The number π is a mathematical constant, approximately equal to 3.14159, that is the ratio of a circle's circumference to its diameter. It appears in many formulae across mathematics and physics, and some of these formulae are commonly used for defining π, to avoid relying on the definition of the length of a curve.
The number π is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as 22/7 are commonly used to approximate it. Consequently, its decimal representation never ends, nor enters a permanently repeating pattern. It is a transcendental number, meaning that it cannot be a solution of an algebraic equation involving only finite sums, products, powers, and integers. The transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge. The decimal digits of π appear to be randomly distributed, but no proof of this conjecture has been found.
For thousands of years, mathematicians have attempted to extend their understanding of π, sometimes by computing its value to a high degree of accuracy. Ancient civilizations, including the Egyptians and Babylonians, required fairly accurate approximations of π for practical computations. Around 250 BC, the Greek mathematician Archimedes created an algorithm to approximate π with arbitrary accuracy. In the 5th century AD, Chinese mathematicians approximated π to seven digits, while Indian mathematicians made a five-digit approximation, both using geometrical techniques. The first computational formula for π, based on infinite series, was discovered a millennium later. The earliest known use of the Greek letter π to represent the ratio of a circle's circumference to its diameter was by the Welsh mathematician William Jones in 1706. The invention of calculus soon led to the calculation of hundreds of digits of π, enough for all practical scientific computations. Nevertheless, in the 20th and 21st centuries, mathematicians and computer scientists have pursued new approaches that, when combined with increasing computational power, extended the decimal representation of π to many trillions of digits. These computations are motivated by the development of efficient algorithms to calculate numeric series, as well as the human quest to break records. The extensive computations involved have also been used to test the correctness of new computer processors.
Because it relates to a circle, π is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses and spheres. It is also found in formulae from other topics in science, such as cosmology, fractals, thermodynamics, mechanics, and electromagnetism. It also appears in areas having little to do with geometry, such as number theory and statistics, and in modern mathematical analysis can be defined without any reference to geometry. The ubiquity of π makes it one of the most widely known mathematical constants inside and outside of science. Several books devoted to π have been published, and record-setting calculations of the digits of π often result in news headlines.
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:
π = 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.
In modern mathematics, this definition is not fully satisfactory for several reasons. Firsly, it lacks a rigorous definition of the length of a curved line. Such a definition requires at least the concept of a limit, or, more generally, the concepts of derivatives and integrals. Also, diameters, circles and circumferences can be defined in Non-Euclidean geometries, but, in such a geometry, the ratio C / d need not to be a constant, and need not to equal to π. Also, there are many occurrences of π in many branches of mathematics that are completely independent from geometry, and in modern mathematics, the trend is to built geometry from algebra and analysis rather than independently from the other branches of mathematics.
https://en.wikipedia.org/wiki/Pi
Archimedes’ Method of Approximating Pi
Since the true value of pi could not be measured directly, Archimedes developed a geometric technique using polygons to establish upper and lower bounds for its value. His method relied on inscribing and circumscribing regular polygons around a circle and calculating their perimeters. By progressively increasing the number of sides, he was able to narrow the range within which pi must lie. This approach was a precursor to the concept of limits, which later became a fundamental idea in calculus.
The Inscribed and Circumscribed Polygon Method
In his work Measurement of a Circle Archimedes considered a circle with diameter d and radius r. He inscribed a regular hexagon inside the circle and circumscribed another hexagon outside it. By calculating the perimeters of these polygons, he obtained lower and upper estimates for the circumference of the circle. Since the ratio of the circumference to the diameter is pi (C / d = pi), these perimeters provided bounds for pi.
He then systematically increased the number of sides of the polygons, doubling them from 6-sided to 12-sided, 24-sided, 48-sided, and finally 96-sided polygons. As the number of sides increased, the perimeters of the inscribed and circumscribed polygons became closer to the true circumference of the circle, refining the estimate of pi.
Using this method, Archimedes established the following inequality:
223/71 < pi < 22/7
This meant that pi was approximately 3.1408 < pi < 3.1429, a remarkably accurate estimate for the time.
Mathematical Process Behind Archimedes’ Approximation
To derive these values, Archimedes used the Pythagorean theorem and properties of similar triangles to calculate the side lengths of the polygons. By repeatedly applying trigonometric relationships (though without the formal notation used today), he determined the perimeters of each successive polygon. His method can be broken down as follows:
- For an inscribed n-sided polygon:
The perimeter Pi provides a lower bound for the circle’s circumference.
Formula: Pi = n * s~i~, where s~i~ is the side length.
- For a circumscribed n-sided polygon:
The perimeter P~c~ gives an upper bound for the circumference.
Formula: P~c~ = n * s~c~, where sc is the side length.
- Refining the estimate:
Archimedes doubled the number of sides, recalculating the new perimeters iteratively.
The values of P~i~ and P~c~ converged toward the true circumference of the circle, P~i~ < C < P~c~.
By the time he reached a 96-sided polygon, his estimates were precise to two decimal places. This level of accuracy was unprecedented and remained the best approximation of pi for nearly 1,000 years.
Circle circumscribed and inscribed by a square where n=4.
The Limitations of Archimedes’ Approach
Archimedes' method had several inherent limitations. First, the computational intensity of his approach increased significantly as the number of sides in his polygons grew. Without the tools of modern algebra or trigonometry, he had to rely solely on geometric reasoning, making the process increasingly complex. Additionally, his method could only provide an approximation of pi rather than an exact value. Since pi is an irrational number that cannot be expressed as a finite fraction, Archimedes' approach was necessarily limited in its precision. Another challenge was the laborious nature of manual computation. Each successive step required extensive geometric derivations, making further refinements impractical beyond a certain point. Despite these limitations, Archimedes' work demonstrated a systematic method for refining numerical approximations and laid the foundation for future mathematical advancements.
Implications of Archimedes’ Work on Pi
Archimedes' method of approximating pi was groundbreaking, not only for its accuracy but also for its influence on the development of mathematical techniques. His approach established a systematic way of refining numerical approximations, which later became essential in calculus and numerical analysis. His work remained the most accurate estimate of pi for over a millennium and laid the foundation for future mathematicians to further refine the calculation of pi.
Archimedes’ method set the stage for many mathematicians across different cultures to refine and improve the approximation of pi. In the 3rd century CE, the Chinese mathematician Liu Hui built upon Archimedes' technique and extended it to a 3072-sided polygon, achieving a more precise approximation of pi at 3.14159. Two centuries later, Zu Chongzhi improved on this result, determining that pi was approximately 355/113 (3.1415929), an extraordinarily precise fraction that remained the most accurate estimate for over a thousand years.
In the Islamic Golden Age, mathematicians such as Al-Khwarizmi and Al-Kashi expanded on these ideas using decimal notation and further refinements of the polygonal method. The Renaissance period saw renewed interest in Archimedes' approach, with European scholars like Ludolph van Ceulen extending the method to polygons with millions of sides. This allowed for calculations of pi accurate to more than 30 decimal places. Despite these advancements, Archimedes’ geometric method remained the dominant approach for approximating pi until the development of calculus in the 17th century.
Ludolph van Ceulen (8 January 1540 – 31 December 1610) was a German-Dutch mathematician from Hildesheim known for the Ludolphine number, his calculation of the mathematical constant pi to 35 digits.
Ludolph van Ceulen spent a major part of his life calculating the numerical value of the mathematical constant π, using essentially the same methods as those employed by Archimedes some seventeen hundred years earlier. He published a 20-decimal value in his 1596 book Van den Circkel ("On the Circle"), which was published before he moved to Leiden, and he later expanded this to 35 decimals.
Van Ceulen's 20 digits is more than enough precision for any conceivable practical purpose. Even if a circle was perfect down to the atomic scale, the thermal vibrations of the molecules of ink would make most of those digits physically meaningless. Future attempts to calculate π to ever greater precision have been driven primarily by curiosity about the number itself.
https://en.wikipedia.org/wiki/Ludolph_van_Ceulen
The above image is the title page of Vanden Circkel, a book about the circle and π by Ludolph Van Ceulen (1540–1610). Published in 1596 in Dutch, it contains the longest decimal approximation of π at the time—20 decimal places. In fact, below the portrait of Van Ceulen, the engraving on the title page has a circle with diameter of 10^20^. Across the top semicircle is “314159265358979323846 te cort” (too short), and “314159265358979323847 te lanck” (too long) is along the bottom semicircle. Later, Van Ceulen would determine π to 35 decimal places. A modified Latin version of the work was published in 1619, images of which can also be found on Convergence here and here.
Part of what little is known of Van Ceulen’s life before 1578 comes from the Preface of Vanden Circkel. Starting in 1566, he earned a living as a mathematics teacher, and in 1580 he opened his first fencing school. A few years later Archimedes’ method of approximating π was translated from the Greek for him, and Van Ceulen proceeded to use the technique to improve on approximations of π, publishing Vanden Circkel in 1596. Below are images from folio 1 and folio 7.
Chapter 21 is devoted to analyzing a work of Joseph Justice Scaliger (1540–1609) called Cyclometrica Elementa (Elements of Circle Measurement), which had several incorrect results, including a “proof” that the area of a circle is equal to 6/5 of the area of an inscribed regular hexagon, which results in π=(9/5)√3 or approximately 3.117691454. Van Ceulen doesn’t mention Scaliger by name, but rather calls him a “highly learned man”. Below is Folio 63a.
https://old.maa.org/press/periodicals/convergence/mathematical-treasure-van-ceulen-s-vanden-circkel
Van Ceulen is famed for his calculation of π to 35 places which he did using polygons with 2^62^ sides. Having published 20 places of π in his book of 1596, the more accurate results were only published after his death. In 1615 his widow Adriana Simondochter published a posthumous work by Van Ceulen entitled De arithmetische en geometrische fondamenten. This contained his computation of 33 decimal places for π. The complete 35 decimal place approximation was only published in 1621 in Snell's Cyclometricus. Having spent most of his life computing this approximation, it is fitting that the 35 places of π were engraved on Van Ceulen's tombstone. In fact Van Ceulen had purchased a grave in the Pieterskerk on 11 November 1602 but, after Van Ceulen's death on 31 December 1610, his widow Adriana exchanged this grave for another, still in the Pieterskerk, and it was in this second grave that Van Ceulen was buried on 2 January 1611. The tombstone gave both Van Ceulen's lower bound of 3.14159265358979323846264338327950288 and his upper bound of 3.14159265358979323846264338327950289. However, the original tombstone disappeared around 1800 to be replaced by a replica two hundred years later. The original text on the tombstone was known since it had been recorded in a guidebook of 1712 and after that reprinted in many articles. Vajta writes :
On July 5, 2000 a very special ceremony took place in the St Pieterskerk (St Peter's Church) at Leiden, the Netherlands. A replica of the original tombstone of Ludolph Van Ceulen was placed into the Church since the original disappeared. ... It was therefore a tribute to the memory of Ludolph Van Ceulen, when on Wednesday 5 July, 2000 prince Willem-Alexander (heir to the throne), unveiled the memorial tombstone in the St Peter's Church, in Leiden.
In Germany π was called the "Ludolphine number" for a long time.
https://mathshistory.st-andrews.ac.uk/Biographies/Van_Ceulen/