Technical preliminaries
analysis
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Numbers and functions
Number systems
Throughout this article are references to a variety ofnumber systems—that is, collections of mathematical objects (numbers) that can be operated on by some or all of the standard operations of arithmetic: addition, multiplication, subtraction, and division. Such systems have a variety of technical names (e.g.,group,ring, field) that are not employed here. This article shall, however, indicate which operations are applicable in the main systems of interest. These main number systems are:
- a. Thenatural numbers ℕ. These numbers are the positive (and zero) whole numbers 0, 1, 2, 3, 4, 5, …. If two such numbers are added or multiplied, the result is again a natural number.
- b. Theintegers ℤ. These numbers are the positive and negative whole numbers …, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, …. If two such numbers are added, subtracted, or multiplied, the result is again an integer.
- c. Therational numbers ℚ. These numbers are the positive and negative fractionsp/q wherep andq are integers andq ≠ 0. If two such numbers are added, subtracted, multiplied, or divided (except by 0), the result is again a rational number.
- d. Thereal numbers ℝ. These numbers are the positive and negativeinfinite decimals (including terminating decimals that can be considered as having an infinite sequence of zeros on the end). If two such numbers are added, subtracted, multiplied, or divided (except by 0), the result is again a real number.
- e. Thecomplex numbers ℂ. These numbers are of the formx +iy wherex andy are real numbers andi =Square root of√−1. (For further explanation,see the sectionComplex analysis.) If two such numbers are added, subtracted, multiplied, or divided (except by 0), the result is again a complex number.
Functions
In simple terms, afunctionf is a mathematical rule that assigns to a numberx (in some number system and possibly with certain limitations on its value) another numberf(x). For example, the function “square” assigns to each numberx its squarex2. Note that it is the general rule, not specific values, thatconstitutes the function.
The common functions that arise in analysis are usually definable by formulas, such asf(x) =x2. They include the trigonometric functions sin (x), cos (x), tan (x), and so on; the logarithmic function log (x); theexponential function exp (x) orex (wheree = 2.71828… is a specialconstant called the base of natural logarithms); and thesquare root functionSquare root of√x. However, functions need not be defined by single formulas (indeed by any formulas). For example, the absolutevalue function |x| is defined to bex whenx ≥ 0 but −x whenx < 0 (where ≥ indicates greater than or equal to and < indicates less than).
The problem of continuity
The logical difficulties involved in setting upcalculus on a sound basis are all related to one central problem, the notion ofcontinuity. This in turn leads to questions about the meaning of quantities that becomeinfinitely large or infinitely small—concepts riddled with logical pitfalls. For example, a circle of radiusr has circumference 2πr and area πr2, where π is the famous constant 3.14159…. Establishing these two properties is not entirelystraightforward, although an adequate approach was developed by the geometers of ancient Greece, especiallyEudoxus andArchimedes. It is harder than one might expect to show that the circumference of a circle is proportional to its radius and that its area is proportional to the square of its radius. The really difficult problem, though, is to show that the constant ofproportionality for the circumference is precisely twice the constant of proportionality for the area—that is, to show that the constant now called π really is the same in both formulas. This boils down to proving atheorem (first proved by Archimedes) that does not mention π explicitly at all: the area of a circle is the same as that of a rectangle, one of whose sides is equal to the circle’s radius and the other to half the circle’s circumference.
Approximations in geometry
A simple geometric argument shows that such an equality must hold to a high degree ofapproximation. The idea is to slice the circle like a pie, into a large number of equal pieces, and to reassemble the pieces to form an approximate rectangle (seefigure). Then the area of the “rectangle” is closely approximated by its height, which equals the circle’s radius, multiplied by the length of oneset of curved sides—which together form one-half of the circle’s circumference. Unfortunately, because of the approximations involved, this argument does not prove the theorem about the area of a circle. Further thought suggests that as the slices get very thin, theerror in the approximation becomes very small. But that still does not prove the theorem, for an error, however tiny, remains an error. If it made sense to talk of the slices being infinitesimally thin, however, then the error would disappear altogether, or at least it would becomeinfinitesimal.
Actually, there exist subtle problems with such a construction. It might justifiably be argued that if the slices are infinitesimally thin, then each has zero area; hence, joining them together produces a rectangle with zero total area since 0 + 0 + 0 +⋯ = 0. Indeed, the very idea of aninfinitesimal quantity is paradoxical because the only number that is smaller than every positive number is 0 itself.
The same problem shows up in many different guises. When calculating the length of the circumference of a circle, it is attractive to think of the circle as a regular polygon with infinitely many straight sides, each infinitesimally long. (Indeed, a circle is the limiting case for a regular polygon as the number of its sides increases.) But while this picture makes sense for some purposes—illustrating that the circumference is proportional to the radius—for others it makes no sense at all. For example, the “sides” of the infinitely many-sided polygon must have length 0, which implies that the circumference is 0 + 0 + 0 + ⋯ = 0, clearly nonsense.