Pi (constant)

Constant Pi ( π ). Constant that relates the perimeter of any circumference to the length of its diameter π = L / D. This is not an exact number but it is one of the so-called irrational, it has infinite decimal places.

Summary

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• 1 Story
  • 1 Numerical nature
  • 2 Experts who continued the approach
  • 3 Modern era
• 2 Presence of Pi in usual formulas
• 3 Different ways to determine (π)
  • 1 Buffon’s method
  • 2 Pendulum method
  • 3 Statistical methods
  • 4 Through Computing
• 4 Curiosities
• 5 References
• 6 Sources

History

Already in antiquity, it was suggested that all circles conserved a close dependence between the contour and its radius, but only since the 17th century did the correlation become a digit and was identified with the name “Pi” (from periphereia, a name that the Greeks gave the perimeter of a circle), Throughout history, this illustrious figure has been assigned various amounts. In the Bible it appears with the value 3, in Babylon 3 1/8; the Egyptians gave him 4 (8/9) ² ; and in China 3.1724.

However, it was in Greece where the correspondence between the radius and the length of a circumference began to consolidate itself as one of the most outstanding puzzles to be solved. A contemporary of Socrates , Antiphon , inscribed in the circle a square, then an octagon, and devised multiplying the number of sides until such time as the obtained polygon almost fitted with the ring. Archimedes gathers and amplifies these results. Prove that the area of ​​a circle is half the product of its radius times the circumference and that the ratio of the perimeter to the diameter is between 3.14084 and 3.14285.

This notation was first used in 1706 by the Welsh mathematician William Jones and popularized by the mathematician Leonard Euler in his “Introduction to Infinitesimal Calculus” in 1748 ^[1] . It was previously known as Ludoph’s constant (after the mathematician Ludolph van Ceulen ) or as Archimedes’ constant.

The computed value of this constant has been known with different precisions in the course of history, in this way in one of the oldest documented references as the Bible appears indirectly associated with the natural number 3 and in Mesopotamia mathematicians used it like 3 and an added fraction of 1/8. Euclid specifies in his Elements the necessary steps to the limit and investigates a system consisting of doubling, like Antiphon, the number of sides of regular polygons and demonstrating the convergence of the procedure. The first theoretical calculation seems to have been carried out by Archimedes, something that many people do not know today, that ( π) does not equal 22/7, and made no claim to have discovered the exact value. If we take their best approximation as the mean of these two limits, we obtain 3.1418, an error of approximately 0.0002.

It is important to realize that the use of trigonometry here is not historical: Archimedes did not have the advantages of algebraic and trigonometric notation and had to derive in a purely geometric way. Plus it didn’t even have the advantage of our decimal notation for numbers, so the calculation was by no means a trivial task. So it was a fabulous feat of imagination and calculation and the wonder is not that it stopped at 96-sided polygons, it went further.

Numerical nature

• It is an irrational number, since it cannot be represented by a fraction; It is represented by a decimal whose integer part is 3, and the decimal part contains an infinity of digits that do not have periodicity. It is therefore a real number.
• It is not a solution of an algebraic equation of a variable of the form a _0x ⁿ+ a ₁ x ^n-1 + … + a _n-1 x + a _n = 0, where a _i is an integer, i = 0, … n. ^[2] . So it is a transcendent number.

Experts who continued the approach

Except for Zu Chongzhi, about whom we know practically nothing and who are highly unlikely to know of Archimedes’ work, there was no theoretical advance in these improvements, only increased energy in calculation. Note how, as in all scientific questions, leadership passed from Europe to the East from the millennium from 400 to 1400 .

In the 2nd century Ptolemy uses polygons of up to 720 sides and a circumference of 60 radius units to get a little closer, and gives the value 3 + 8/60 + 30/3600 = 377/120 = 3’14166.

Modern era

As mathematics has been developed, in its various branches, algebra, calculus, etc., different devices have been built that allow its value to be refined more and more. One of the most curious cases in history was that of the English mathematician William Shanks , who after a job that took almost twenty years, obtained 707 decimal places in 1853 . Unfortunately, Shanks made a mistake in the 528th decimal, and from there they are all wrong.

Ferguson, in 1947 , obtained a value with 808 decimal places.

Using the Pegasus computer, in 1957 , a figure of 7,840 decimal places was achieved.

Later, in 1961 , using an IBM 7090 computer, 100,000 decimal places were reached.

Then, in 1967 , with a CDC 6600, 500,000 decimal places were reached.

In 1987 , with a Cray-2, a figure was obtained with 100,000,000 decimals for Pi.

And finally, in 1995 , at the University of Tokyo , a pi value of 3.14 was reached … and 4,294,960,000 decimals were added.

Presence of Pi in usual formulas

Length

• From the circumference: L = 2 × π × r. where r is the radius.

Areas

1. From the circle: A = π r ², where r is the radius
2. From the ellipse: A = π ab, where a and b are the semi-axes.
3. From the sphere: A = 4π r ²
4. Del toro: A = 4π ²Being R, the maximum distance of the circumference of radius r, to the center of rotation ^[3]

Volumes

1. From the sphere: V = 4/3 π r ³, where r is the radius
2. From the ellipsoid: V = 4/3 π abc, where a, b and c are semi-axes.
3. Del toro: V = 2π ²Rr ² ,

Different ways of determining ( π )

Buffon’s method

In the 18th century Georges Louis Leclerc, Count of Buffon, French naturalist, devised an ingenious method. called “Buffon’s Needle” which relates the number pi to the launch of a needle on a scratched surface. Buffon showed that if we randomly drop a needle of length L on a surface on which there are drawn parallel lines separated by a distance D (the calculation can be repeated using a tile floor and a needle), the probability that the needle cut to a line is: L * ( π ) / 2D. With a large number of runs, an acceptable value of ( π ) is achieved.

As mathematics has been developed , in its various branches, algebra, calculus, etc., different devices have been built that allow its value to be refined more and more.

Pendulum method

Using the period of a pendulum to make an estimate

Statistical methods

The proposed system is similar to that of the Earl of Buffon, based on probability. Suppose a circumference of radius 1, inscribed in a square If we randomly create pairs of numbers (x, y) between zero and one, if 1 = x ² + y ² the point generated by x and y will be inside the circle while if x ² + and ² different from 1 the points will logically be in the square but outside the roundabout. The probability that the points are within the circumference will be given by the relationship between the area of ​​the circle and the surface of the square. With a significant series of repetitions the ratio between those who fall in and out of the circle tends to ( π) / 4, and thus we obtain the value of ( π ) in a statistical way.

Through Computing

A program has been created that randomly generates the pairs of digits. Specifically, it creates 10 million points and determines the number ( π ) each million rolls. Being a statistical operation, sometimes we approach the correct value (known with thousands of figures) and sometimes we move away. With this technique we determine 3 correct decimal places obtaining an error close to 0.02%.

Curiosities

The March 14 is celebrated worldwide on Pi Day , a festival that was born in the United States in 1988 and has spread worldwide to promote mathematical disclosure. The date was chosen because its 3/14 notation is similar to the number π rounded to two decimal places (3.14).

In addition to this coincidence, the conference honors the German scientist Albert Einstein on his birth just then, in 1879.

To be more precise, just like those experienced in mathematics, the festivities will start at 1:59 local time for each nation, coinciding with the other digits that follow: 3.14159. ^[4]

• Equality e ^iπ+1 = 0, is an expression that gathers e, the base of the natural logarithms, i the imaginary unit, π the ratio between the circumference and its diameter, 0 and 1 binary digits sufficient to write all the real numbers and complex.