Pi in the context of "Mathematical constant"

A number is a mathematical object used to count, measure, and label. The most basic examples are the natural numbers: 1, 2, 3, 4, 5, and so forth. Individual numbers can be represented in language with number words or by dedicated symbols called numerals; for example, "five" is a number word and "5" is the corresponding numeral. As only a limited list of symbols can be memorized, a numeral system is used to represent any number in an organized way. The most common representation is the Hindu–Arabic numeral system, which can display any non-negative integer using a combination of ten symbols, called numerical digits. Numerals can be used for counting (as with cardinal number of a collection or set), labels (as with telephone numbers), for ordering (as with serial numbers), and for codes (as with ISBNs). In common usage, a numeral is not clearly distinguished from the number that it represents.

In mathematics, the notion of number has been extended over the centuries to include zero (0), negative numbers, rational numbers such as one half ${\displaystyle \left({\tfrac {1}{2}}\right)}$, real numbers such as the square root of 2 ${\displaystyle \left({\sqrt {2}}\right)}$, and π, and complex numbers which extend the real numbers with a square root of −1, and its combinations with real numbers by adding or subtracting its multiples. Calculations with numbers are done with arithmetical operations, the most familiar being addition, subtraction, multiplication, division, and exponentiation. Their study or usage is called arithmetic, a term which may also refer to number theory, the study of the properties of numbers.

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.

Approximations for the mathematical constant pi (π) in the history of mathematics reached an accuracy within 0.04% of the true value before the beginning of the Common Era. In Chinese mathematics, this was improved to approximations correct to what corresponds to about seven decimal digits by the 5th century.

Further progress was not made until the 14th century, when Madhava of Sangamagrama developed approximations correct to eleven and then thirteen digits. Jamshīd al-Kāshī achieved sixteen digits next. Early modern mathematicians reached an accuracy of 35 digits by the beginning of the 17th century (Ludolph van Ceulen), and 126 digits by the 19th century (Jurij Vega).

In geometry, a square is a regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal sides. As with all rectangles, a square's angles are right angles (90 degrees, or π/2 radians), making adjacent sides perpendicular. The area of a square is the side length multiplied by itself, and so in algebra, multiplying a number by itself is called squaring.

Equal squares can tile the plane edge-to-edge in the square tiling. Square tilings are ubiquitous in tiled floors and walls, graph paper, image pixels, and game boards. Square shapes are also often seen in building floor plans, origami paper, food servings, in graphic design and heraldry, and in instant photos and fine art.

A circular arc is the arc of a circle between a pair of distinct points. If the two points are not directly opposite each other, one of these arcs, the minor arc, subtends an angle at the center of the circle that is less than π radians (180 degrees); and the other arc, the major arc, subtends an angle greater than π radians. The arc of a circle is defined as the part or segment of the circumference of a circle. A straight line that connects the two ends of the arc is known as a chord of a circle. If the length of an arc is exactly half of the circle, it is known as a semicircular arc.

Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square with the area of a given circle by using only a finite number of steps with a compass and straightedge. The difficulty of the problem raised the question of whether specified axioms of Euclidean geometry concerning the existence of lines and circles implied the existence of such a square.

In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem, which proves that pi (${\displaystyle \pi }$) is a transcendental number.That is, ${\displaystyle \pi }$ is not the root of any polynomial with rational coefficients. It had been known for decades that the construction would be impossible if ${\displaystyle \pi }$ were transcendental, but that fact was not proven until 1882. Approximate constructions with any given non-perfect accuracy exist, and many such constructions have been found.

In mathematics, the irrational numbers are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable, meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself.

Among irrational numbers are the ratio π of a circle's circumference to its diameter, Euler's number e, the golden ratio φ, and the square root of two. In fact, all square roots of natural numbers, other than of perfect squares, are irrational.

In mathematics, a limiting case of a mathematical object is a special case that arises when one or more components of the object take on their most extreme possible values. For example:

• In statistics, the limiting case of the binomial distribution is the Poisson distribution. As the number of events tends to infinity in the binomial distribution, the random variable changes from the binomial to the Poisson distribution.
• A circle is a limiting case of various other figures, including the Cartesian oval, the ellipse, the superellipse, and the Cassini oval. Each type of figure is a circle for certain values of the defining parameters, and the generic figure appears more like a circle as the limiting values are approached.
• Archimedes calculated an approximate value of π by treating the circle as the limiting case of a regular polygon with 3 × 2 sides, as n gets large.
• In electricity and magnetism, the long wavelength limit is the limiting case when the wavelength is much larger than the system size.
• In economics, two limiting cases of a demand curve or supply curve are those in which the elasticity is zero (the totally inelastic case) or infinity (the infinitely elastic case).
• In finance, continuous compounding is the limiting case of compound interest in which the compounding period becomes infinitesimally small, achieved by taking the limit as the number of compounding periods per year goes to infinity.

A limiting case is sometimes a degenerate case in which some qualitative properties differ from the corresponding properties of the generic case. For example: