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Discover the history of transcendental numbers
This number cannot be written as a finite integer, a fraction, or an irrational number. To date, people have accepted it as a transcendental number.
16 of the Greek alphabet. It is defined as a constant, which is the ratio of the circumference of a circle to its diameter. The name pi comes from the word peripheria (perijeria) which means circumference of the circle. But it does not have an exact name, usually people The so-called p, c, or p-script was used around the mid-18th century, after Euler published his analytical treatise in 1748. The intention to use the symbol was to commemorate the Greek mathematicians who were first find out the approximate number of pi At the end of the 20th century the number was calculated with precision to the 200 billionth number (200 000 000 000) September 11, 2000: the one million billionth odd number (1,000,000,000,000,000) .000000) is zero The simplest definition given to this famous number is: it is ratio of the area of the disk to the square of the radius. For example, the area of the circular disk of the figure below is equal to the area of the square. The same number is found again in the calculation of the circumference of the circle, which is 2p times its radius. As Archimède has observed, that number is for these two calculations. And it wouldn’t be surprising if we ran into the same number again and again.* The area of the rim lying between two circles of roughly equal radii can be calculated in two ways: Subtract the area of the great circle. small disc area Since the radii of the two circles are roughly equal, the area of the rim is the product of the circumference of one of the circles and the thickness of the rim.
2. Methods of calculating Pi
Approximate calculation. The oldest method.Draw a circle with radius 1 unit and two equilateral polygons inscribed and circumscribed about the circle.If the polygon is a square then the number of circumferences of the circle. will be between the circumference of the inscribed and circumscribed square, meaning the value of Pi will be :2
2,828 Increasing the number of sides to 6 gives us a better result: 3 (Because the hexagon’s side is equal to the circle’s radius) and 2
= 3.461…: 33 When we calculate the perimeter of polygons with thousands of sides, and divide the result by the diameter of the circle, we get the best approximation of
Numbers easy to remember: are the first odd numbers, 2 3s, 2 5s, 2 1s and the sum of the two numbers of the diagonal numerator and denominator will be 6.
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Babylonians calculate the number p by comparing the circumference of a circle with the polygon inscribed in the circle, which is 3 times the diameter of the circle. They approximate: p = 3 + 1/8 (ie 3.125) Archimède, using a polygon with 96 sides, calculated a smaller approximation (inférieur) of 3 + (10/71) = 3.1408 … and the larger approximation is 3 + (1/7) = 3.1429… That is: 3.1408… To determine the value of Pi, one can try to draw a circular disk and a square of the same area. Analyze using rulers and compasses. And also using rulers and compasses, we draw a line segment of length Pi, and then deduce the exact value of this number. But this way of drawing is not possible: In 1837, Pierre Wantzel proved that one can only draw straight lines with rulers and compasses when the length is an algebraic number, i.e. an answer from an algebraic equation where the coefficients are integers, and in 1882, Ferdinand von Lindermann proved that the number Pi is not an algebraic number.
Pi is found in many other branches of mathematics
*For example, when we measure an angle, we have to choose a unit by arbitrarily specifying a full 360 round, then with the unit “degrees” it will have a measure of 1/360 of a turn. If we use a round value of 2p, the unit of measure will be called radians and have a value of 1/(2p). Measuring angles in radians has more advantages: for example the length of the part of a circle bounded by angle a will be equal to when we measure the angle in radians, but if measured in degrees, it will be (2pra)/360* Equivalent Similarly, the ratio (sinx)/x approaches 1 as x approaches 0 if we calculate angles in radians, but advances to 180/p if we calculate angles in degrees.* Using radians to measure angles gives many characteristics , for example, according to Euler’s theorem, the exponentiel of the complex number 2ip is equal to 1. And also from the results of using radians to calculate the angle, one finds Pi in unexpected places: for example, the sum of infinites (Leibniz série de Leibniz series)1 – (1/3) + (1/5) – (1/7) – … has a value equal to p/4.* Integral:meaning the area under the curve of the square The process f(x) = 1/(1+ x2) between 0 and 1 is also p/4. These two results are not easily interpreted because the tangent to the angle p/4 equals 1The number Pi also appears in the value of the sum.1 + (1/22) + (1/32) + (1/42) + … is equal to p/6
Odd numbers of Pi
The number Pi summarizes a history of ancient mathematics more than 4000 years that covers Analytic Geometry or Algebra. Mathematicians have admired it since the time of Ancient Civilization and especially the Greeks in the matter of geometry. The oldest value of the number Pi that humans have used and has been certified from a tabletLater, ongoing research works:* Archimède calculated Pi = 3.142 with an accuracy of 1/1000. The formula is: 3 + 10/71 Archimedes have been used for 2000 years.* In the Bible, about 550 B.C., this number was hidden in a verse in an ancient Babylonian tablet (of the land). Iraque) has an angular script (écriture cunéiforme), was discovered in 1936 and the age of the tablet is 2000 years before God. After many curious minds searching, the number Pi = 3.141509* Around 1450, Al”Kashi calculated the number Pi with 14 odd numbers thanks to Archimède’s polygon methodIt was the first time in the history of multiplication. type found the number Pi with more than 10 odd numbers.* In 1609 Ludolph von Ceulen, thanks to Archimède’s method, was able to calculate the number Pi with 34 odd numbers, which was engraved on his tombstone. Impossible Calculate the exact value of Pi. Late in the 18th century, Johann Heinrich Lambert (1728-1777) and Adrien-Marie Legendre (1752-1833) proved that there was no such thing as a fraction for calculating Pi. 19, Lindemann proves that Pi cannot be a solution of an algebraic equation with integer coefficients (eg y = ax2 +bx + c where a, b, c are integers)* Next Ludolph von Ceulen thanks to the diligent research of mathematicians: Newton (1643-1727) Leibniz (1646-1716) Gregory (1638-1675) Scientists Euler (1707-1783), Gauss, Leibniz, Machin , Newton, Viète look for formulas to approximate numerical value of p for exactness. And the simplest formula discovered by Leibniz in 1674 is: p/4 = 1-1/3 + 1/5 – 1/7 + …Carl Louis Ferdinand von Lindemann (1852-1939)Srinivasa Aiyangar Ramanujan (1887-1920 )Williams Shanks (1812-1882) calculated in 1874 with 707 odd numbers It took until the 18th and early 20th centuries for Pi to be calculated with an accuracy of 1000 odd numbers. In 1995, Hyroyuki Gotu took over. world record : find 42 195 odd numbers.
Where does the symbol π (Pi) come from?
According to the mathematician and historian – Florian Cafori (1859-1930), the first person to use Greek numeral notation in geometry was Mr William Oughtred (1575-1660). To indicate circumference, English is “periphery, he used the Greek word: Pi (π). To indicate diameter, English is “diameter” he used the Greek word: Delta.
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In 1760 Mr. William Jones (1675-1749) in his book Synopsis Palmariorum Matheseos, he also used the letter Pi (π) to indicate the ratio of the circumference divided by the diameter of the circle. It must wait until the famous mathematician is Mr. Leonard Euler, the Swiss, then the notation Pi (π) became widely used, and is universally recognized and used as the ratio of circumference divided by diameter of a circle; it was 1748, Leonard Euler wrote in his book Introductio in analysin infinitorum.