3/14 pi day Key Lime for me today

Definition

The circumference of a circle is slightly more than three times as long as its diameter. The exact ratio is called π.

π is commonly defined as the ratio of a circle‘s circumference C to its diameter d:[1]

$\pi = \frac{C}{d}$

The ratio C/d is constant, regardless of the circle’s size. For example, if a circle has twice the diameter of another circle it will also have twice the circumference, preserving the ratio C/d. This definition of π is not universal, because it is valid only in flat (Euclidean) geometry; it is not valid in curved (non-Euclidean) geometries.[1] For this reason, some mathematicians prefer definitions of π based on calculus or trigonometry that do not rely on the circle. One such definition is: π is twice the smallest positive x for which cos(x) equals 0.[1][2]

Name

The symbol used by mathematicians to represent the ratio of a circle’s circumference to its diameter is the Greek letter π. That letter (and therefore the number π itself) can be denoted by the Latin word pi.[3] In English, π is pronounced as “pie” ( , /ˈpaɪ/).[4] The lower-case letter π (or π in sans-serif font) is not to be confused with the capital letter Π, which denotes a product of a sequence.

The earliest known use of the Greek letter π to represent the ratio of a circle’s circumference to its diameter was by mathematician William Jones in his 1706 work Synopsis Palmariorum Matheseos; or, a New Introduction to the Mathematics.[5] The Greek letter first appears there in the phrase “1/2 Periphery (π)” in the discussion of a circle with radius one. Jones may have chosen π because it was the first letter in the Greek spelling of the word periphery.[6] However, he writes that his equations for π are from the “ready pen of the truly ingenious Mr. John Machin”, leading to speculation that Machin may have employed the Greek letter before Jones.[7] It had indeed been used earlier for geometric concepts.[7] William Oughtred used π and δ, the Greek letter equivalents of p and d, to express ratios of periphery and diameter in the 1647 and later editions of Clavis Mathematicae.

After Jones introduced the Greek letter in 1706, it was not adopted by other mathematicians until Euler started using it, beginning with his 1736 work Mechanica. Before then, mathematicians sometimes used letters such as c or p instead.[7] Because Euler corresponded heavily with other mathematicians in Europe, the use of the Greek letter spread rapidly.[7] In 1748, Euler used π in his widely read work Introductio in analysin infinitorum (he wrote: “for the sake of brevity we will write this number as π; thus π is equal to half the circumference of a circle of radius 1″) and the practice was universally adopted thereafter in the Western world.[7]

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