π is a mathematical constant that symbolizes the ratio of a circle s circumference to its

Transcription

1 Ziya Chen Math 4388 Shanyu Ji Origin of π π is a mathematical constant that symbolizes the ratio of a circle s circumference to its diameter, which is approximately We have been using this symbol for as long as we were involved with mathematics. But when was π developed, when was it first used, and how did people applied it in the beginning? Those are probably some of the questions we never bother to think of. π actually had existed since the ancient time of Babylon around B.C. The approximation of π back then was not very accurate, however, it did get more precise with the help of a Greek mathematician named Archimedes and a Chinese mathematician named Liu Hui. The development of π occurred in almost every major ancient civilizations and each one had their own unique way of finding its value. One of the earliest calculations of π was from Babylon. They calculated π by using the formula for circumference and area of a circle, where C = 2πr and A = πr 2, respectively. From the tablet that was obtained during that period of time, we could see the Babylonians simple take the circumference of the circle equal to 3. Then they calculated the area of circle by: 3 = 2πr r = 3 2π A = π ( 3 2π ) 2 A = 9 4π

2 After getting the area, they set 45 = 9 (60 is the base that Babylonians used in their number 60 4π system; 45 is from the tablet). In the end, π just simply turned out to be 3. Although 3 was a bit off, but it was good enough for the Babylonians when they applied it in the geometry formula in order to construct their architectural projects. Another tablet discovered around B.C. showed one more approximation close to for π and that was the closest number the Babylonians got to for π. Egypt was also able to get a close calculation of π The method of acquiring π was recorded in a book called, Rhind Papyrus (1650 BC), which consisted of a collection of about 87 mathematical problems. The recorded statement that involves π calculation was A square of side 8 has the area of a circle of diameter 9 1. The Egyptians were able to solve for π by connecting this with the formula for area of circle, A = πr 2. First step was to cut off 1/9 of the diameter from the original diameter length. So let diameter = 2r, the Egyptians obtained 2r [ 1 ] 2r = 9 [8 ] 2r. Then to make this length into the side of a square, they squared it. So 9 [ 8 9 2]2 r 2 = r2 = πr 2 = A, and from this, the Egyptians concluded that π = One of the hints that was given in Rhind Papyrus (1650 BC) on obtaining this calculation method was a picture of octagon inscribed in a 9x9 square. It is easy to see that the area of the circle is very close to the area of octagon. Thus, the Egyptians deduced the area of octagon as a way of calculation the area of a circle, which then led to the calculation of π. Like the Babylonians, Egyptians also used π in its geometry calculation for architectural purposes. 1 Rhind Papyrus (1650 BC)

3 The first person who actually preformed a precise and accurate approximation of π was Archimedes. Archimedes was born in Syracuse, a city in Greece, around 287 B.C. He was one of the most famous mathematicians in the world who helped greatly with the advance of mathematics during ancient time. Archimedes was able to bring the method of exhaustion to full maturity and used it to approximate the value for π. Method of exhaustion is a method used to find the area of certain shape by inscribing a sequence of polygons. The area of the polygons would converge to the containing shape s area. Hence, Archimedes filled the circle with n- polygon, where n was denoted as number of sides, to calculate π. n would have to become greater and greater in order to get closer to the actual value of π. The formula resulted was π A n r 2, where A n was the area of n- polygon. Another way that Archimedes used to approximate π was to apply the method of exhaustion on the perimeters of the inscribed and circumscribed n-polygons and expressed them in inequality. Let p n and P n be denoted as the perimeter of the inscribed and circumscribed n- polygons, respectively. Then we have, p 6 < p 12 < < p n < π < P n < < P 12 < P 6. Therefore, by using the method of exhaustion, Archimedes was able to get the inequality < π < , which was very close the true value of π. On the other side of the world, isolated due to geographical reasons, lies China. Similar to the other ancient civilizations, China also had someone who could approximate π, an accurate one. His name is Liu Hui, a famous mathematician from the Wei Kingdom around A.D Before Liu s time, China first used π 3 like the Babylonians. Then another mathematician named Zhang Heng before Liu rendered to π However, Liu was not satisfied

4 with this approximation; he thought the value was too big, so he began his own calculation. First, Liu obtained an inequality with the relation between the area of inscribed polygons and area of circle. He inscribed a polygon with n side and 2n side, and denoted their area as A n and A 2n, respectively. Then let the differences in area between the two polygons be D 2n = A 2n A n. Thus, A 2n < area of circle < A 2n + D 2n, and if r = 1, he acquired A 2n < π < A 2n + D 2n. After this, he performed iterative algorithm with r = 10, and a 48-gon and 96-gon. Liu calculated that the area of 48-gon = A 96 = and area of 96-gon = A 192 = Then by applying the difference formula, we have D 192 = into Liu s inequality, he obtained < 100π < < π < = Lastly, plugging He did not simply stop here. Later he was able to discover a quicker and more accurate method in approximating π. Liu found out that the proportion of the difference in area of successive order polygons was approximately 1/4 2. Let F be the proportion of the difference, so the area of unit circle is = π A F D 192, in which F = (1 4 )2 + ( 1 4 )3 + = = 1 3. By plugging in, it led to π He also did the same calculation on a 1536-gon, and again, he received the same result of Liu was satisfied at this point and was content to be able to approximate π to an accuracy of 5 digits. π is not something we could obtain easily just by doing simple calculation. Since π is more like an approximation, it requires rigorous algorithm in order to get a more precise estimation to its actual value. Mathematicians like Archimedes and Liu Hui were able to get a good approximation through a series of calculation involving area of inscribed n-polygons inside 2 Yoshio Mikami: Ph.D. Dissertation 1932

5 a circle. The origin of π started off simple and inaccurate, but as time progressed, we were able to see how the calculations have evolved along with accuracy to attain the value of π we use today.

6 Works Cited Allen, Donald G. "π A Brief History." (n.d.): n. pag. Web. 7 Oct Carother, Neal. "Archimedes' Method of Exhaustion." Archimedes' Method. N.p., n.d. Web. 07 Oct Dyer, Jason. "On the Ancient Babylonian Value for Pi." The Number Warrior. N.p., 14 Oct Web. 07 Oct Dyer, Jason. "On the Ancient Egyptian Value for Pi." The Number Warrior. N.p., 05 Mar Web. 07 Oct Encyclopedia. "Liu Hui's π Algorithm." N.p., n.d. Web. 7 Oct "Pi Day: History of Pi Exploratorium." Pi Day: History of Pi Exploratorium. The Museum of Science, Art and Human Perception, Web. 07 Oct