Mathematics

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.

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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.