The Oxford School

Math week 9*-12* Sept.

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.
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.
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.
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.”
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.
Branches of Mathematics
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.
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.
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.

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.