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Topic Last Updated on 10-07-2024
Post on topic: Pi Number.
А Grand Competition
If one were to measure the circumference of a circle with a rope, it would turn out that it’s equal to approximately three times its own diameter — people determined this as far back as ancient times. The amazing thing is that this ratio is true for any circle, no matter what size it may be, from a button to a wheel. In other words, all circumferences can be expressed by a specific constant, which is known to be slightly greater than three. For hundreds of years, it plagued the minds of great thinkers since it clearly was a value of great significance, and they had almost succeeded in calculating it, but the search dragged on for thousands of years.
The Ancient Babylonians defined the value of π as equal to 3. This comes from the formula for the area of a circle S = l²/12 (where l i is the circumference of the circle), which was found in the calculations of the Babylonians. This is a very rough estimate that leaves a lot of room for errors. Of course, they didn’t use the in designation π in Babylon — with this character we refer to the ratio of the circumference to its diameter.
Egyptians use The pi number
The Egyptians used geometric calculations for land surveying and architecture when they came across the number pi. After all, the fundamental instrument for measuring the length of an area was the wheel. In one turn, the wheel’s diameter d leaves a mark of the length πd, while the height can be conveniently measured by the lengths of the diameters and the width of the wheel. This explains the next astounding fact.
The Great Pyramid and π’s Mystery in Construction
The Great Pyramid of Giza represents a perfect quadrangular pyramid, the length of the side of the base of which is а = 755.75 ft, and the height is h = 481.4 ft. A doubled ratio of these lengths (2a/h) is equal to 3.1759…, which is very close to the value of π. This ratio occurs naturally if it is produced by measuring a wheel. For example, if one builds a wall out of rectangular blocks, each of which is equal in height to the diameter of a wheel (D) and in width to its rotation — that is, the circumference of a circle (2πR = πD) — we will find the exact same “mystical” ratio of the height of the wall to its length (πD/D=π).
Archimedes’ Approximation
A more accurate value of π was derived in Ancient Greece when geometers attempted to solve the classical problem of squaring the circle: using only a compass and a ruler, they needed to construct a square with the same area as a given circle. Algebraically speaking, this can be written down as a² = πR², where а is the side of the square and R is the radius of the circle. Consequently, a=R√π, meaning that this equation results in the creation of a part of the length √π.
Nonetheless, Archimedes found a sufficient approximation of the constant π ≈ 22/7 ≈ 3.142857. He indicated the bounds of the open interval to which π belongs: “A premier of any circle is equal to a tripled diameter with excess, which is less than one-seventh of the diameter, but more than fourteen and eight hundredths.” In his work, Measurement of a Circle, Archimedes proved the validity of the inequality: 3(1/7) < π < 3(10/71).
The scientist reached this conclusion by calculating the areas of two polygons: one inscribed within the circle and one within which the circle was circumscribed.
THE UNSOLVABLE PROBLEM OF SQUARING THE CIRCLE
Why were the Greeks so concerned by the quadrature of a circle? It’s a matter of the paradox of the problem. According to the Greek mathematicians, if a figure has an area, then there exists a square with the same area. A circle obviously has an area, but finding an equal square was an impossible task. Methods of advanced mathematics proved that the equation in its current form was not solvable. Johan Lambert proved that π is irrational in his 1768 work, Mémoires sur quelques propriétés remarquables des quantités transcendantes, circulaires et logarithmiques. Meanwhile, the complete proof of the impossibility of the equation for the quadrature of a circle with the help of a ruler and compass was presented by Ferdinand von Lindemann in 1882 when he proved the transcendence of pi number. In other words, he showed that π couldn’t be the root of any polynomial with rational coefficients.
Pi Number | THE ARCHIMEDES METHOD
If one were to inscribe a figure with the area S₁ into a circle and outline it with a figure with an area of S₂, then the area of the circle would satisfy the inequality S₁<Ssquare<S₂, which allows one to estimate the area. If one were to use equilateral polygons, by increasing the number of the sides, the polygon would become more like a circle, and the area of the circle would be closer to that of the polygon.
That being said, the value of π is determined as a ratio of the perimeter of the figure to the diameter of the circle.
Solving the formula of the area of a circle
Let’s say we have a certain circumference with the radius R. We then cut it into 4 equal sections and join them according to the diagram (A). The sum of the length of the arcs of two sectors that are shown in red is equal to half of the circumference.
If we cut the circumference into an endless amount of pieces in a similar manner and put them together, then we’ll get a rectangle— it’s obvious that its area is equal to the area of the circle. Consequently, the area of the circle is S = R×l/2. Now all that’s left is to derive the length of the circumference of the circle from the total of the value of π = l/D = l/2R and place it in this formula:
“Whoever has the desire — let them go further”
In the 1990s, an American student, Colin Percival, launched the global project PiHex. Any inhabitant of the Earth with access to the Internet could become a participant in collective calculations for specific bits of pi number — the project encompassed 1,885 computers that used 1.3 million processor hours. Calculations on each device were conducted using “idle” time slices (CPU time that no other computer program is using). By September 11, 2000, the date of the project’s end, participants derived the quadrillionth (40,000,000,000,000th) binary digit of π. This digit turned out to be zero.
