. Cantor’s theory was considered abstract and attacked as a “disease from which mathematics will soon recover.” However, it now forms part of the foundations of mathematics. The application of set theory greatly advanced mathematics in the 20th century.

D.3.d.

Non-Euclidean Geometry

By the 19th century Euclidean geometry was the most solidly constructed branch of mathematics because many properties of the number system were still proved geometrically. Nevertheless, one of Euclid’s axioms known as the parallel postulate makes an assertion that experience cannot substantiate. This axiom states that through a point outside a given line it is possible to draw only one line parallel to the given line. Over the centuries mathematicians had tried to frame a substitute axiom, but each substitute proved to have a flaw. Nor could they deduce the questionable axiom from the other nine axioms.

One could start by assuming either that there was no parallel to a line through a point, or that there was more than one line through the point. The parallel postulate could be proved if, by adopting either of these assumptions in place of the Euclidean postulate, one could show that the new set of ten axioms led to a contradiction.

During the early 19th century Gauss concluded that it was possible to replace the parallel postulate and still produce a consistent, though non-Euclidean, geometry. The substitute postulate would state that an infinite number of parallel lines could be drawn through a given point. But Gauss did not publish his ideas because he feared they would expose him to ridicule.

Credit for the creation of non-Euclidean geometry is given to Russian mathematician Nikolay Lobachevsky and Hungarian mathematician János Bolyai. Each man published an organized presentation of a geometry that allows an infinite number of parallel lines through a given point. In the 1850s German mathematician Georg F. B. Riemann introduced the idea of a geometry in which there are no parallel lines. Riemann also introduced the idea of a curved line called a geodesic as the shortest distance between two points. Riemann’s geometrical interpretation of the theory of functions proved important for Einstein’s discoveries on relativity. Non-Euclidean geometry was not taken seriously until Einstein’s theory of special relativity in 1905 awakened the scientific world to the reality of Riemannian curved space.

Non-Euclidean geometry was the most impressive intellectual creation of the 19th century. It showed that mathematics could no longer be regarded as a body of unquestionable truths and that the observable world could not provide all the answers. Mathematicians were henceforth liberated to explore whatever ideas attracted them, and they turned more and more toward abstraction and theory. Individual mathematicians felt free to define their notions and to set up their axioms as they pleased, subject only to the limitation that the axioms do not give rise to theorems that contradict one another. The enormous expansion in mathematical activity in the 20th century was largely the consequence of this new freedom.

D.3.e.

Advances in Algebra

Algebra underwent a transformation during the 19th century, moving from the solution of polynomial equations to a study of the structure of algebraic systems. A first step in this direction was the publication of Treatise on Algebra (1830) by George Peacock, an English mathematician. Peacock attempted to provide algebra with the logical foundation Euclid had given geometry.

The creation of different systems of algebra began with Irish mathematician William Rowan Hamilton. In searching for general properties of complex numbers, Hamilton in 1843 discovered quaternions, a class of complex numbers that break the commutative law in algebra. This law states that a x b = b x a. Hamilton’s quaternions paved the way for the study of new algebraic systems.

Immediately after Hamilton’s discovery, German mathematician Hermann Grassmann and American mathematician and physicist J. Willard Gibbs began the analysis of three-dimensional vectors. From his investigations, Grassmann developed what is now called exterior algebra, which he applied to spaces of n (indefinitely many) dimensions. Gibbs used ideas of Grassmann to produce a system of vector analysis that could be applied to physics. He published his Elements of Vector Analysis in three parts from 1881 to 1884.

Another major step in algebra during the 19th century was the development of the theory of groups, which had its beginnings in the work of Lagrange. Norwegian mathematician Niels Henrik Abel demonstrated that it was impossible to solve by elementary algebra any equation of degree greater than four. Évariste Galois of France introduced the group concept to the solution of algebraic equations, showing that equations have associated groups of substitutions that govern their solubility. Galois’s work signaled a new direction in mathematics.

Just as Descartes had applied the algebra of his time to the study of geometry, so too did German mathematician Felix Klein and Norwegian mathematician Marius Sophus Lie apply the new algebra of the 19th century to geometry. Klein continued the group theory work of Galois, studying the properties that remained constant in a geometry when it underwent a group of transformations. Lie, too, worked in group theory, applying it not only to geometry but also to differential equations and other areas of mathematics.

Among the mathematicians who used the discoveries of Hamilton and Grassman was George Boole in England. Boole claimed mathematics could be investigated in terms of logic and provided symbolic notation for mathematical operations. Boole’s major contribution to mathematics is Boolean algebra, an algebra of sets that later formed the basis of symbolic logic and computer technology.

E.

20th-Century Mathematics

During the 20th century mathematics made rapid advances on all fronts. The foundations of mathematics became more solidly grounded in logic, while at the same time mathematics advanced the development of symbolic logic. Philosophy was not the only field to progress with the help of mathematics. Physics, too, benefited from the contributions of mathematicians to relativity theory and quantum theory. Indeed, mathematics achieved broader applications than ever before, as new fields developed within mathematics (computational mathematics, game theory, and chaos theory) and other branches of knowledge, including economics and physics, achieved firmer grounding through the application of mathematics. Even the most abstract mathematics seemed to find application, and the boundaries between pure mathematics and applied mathematics grew ever fuzzier.

Mathematicians searched for unifying principles and general statements that applied to large categories of numbers and objects. In algebra, the study of structure continued with a focus on structural units called rings, fields, and groups, and at mid-century it extended to the relationships between these categories. Algebra became an important part of other areas of mathematics, including analysis, number theory, and topology, as the search for unifying theories moved ahead. Topology—the study of the properties of objects that remain constant during transformation, or stretching—became a fertile research field, bringing together geometry, algebra, and analysis. Because of the abstract and complex nature of most 20th-century mathematics, most of the remaining sections of this article will discuss practical developments in mathematics with applications in more familiar fields.

Until the 20th century the centers of mathematics research in the West were all located in Europe. Although the University of Göttingen in Germany, the University of Cambridge in England, the French Academy of Sciences and the University of Paris, and the University of Moscow in Russia retained their importance, the United States rose in prominence and reputation for mathematical research, especially the departments of mathematics at Princeton University and the University of Chicago.

E.1.

The Hilbert Problems

At the Second International Congress of Mathematicians held in Paris in 1900, German mathematician David Hilbert spoke to the assembly. Hilbert was a professor at the University of Göttingen, the former academic home of Gauss and Riemann. Hilbert’s speech at Paris was a survey of 23 mathematical problems that he felt would guide the work being done in mathematics during the coming century. These problems stimulated a great deal of the mathematical research of the 20th century, and many of the problems were solved. When news breaks that another “Hilbert problem” has been solved, mathematicians worldwide impatiently await further details.

Hilbert contributed to most areas of mathematics, starting with his classic Grundlagen der Geometrie (Foundations of Geometry), published in 1899. Hilbert’s work created the field of functional analysis (the analysis of functions as a group), a field that occupied many mathematicians during the 20th century. He also contributed to mathematical physics. From 1915 on he fought to have Emmy Noether, a noted German mathematician, hired at Göttingen. When the university refused to hire her because of objections to the presence of a woman in the faculty senate, Hilbert countered that the senate was not the changing room for a swimming pool. Noether later made major contributions to ring theory in algebra and wrote a standard text on abstract algebra.

E.2.

Mathematics and Logic

In some ways pure mathematics became more abstract in the 20th century, as it joined forces with the field of symbolic logic in philosophy. The scholars who bridged the fields of mathematics and philosophy early in the century were Alfred North Whitehead and Bertrand Russell, who worked together at Cambridge University. They published their major work, Principia Mathematica (Principles of Mathematics), in three volumes from 1910 to 1913. In it they demonstrated the principles of mathematical logic and attempted to show that all of mathematics could be deduced from a few premises and definitions by the rules of formal logic. In the late 19th century, German mathematician Gottlob Frege had provided the system of notation for mathematical logic and paved the way for the work of Russell and Whitehead. Mathematical logic influenced the direction of 20th-century mathematics, including the work of Hilbert.

Hilbert proposed that the underlying consistency of all mathematics could be demonstrated within mathematics. But logician Kurt Gödel in Austria proved that the goal of establishing the completeness and consistency of every mathematical theory is impossible. Despite its negative conclusion Gödel’s Theorem, published in 1931, opened up new areas in mathematical logic. One area, known as recursion theory, played a major role in the development of computers.

E.3.

Mathematics and Physics

Several revolutionary theories, including relativity and quantum theory, challenged existing assumptions in physics in the early 20th century. The work of a number of mathematicians contributed to these theories. Among them was Noether, whose gender had denied her a paid position at the University of Göttingen. Noether’s mathematical formulations on invariants (quantities that remain unchanged as other quantities change) contributed to Einstein’s theory of relativity. Russian mathematician Hermann Minkowski contributed to relativity the notion of the space-time continuum, with time as a fourth dimension. Hermann Weyl, a student of Hilbert’s, investigated the geometry of relativity and applied group theory to quantum mechanics. Weyl’s investigations helped advance the field of topology. Early in the century Hilbert quipped, “Physics is getting too difficult for physicists.”

Hungarian-born American mathematician John von Neumann built a solid mathematical basis for quantum theory with his text Mathematische Grundlagen der Quantenmechanik (1932, Mathematical Foundations of Quantum Mechanics). This investigation led him to explore algebraic operators and groups associated with them, opening a new area now known as Neumann algebra. Von Neumann, however, is probably best known for his work in game theory and computers.

During World War II (1939-1945) mathematicians and physicists worked together on developing radar, the atomic bomb, and other technology that helped defeat the Axis powers. Polish-born mathematician Stanislaw Ulam solved the problem of how to initiate fusion in the hydrogen bomb. Von Neumann participated in numerous U.S. defense projects during the war.

Mathematics plays an important role today in cosmology and astrophysics, especially in research into big bang theory and the properties of black holes, antimatter, elementary particles, and other unobservable objects and events. Stephen Hawking, among the best-known cosmologists of the 20th century, in 1979 was appointed Lucasian Professor of Mathematics at Trinity College, Cambridge, a position once held by Newton.

E.4.

Mathematics and Economics

Mathematics formed an alliance with economics in the 20th century as the tools of mathematical analysis, algebra, probability, and statistics illuminated economic theories. A specialty called econometrics links enormous numbers of equations to form mathematical models for use as forecasting tools.

Game theory began in mathematics but had immediate applications in economics and military strategy. This branch of mathematics deals with situations in which some sort of decision must be made to maximize a profit—that is, to win. Its theoretical foundations were supplied by von Neumann in a series of papers written during the 1930s and 1940s. Von Neumann and economist Oskar Morgenstern published results of their investigations in The Theory of Games and Economic Behavior (1944). John Nash, the Princeton mathematician profiled in the motion picture A Beautiful Mind, shared the 1994 Nobel Prize in economics for his work in game theory.

E.5.

Mathematics and Computers

Mathematicians, physicists, and engineers contributed to the development of computers and computer science. But the early, theoretical work came from mathematicians. English mathematician Alan Turing, working at Cambridge University, introduced the idea of a machine that could perform mathematical operations and solve equations. The Turing machine, as it became known, was a precursor of the modern computer. Through his work Turing brought together the elements that form the basis of computer science: symbolic logic, numerical analysis, electrical engineering, and a mechanical vision of human thought processes.

Computer theory is the third area with which von Neumann is associated, in addition to mathematical physics and game theory. He established the basic principles on which computers operate. Turing and von Neumann both recognized the usefulness of the binary arithmetic system for storing computer programs.

The first large-scale digital computers were pioneered in the 1940s. Von Neumann completed the EDVAC (Electronic Discrete Variable Automatic Computer) at the Institute of Advanced Study in Princeton in 1945. Engineers John Eckert and John Mauchly built ENIAC (Electronic Numerical Integrator and Calculator), which began operation at the University of Pennsylvania in 1946. As increasingly complex computers are built, the field of artificial intelligence has drawn attention. Researchers in this field attempt to develop computer systems that can mimic human thought processes.

Mathematician Norbert Wiener, working at the Massachusetts Institute of Technology (MIT), also became interested in automatic computing and developed the field known as cybernetics. Cybernetics grew out of Wiener’s work on increasing the accuracy of bombsights during World War II. From this came a broader investigation of how information can be translated into improved performance. Cybernetics is now applied to communication and control systems in living organisms.

Computers have exercised an enormous influence on mathematics and its applications. As ever more complex computers are developed, their applications proliferate. Computers have given great impetus to areas of mathematics such as numerical analysis and finite mathematics. Computer science has suggested new areas for mathematical investigation, such as the study of algorithms. Computers also have become powerful tools in areas as diverse as number theory, differential equations, and abstract algebra. In addition, the computer has made possible the solution of several long-standing problems in mathematics, such as the four-color theorem first proposed in the mid-19th century.

The four-color theorem stated that four colors are sufficient to color any map, given that any two countries with a contiguous boundary require different colors. Mathematicians at the University of Illinois finally confirmed the theorem in 1976 by means of a large-scale computer that reduced the number of possible maps to slightly less than 2,000. The program they wrote ran thousands of lines in length and took more than 1,200 hours to run. Many mathematicians, however, do not accept the result as a proof because it has not been checked. Verification by hand would require far too many human hours. Some mathematicians object to the solution’s lack of elegance. This complaint has been paraphrased, “a good mathematical proof is like a poem—this is a telephone directory.'

F.

The Future of Mathematics

Hilbert inaugurated the 20th century by proposing 23 problems that he expected to occupy mathematicians for the next 100 years. A number of these problems, such as the Riemann hypothesis about prime numbers, remain unsolved in the early 21st century. Hilbert claimed, “If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proven?”

The existence of old problems, along with new problems that continually arise, ensures that mathematics research will remain challenging and vital through the 21st century. Influenced by Hilbert, the Clay Mathematics Institute at Harvard University announced the Millennium Prize in 2000 for solutions to mathematics problems that have long resisted solution. Among the seven problems is the Riemann hypothesis. An award of $1 million awaits the mathematician who solves any of these problems.