Mathematics is one of the most useful and fascinating divisions of human knowledge. It includes many topics of study. For this reason, the term mathematics is difficult to define. It comes from a Greek word meaning “inclined to learn.”
Most of the basic mathematics taught in school involves the study of number, quantity, form, measurement, and relations. Arithmetic, for example, concerns problems with numbers. Algebra involves solving equations (mathematical statements of equality) in which letters represent unknown quantities. Geometry concerns the properties and relationships of figures in space.
The most important skills in mathematics are careful analysis, clear reasoning, defining terms precisely, and pattern recognition. Along with rigorous and creative thinking, these skills can help us solve some of the deepest puzzles the world presents. Mathematics is based upon axioms (fundamental statements assumed to be true) and logic. Starting from widely accepted statements, mathematicians use logic to draw conclusions and develop mathematical systems.
The importance of mathematics
The work of mathematicians may be divided into pure mathematics and applied mathematics. Pure mathematics seeks to advance mathematical knowledge for its own sake rather than for any immediate practical use. For example, a mathematician may create a system of geometry for an imaginary world where objects have more dimensions than just length, width, and depth. Applied mathematics seeks to develop mathematical techniques for use in science, technology, and other fields. Technology consists of the tools, materials, techniques, and sources of power that make our lives and work easier.
The boundary between pure and applied mathematics is not always clear-cut. Ideas developed in pure mathematics often have far-reaching applications, and work in applied mathematics frequently leads to research in pure mathematics.
Nearly every part of our lives involves mathematics. Most electronic gadgets, such as watches, cell phones, game consoles, personal computers, and ATM’s, have mathematical foundations. Mathematics plays an essential role in the development of modern technology.
In everyday life,
we use mathematics for such simple tasks as telling time from a clock or counting our change after making a purchase. We also use mathematics for such complex tasks as making up a household budget or figuring our income tax. Cooking, driving, gardening, sewing, and many other common activities involve mathematical calculations. Mathematics is also a key aspect of many games, hobbies, and sports.
In science.
Mathematics is an essential part of the scientific world. It helps scientists describe natural phenomena, design experiments, and analyze data. Scientists use mathematical formulas to express their findings precisely and to make predictions based on these findings.
The physical sciences, such as astronomy, chemistry, and physics, rely heavily on mathematics. Biological processes are best studied and understood using mathematical models and analysis. Such social sciences as economics, psychology, and sociology also depend greatly on statistics and other kinds of mathematics. For example, some economists create mathematical models of economic systems. These models are sets of formulas used to predict how a change in one part of the economy might affect other parts.
In industry.
Mathematics helps industries design, develop, and test products and manufacturing processes. Mathematics is necessary in designing bridges, buildings, dams, highways, tunnels, and other architectural and engineering projects.
In business.
Mathematics is used in transactions that involve buying and selling. Mathematics is used to help increase profits and lower costs. Businesses need mathematics to keep records of such things as inventory and employees’ hours and wages. Bankers use mathematics to handle and invest funds. Mathematics helps insurance companies calculate risks and compute the rates charged for insurance coverage.
Branches of mathematics
Mathematics has many branches. They may differ in the types of problems involved and in the practical application of their results. However, mathematicians working in different branches often use many of the same basic concepts and operations. This section discusses several of the main kinds of mathematics.
Arithmetic
includes the study of whole numbers, fractions and decimals, and the operations of addition, subtraction, multiplication, and division. It forms the foundation for other kinds of mathematics by providing such basic skills as counting and grouping objects, and measuring and comparing quantities.
Algebra,
unlike arithmetic, is not limited to work with specific numbers. Algebra involves solving problems with equations in which letters, such as x and y, stand for unknown quantities. Algebraic operations also use negative numbers and imaginary numbers (the square roots of negative numbers).
Geometry
is concerned with the properties and relationships of figures in space. Plane geometry deals with squares, circles, and other figures that lie on a plane. Solid geometry involves such figures as cubes and spheres, which have three dimensions.
About 300 B.C., Euclid, a Greek mathematician, stated the definitions and assumptions of the system of geometry that describes the world as we usually experience it. But later mathematicians developed alternative systems of geometry that rejected Euclid’s assumption about the nature of parallel lines. Such non-Euclidean geometries have proved useful, for example, in the theory of relativity—one of the outstanding achievements of scientific thought.
Analytic geometry and trigonometry.
Analytic geometry relates algebra and geometry. It provides a way to represent an algebraic equation as a line or curve on a graph. Analytic geometry also makes it possible to write equations that exactly describe many curves. For example, the equation x = y2 describes a curve called a parabola.
Trigonometry is used widely by astronomers, navigators, and surveyors to calculate angles and distances when direct measurement is impossible. It deals with the relations between the sides and angles of triangles, especially right triangles (triangles that have a 90° angle). Certain relations between the lengths of two sides of a right triangle are called trigonometric ratios. Using trigonometric ratios, a person can calculate the unknown angles and lengths in a triangle from the known angles and lengths. Formulas involving trigonometric ratios describe curves that physicists and engineers use to analyze the behavior of heavenly bodies, heat, light, sound, and other natural phenomena.
Calculus and analysis
have many practical uses in engineering, physics, and other sciences. Calculus provides a way of solving many problems that involve motion or changing quantities. Differential calculus seeks to determine the rate at which a varying quantity changes. It is used to calculate the slope of a curve and the changing speed of a rocket or drone. Integral calculus tries to find a quantity when the rate at which it is changing is known. It is used to calculate the area of a curved figure or the amount of work done by a varying force. Unlike algebra, calculus includes operations with infinitesimals (quantities that are not zero but are smaller than any assignable quantity).
Analysis involves various mathematical operations with infinite quantities and infinitesimals. It includes the study of infinite series, sequences of numbers or algebraic expressions that go on indefinitely. The concept of infinite series has important applications in such areas as the study of heat and of vibrating strings.
Probability and statistics.
Probability is the mathematical study of the likelihood of events. It is used to determine the chances that an uncertain event may occur. For example, using probability, a person can calculate the chances that three tossed coins will all turn up heads.
Statistics is the branch of mathematics concerned with the collection and analysis of large bodies of data to identify trends and overall patterns. Statistics relies heavily on probability, computer-assisted computations, and data analysis. Statistical methods provide information to government, business, and science. For example, physicists use statistics to study the behavior of the many molecules in a sample of gas. Social scientists use statistics to better understand quantitative (numerical) relationships between income and education levels, which then helps inform policy decisions.
Set theory and logic.
Set theory deals with the nature and relations of sets. A set is a collection of items, which may be numbers, ideas, or objects. The study of sets is important in investigating most basic mathematical concepts.
In the field of logic—the branch of philosophy that deals with the rules of correct reasoning—mathematicians have developed symbolic logic. Symbolic logic is a formal system of reasoning that uses mathematical symbols and methods. Mathematicians have devised various systems of symbolic logic that have been important in the development of computers.
History
Early civilization.
Prehistoric people probably first counted with their fingers. They also had various methods for recording such quantities as the number of animals in a herd or the days since the full moon. To represent such amounts, they used a corresponding number of pebbles, knots in a cord, or marks on wood, bone, or stone. They also learned to use regular shapes when they molded pottery or carved arrowheads.
By about 3000 B.C., mathematicians of ancient Egypt used a decimal system (a system of counting in groups of 10) without place values. The Egyptians pioneered in geometry, developing formulas for finding the area and volume of simple figures. Egyptian mathematics had many practical applications, ranging from surveying fields after the annual floods to making the intricate calculations necessary to build the pyramids.
Around the same time, the people of ancient Mesopotamia, in what is now Iraq, developed a sexagesimal system—a system based on groups of 60. Today, we use such a system to measure time in hours, minutes, and seconds. Historians do not know exactly how this system developed. They think it may have arisen from the use of weights and measures based on groups of 60. The system had important uses in astronomy, and also in commerce, because 60 can be divided easily.
The Greeks.
Ancient Greek scholars became the first people to explore pure mathematics, apart from practical problems. They made important advances by introducing the concepts of logical deduction and proof to create a systematic theory of mathematics. According to tradition, one of the first to provide mathematical proofs based on deduction was the philosopher Thales, who worked in geometry about 600 B.C.
The Greek philosopher Pythagoras, who lived about 550 B.C., explored the nature of numbers, believing that everything could be understood in terms of whole numbers or their ratios. However, about 400 B.C., the Greeks discovered irrational numbers (numbers that cannot be expressed as a ratio of two whole numbers), and they recognized that Pythagorean ideas were incomplete. About 370 B.C., Eudoxus of Cnidus, a Greek astronomer, formulated a theory of proportions to resolve problems associated with irrational numbers. He also developed the method of exhaustion, a way of determining areas of curved figures, which foreshadowed integral calculus.
Euclid, one of the foremost Greek mathematicians, wrote the Elements about 300 B.C. In this book, Euclid constructs an entire system of geometry by means of basic axioms, abstract definitions, and logical deductions. During the 200’s B.C., the Greek mathematician Archimedes extended the method of exhaustion. Using a 96-sided figure to approximate a circle, he calculated a highly accurate value for pi (the ratio of a circle’s circumference to its diameter). Ptolemy, an astronomer in Alexandria, Egypt, applied geometry and trigonometry to astronomy about A.D. 150 in a 13-part work on planetary motions. It became known as the Almagest, meaning the greatest.
Chinese mathematics
originally developed to aid record keeping, land surveying, and building. By the 100’s B.C., the Chinese had devised a decimal system of numbers that included fractions, zero, and negative numbers. They solved arithmetic problems with the aid of special sticks called counting rods. The Chinese also used these devices to solve equations—even groups of simultaneous equations in several unknowns.
Perhaps the best-known early Chinese mathematical work is the Jiu Zhang Suan Shu (Nine Chapters on the Mathematical Art), a handbook of practical problems that was compiled in the first two centuries B.C. In 263 A.D., the Chinese mathematician Liu Hui wrote a commentary on the book. Among Liu Hui’s greatest achievements was his analysis of a mathematical statement called the Gou-Gu theorem. The theorem, known as the Pythagorean theorem in the West, describes a special relationship that exists between the sides of a right triangle. Liu Hui also calculated the value of pi more accurately than ever before. He did so by using a figure with 3,072 equal sides to approximate a circle.
In the Middle East.
Starting in 750, an Islamic empire called the Abbāsid caliphate ruled much of the Middle East. Scholars in the caliphate translated and preserved the works of ancient Greek mathematicians and made their own original contributions as well. A book written about 825 by the Persian mathematician Al-Khwārizmī described a numeration system developed in India. This decimal system, which used place values and zero, became known as the Arabic numeral (or Hindu-Arabic numeral) system. Al-Khwārizmī also wrote an influential book about algebra. The word algebra comes from the Arabic title of this book. In addition, the word algorithm is derived from his name.
In the mid-1100’s, a Latin translation of al-Khwārizmī’s book on arithmetic introduced the Arabic numeral system to Europe. In 1202, Leonardo Fibonacci, an Italian mathematician, published a book on algebra that helped promote this system. Arabic numerals gradually replaced Roman numerals in Europe.
Middle Eastern astronomers of the 900’s made major contributions to trigonometry. In the 1000’s, an Arab physicist known as Alhazen applied geometry to optics. The Persian poet and astronomer Omar Khayyam wrote an important book on algebra about 1100. In the 1200’s, Nasir al-Din al-Tusi, a Persian mathematician, created ingenious mathematical models for use in astronomy.
The Renaissance.
During the 1400’s and 1500’s, European explorers sought new overseas trade routes, stimulating the application of mathematics to navigation and commerce. Artists created a system of mathematical perspective that gave their paintings an illusion of depth and distance. The invention of printing with movable type in the mid-1400’s resulted in speedy and widespread communication of mathematical knowledge.
The Renaissance also brought major advances in pure mathematics. In a book published in 1533, a German mathematician known as Regiomontanus established trigonometry as a field separate from astronomy. French mathematician François Viète made advances in algebra in a book published in 1591.
Mathematics and the scientific revolution.
By 1600, the increased use of mathematics and the growth of the experimental method were contributing to revolutionary advances in knowledge. In 1543, Nicolaus Copernicus, a Polish astronomer, published an influential book in which he argued that the sun, not Earth, is the center of the universe. In 1614, John Napier, a Scottish mathematician, published his discovery of logarithms, numbers that can be used to simplify such complicated calculations as those used in astronomy. Galileo, an Italian astronomer of the late 1500’s and early 1600’s, found that many types of motion can be analyzed mathematically.
In 1637, French philosopher René Descartes proposed mathematics as the perfect model for reasoning. He invented analytic geometry. Another French mathematician of the 1600’s, Pierre de Fermat, founded modern number theory. He and French philosopher Blaise Pascal explored probability theory. Fermat’s work with infinitesimals helped lay a foundation for calculus.
The English scientist Sir Isaac Newton invented calculus in the mid-1660’s. His discovery was published in 1687. Working independently, the German philosopher and mathematician Gottfried Wilhelm Leibniz also invented calculus, in the mid-1670’s. He published his findings in 1684 and 1686.
Developments in the 1700’s.
A remarkable family of Swiss mathematicians, the Bernoullis, made many contributions to mathematics during the late 1600’s and the 1700’s. Jakob Bernoulli did pioneering work in analytic geometry and wrote about probability theory. Jakob’s brother Johann also worked in analytic geometry and in mathematical astronomy and physics. Johann’s son Nicolaus helped advance probability theory. Johann’s son Daniel used mathematics to study the motion of fluids and the properties of vibrating strings. 
During the mid-1700’s, Swiss mathematician Leonhard Euler advanced calculus by showing that the operations of differentiation and integration were opposites. Beginning in the late 1700’s, French mathematician Joseph-Louis Lagrange worked to develop a firmer foundation for calculus. He was suspicious of relying on assumptions from geometry and, instead, developed calculus entirely in terms of algebra. There were also great advances in the study of differential equations, in which the solution involves a function (relation between two variables) or set of functions rather than a number or set of numbers. Calculus of variations, used for finding a function or curve that optimizes a particular situation, was also developed. For example, the tautochrone problem finds the curve for which two balls, when placed on the curve, roll to the bottom of it in the same amount of time no matter how high up either ball is originally placed.
In the 1800’s,
public education expanded rapidly, and mathematics became a standard part of university education. Many of the great works in mathematics of the 1800’s were written as textbooks. In the 1790’s and early 1800’s, French mathematician Adrien Marie Legendre wrote particularly influential textbooks and did work in calculus, geometry, and number theory. Important calculus textbooks by French mathematician Augustin Louis Cauchy were published in the 1820’s. Cauchy and Jean Baptiste Fourier, another French mathematician, made significant advances in mathematical physics.
Carl Friedrich Gauss, a German mathematician, proved the fundamental theorem of algebra, which states that every equation has at least one root. His work with imaginary numbers led to their increased acceptance. In the 1810’s, Gauss developed a non-Euclidean geometry but did not publish his discovery. Working separately, Janos Bolyai of Hungary and Nikolai Lobachevsky of Russia also developed non-Euclidean geometries. They published their discoveries about 1830. In the mid-1800’s, Georg Friedrich Bernhard Riemann of Germany developed another non-Euclidean geometry.
During the early 1800’s, the works of German mathematician August Ferdinand Möbius helped develop a study in geometry that became known as topology. Topology explores the properties of a geometrical figure that do not change when the figure is bent or stretched.
In the late 1800’s, German mathematician Karl Theodor Wilhelm Weierstrass worked to establish a more solid theoretical foundation for calculus. In the 1870’s and 1880’s, his student Georg Cantor developed set theory and a mathematical theory of the infinite.
Much exciting work in applied mathematics was done in the 1800’s. In the United Kingdom, Charles Babbage developed early mechanical computing machines, and George Boole created a system of symbolic logic. During the late 1800’s, French mathematician Henri Poincaré contributed to probability theory, celestial mechanics, and the study of electromagnetic radiation.
Philosophies of mathematics.
Many mathematicians have shown concern for the philosophical foundations of mathematics. To eliminate contradictions, some mathematicians have used logic to develop mathematics from a set of axioms. Two British philosophers and mathematicians, Alfred North Whitehead and Bertrand Russell, promoted a philosophy of mathematics called logicism. In their three-volume work, Principia Mathematica (1910-1913), they argued that all propositions (statements) in mathematics can be derived logically from just a few axioms.
David Hilbert, a German mathematician of the early 1900’s, was a formalist. Formalists consider mathematics to be a purely formal system of rules. Hilbert’s work led to the study of imaginary spaces with an infinite number of dimensions.
Beginning in the early 1900’s, Dutch mathematician Luitzen Brouwer championed intuitionism. He believed people understand the laws of mathematics by intuition (knowledge not gained by reasoning or experience).
In the early 1930’s, Austrian mathematician Kurt Gödel demonstrated that for any logical system, there are always theorems that cannot be proved either true or false by the axioms within that system. He found this to be true even of basic arithmetic.
Mathematicians made major advances in the study of abstract mathematical structures during the 1900’s. One such structure is the group. A group is a collection of items, which may be numbers, and rules for some operation with these items, such as addition or multiplication. Group theory is useful in many areas of mathematics and such fields as subatomic physics.
Since 1939, a group of mathematicians, most of whom are French, have published an influential series of books under the pen name Nicolas Bourbaki. This series takes an abstract approach to mathematics, using axiom systems and set theory.
New areas of mathematical specialization arose during the 1900’s, including systems analysis and computer science. Advances in mathematical logic have been essential to the development of electronic computers. Computers, in turn, enable mathematicians to complete long and complicated calculations quickly. Since the 1970’s, computer-based mathematical models have become widely used to study weather patterns, economic relationships, and many other systems.
Other new areas of mathematical specialization in the late 1900’s included fractal geometry and chaos theory. Fractal geometry deals with complex shapes called fractals. These shapes consist of small-scale and large-scale structures that resemble one another. Certain fractals are also similar to natural objects, such as coastlines or branching trees. Although fractals seem irregular, they have a simple organizing principle. Chaos theory tries to find underlying patterns in variations that seem random, such as changes in the weather or the stock market.
Mathematics flourishes by solving problems and creating new ones. Some of the most difficult and longstanding mathematical problems have been solved since the mid-1900’s. For example, the four color theorem, which states that only four colors are needed to distinguish separate areas on any two-dimensional map, was proved with the aid of computers by American mathematician Kenneth Appel and German American mathematician Wolfgang Haken in 1974. The proof of Fermat’s Last Theorem, which states that if n is greater than 2 there are no whole-number solutions to the equation x n + y n = z n , was published by British mathematician Andrew Wiles in 1995. In 2016, Ukrainian mathematician Maryna Viazovska proved a longstanding conjecture (unproven statement) about the most efficient way to pack spheres in a space containing either 8 or 24 dimensions.
Many interesting conjectures are still awaiting proof or disproof. For example, Goldbach’s conjecture states that every even integer greater than 2 is the sum of two prime numbers. Another example is the twin prime conjecture, which states that there are infinitely many prime numbers that differ by just two. Also, no one has been able to prove that there are infinitely many Mersenne primes, which are prime numbers of the form 2p – 1 where p is a prime number.
Trends in teaching mathematics.
Before the 1950’s, most mathematics courses in elementary, junior high, and high schools in the United States stressed the development of basic computational skills. During the late 1950’s and the 1960’s, new mathematics was introduced. New mathematics is a way of teaching mathematics that stresses understanding concepts rather than memorizing rules and performing repetitious drills. In the 1970’s and 1980’s, educators continued to use new mathematics, but they gave added emphasis to problem solving and computational skills. More recently, educators have become aware of different learning styles and of the importance of taking a student’s life and mathematics background into account when applying different teaching techniques. Today, educators make much greater use of technology and computer programs to assist in learning. Group activities in the classroom have largely replaced lectures and rote learning techniques.
At the college level, educators have moved away from teaching mathematics in the same way to all students. Instead, colleges and universities offer more courses in specialized applications of mathematics in such fields as economics, engineering, and physics. Long-term projects have also been added to further student understanding as well as to incorporate research and communication skills.
Careers
A strong background in mathematics is excellent preparation for a wide variety of careers. Students who wish to study mathematics in college should take high school courses in algebra, geometry, trigonometry, statistics, and calculus, if available. These courses also are useful for study in architecture, engineering, linguistics (the study of language), and all the natural and social sciences.
In college, the basic courses for a major in mathematics include advanced calculus, graph theory, numerical analysis, probability and statistics, theories of real and complex variables, topology, and differential equations. Courses in logic and computer programming also are useful in preparing for many careers.
Mathematicians teach at all levels. High school mathematics teachers must have at least a bachelor’s degree in mathematics. Many mathematicians with a doctor’s degree teach at colleges and universities.
Large numbers of mathematicians work in business, government, or industry. People with a mathematics degree may find work as accountants and auditors, actuaries, data and computer scientists, engineers, financial and stock analysts, medical scientists, software developers, and statisticians. Those with advanced degrees conduct research for the communications, energy, manufacturing, or transportation industries. Mathematicians also work in the computer industry as programmers or as systems analysts who determine the most efficient use of a computer in any given situation. Insurance companies employ mathematicians as actuaries to calculate risks and help design policies. Mathematicians also work as software developers, as data analysts, and in the field of computer science. Some mathematicians work as cryptanalysts (people who work with codes and ciphers) to keep the flow of information secure.
