Limit (mathematics) in the context of "Regular polygon"

Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.

These theories are usually studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis.Analysis may be distinguished from geometry; however, it can be applied to any space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space).

Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.

Originally called infinitesimal calculus or "the calculus of infinitesimals", it has two major branches, differential calculus and integral calculus. The former concerns instantaneous rates of change, and the slopes of curves, while the latter concerns accumulation of quantities, and areas under or between curves. These two branches are related to each other by the fundamental theorem of calculus. They make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. It is the "mathematical backbone" for dealing with problems where variables change with time or another reference variable. It has also been called "the basic instrument of physical science".

The method of exhaustion (Latin: methodus exhaustionis) is a method of finding the area of a shape by inscribing inside it a sequence of polygons (one at a time) whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area between the nth polygon and the containing shape will become arbitrarily small as n becomes large. As this difference becomes arbitrarily small, the possible values for the area of the shape are systematically "exhausted" by the lower bound areas successively established by the sequence members.

The method of exhaustion typically required a form of proof by contradiction, known as reductio ad absurdum. This amounts to finding an area of a region by first comparing it to the area of a second region, which can be "exhausted" so that its area becomes arbitrarily close to the true area. The proof involves assuming that the true area is greater than the second area, proving that assertion false, assuming it is less than the second area, then proving that assertion false, too.

In geometry, the area enclosed by a circle of radius r is πr. Here, the Greek letter π represents the constant ratio of the circumference of any circle to its diameter, approximately equal to 3.14159.

One method of deriving this formula, which originated with Archimedes, involves viewing the circle as the limit of a sequence of regular polygons with an increasing number of sides. The area of a regular polygon is half its perimeter multiplied by the distance from its center to its sides, and because the sequence tends to a circle, the corresponding formula–that the area is half the circumference times the radius–namely, A = ⁠1/2⁠ × 2πr × r, holds for a circle.

In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex or star. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon (effectively a straight line), if the edge length is fixed.

In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a length, duration or temperature. Here, continuous means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion.

The real numbers are fundamental in calculus (and in many other branches of mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives.

In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets.

A topological space is the most general type of a mathematical space that allows for the definition of limits, continuity, and connectedness. Common types of topological spaces include Euclidean spaces, metric spaces and manifolds.

In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures in combinatorics through generating functions. The mathematical properties of infinite series make them widely applicable in other quantitative disciplines such as physics, computer science, statistics and finance.

Among the Ancient Greeks, the idea that a potentially infinite summation could produce a finite result was considered paradoxical, most famously in Zeno's paradoxes. Nonetheless, infinite series were applied practically by Ancient Greek mathematicians including Archimedes, for instance in the quadrature of the parabola. The mathematical side of Zeno's paradoxes was resolved using the concept of a limit during the 17th century, especially through the early calculus of Isaac Newton. The resolution was made more rigorous and further improved in the 19th century through the work of Carl Friedrich Gauss and Augustin-Louis Cauchy, among others, answering questions about which of these sums exist via the completeness of the real numbers and whether series terms can be rearranged or not without changing their sums using absolute convergence and conditional convergence of series.

In kinematics, the speed (commonly referred to as v) of an object is the magnitude of the change of its position over time or the magnitude of the change of its position per unit of time; it is thus a non-negative scalar quantity. The average speed of an object in an interval of time is the distance travelled by the object divided by the duration of the interval; the instantaneous speed is the limit of the average speed as the duration of the time interval approaches zero. Speed is the magnitude of velocity (a vector), which indicates additionally the direction of motion.

Speed has the dimensions of distance divided by time. The SI unit of speed is the metre per second (m/s), but the most common unit of speed in everyday usage is the kilometre per hour (km/h) or, in the US and the UK, miles per hour (mph). For air and marine travel, the knot is commonly used.