Numbers and Mathematics
It is said that mathematics is the base of all other sciences, and that arithmetic, the science of numbers, is the base of mathematics. Numbers consist of whole numbers (integers) which are formed by the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 and by combinations of them. For example，247 —two hundred and forty seven—is a number formed by three digits. Parts of numbers smaller than 1 are sometimes expressed in terms of fractions, but in scientific usage they are given as decimals. This is because it is easier to perform the various mathematical operations if decimals are used instead of fractions.
The main operations are: to add, subtract, multiply and divide; to square, cube or raise to any other power；to take a square, cube or any other root and to find a ratio or proportion between pairs of numbers or a series of numbers. Thus, the decimal, or tenscale, system is used for scientific purposes throughout the world, even in countries whose national systems af weights and measurements are based upon other scales. The other scale in general use nowadays is the binary, or two- scale, in which numbers are expressed by combinations of only two digits, 0 and 1. Thus, in the binary scale, 2 is expressed as 010，3 is given as 011, 4 is represented as 100, etc. This scale is perfectly adapted to the "off-on" pulses of electricity, so it is widely used in electronic computers; because of its simplicity it is often called "the lazy schoolboy's dream".
Other branches of mathematics such as algebra and geometry are also extensively used in many sciences and even in some areas of philosophy. More specialized extensions, such as probability theory and group theory, are now applied to an increasing range of activities，from economics and the design of experiments to war and politics. Finally, a knowledge of statistics is required by every type of scientist for the analysis of data. Moreover, even an elementary knowledge of this branch of mathematics is sufficient to enable the journalist to avoid misleading his readers, or the ordinary citizen to detect the attempts which are constantly made to deceive him.
II. Mathematical Methods in Physics and Engineering
Students of physics and engineering, after having completed the standard course in calculus through differential equations, are faced with the problem of deciding what additional mathematics to take. Some leave mathematics out of their curriculum altogether, only to discover, after they have embarked on a program of graduate study, that they must go back and take undergraduate courses in order to acquire the mathematical background and maturity necessary to understand work being done in modern physics and engineering. Others, in their haste to pick up additional techniques not covered in the elementary calculus course, take courses in advanced engineering mathematics which cover such topics as Fourier series, Laplace transforms, partial differential equations, boundary-value problems, and complex variables. These topics are often presented in a very heuristic fashion, because the students lack a solid background in analysis. In fact, many of these mathematical techniques are taught in the same heuristic manner in courses in physics and engineering when they are needed in specific applications. Eventually, however, most graduate students will need a thorough understanding of applied mathematics if they are going to be able to read the literature in their own field and use mathematics effectively in their own work.
It is my opinion that a student who intends to do graduate work in physics or in some branches of engineering should develop a broad mathematical base for his graduate studies when he is an undergraduate. He should do this by taking a course in what has traditionally been called advanced calculus, followed by an introduction to mathematical physics based on the advanced calculus. Without the burden of presenting applications, advanced calculus can be a course which carefully develops the concepts of function, limit, continuity, differentiation, integration, infinite series, improper integrals, and possibly functions of a complex variable. Then, with this as a background, the applications can be presented in the mathematical physics course, and the students will be capable of understanding thoroughly the mathematics involved. This book has been written to fill the need for a textbook for the latter course.
The student is assumed to have had elementary calculus through differential equations, some elementary mechanics, plus the kind of advanced calculus course described above including vector analysis. Some knowledge of functions of a complex variable is very helpful but not essential. The bulk of material can be understood without prior knowledge of complex variable analysis. The necessary higher algebra is developed from the beginning, so that no more than elementary college algebra is needed.
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