Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
ππ’ and Integration by Okkyung Cho Abstract Pi(Ο) is one of the most widely known mathematical constants. There have been a lot of efforts for the computation of the mathematical constant Ο = 3.14 β― through the ages. With the discovery of Calculus in 1600s, a number of formulas for Ο were discovered. During this time one motivation for computations of Ο was to see if the decimal expansion of Ο repeats. This talk gives several formulas for Ο related to the integration and a proof that Ο is irrational. 1 Pi and Popular culture Star Trek (1967 episode) Kirk asks: βArenβt there some mathematical problems that simply canβt be solved?β Spock answers by telling to a rogue computer: βCompute to the last digit the value of Pi.β 2 Pi and Popular culture Life of Pi (2001 Yann Martelβs book) βMy name is Piscine Molitor Patel known to all Pi Patel. For good measure I added π = 3.14β and I then drew a large circle which I sliced in two with a diameter, to evoke that basic lesson of geometry.β οͺ The Notation of π was introduced by Euler in 1737. 3 ππ ππ‘π² π π’π¬ π§π¨π π Even Maple or Mathematica `knows' this since 1( 1 β π₯ )4 π₯ 4 22 (1) 0<β« ππ₯ = βπ 2 1+π₯ 7 0 The integrand is strictly positive on (0, 1), and the answer in (1) is an area and so strictly positive. In this case, the indefinite integral provides immediate reassurance. We obtain π‘( 1 β π₯)4 π₯ 4 (2 ) β« ππ₯ 2 1+π₯ 0 1 7 2 6 4 3 5 = π‘ β π‘ + π‘ β π‘ + 4π‘ β 4 arctan(π‘) 7 3 3 and the FTOC proves (1). οͺ 22 is one of the early continued fraction approximations. 7 4 One can take this idea a bit further. Note that 1 1 4 4 (3) β« (1 β π₯) π₯ ππ₯ = , 630 0 and we observe that 1( 4 4 ) 1 1 1 β π₯ π₯ 4 4 (4 ) β« (1 β π₯) π₯ ππ₯ < β« ππ₯ 2 2 0 1+π₯ 0 1 < β« (1 β π₯)4 π₯ 4 ππ₯ 0 Combine this with (1) and (3) to derive: 223 22 1 22 1 22 < β <π< β < 71 7 630 7 1260 7 and so obtain Archimedes famous computation: 10 10 3 <π<3 . 71 70 οͺ The derivation above seems first to have been written down in Eureka, the Cambridge student journal in 1971. 5 The Childhood of Pi About 2000 BC, the Babylonians used the 1 approximation 3 = 3.125. At this same time or 8 earlier, according to an ancient papyrus, Egyptians assumed a circle with diameter nine has the same area as a square of side eight, 256 which implies π = = 3.1604 β― 81 Some have argued that the ancient Hebrews used π = 3. (I Kings 7:23) 6 Uniqueness of π Archimedes (250 BC) was the first to show that the βtwo Piβsβ are the same: π¨πππ = π1 π 2 and π·ππππππππ = 2π2 π οͺ Archimedes' construction for the uniqueness of π, taken from his Measurement of a Circle. 7 Archimedes Method The first rigorous mathematical calculation of π was also due to Archimedes, who used brilliant scheme on building inscribed and circumscribed polygons 6 β¦ 12 β¦ 24 β¦ 48 β¦ 96 ππ π to obtain the bounds: π < π < 3 . ππ π 8 Precalculus π Calculation Variations of Archimedesβ geometric scheme were the basis for all high-accuracy calculations of π for the next 1800 years. Name Babylonians Egyptians Hebrews (1 Kings7:23) Archimedes Ptolemy Liu Hui Tsu Chβung Chi Al-Kashi Romanus Van Ceulen (Ludolphβs number* ) Yea 2000?r BC 2000? BC 550? BC 250? BC 150 263 480? 1429 1593 1615 Digits 1 1 1 3 3 5 7 14 15 35 * The last great Archimedean calculation, performed by Van Ceulen using 262 -gons - to 39 places with 35 correct - was published posthumously. 9 Piβs Adult Life with Calculus In the later 17th century, Newton and Leibniz founded the calculus, and this powerful tool was quickly exploited to find new formulae for Ο. One early calculus-based formula comes from the integral: π₯ 1 β1 tan π₯ = β« ππ‘ 2 0 1+π‘ π₯ = β« (1 β π‘ 2 + π‘ 4 β π‘ 6 + β― )ππ‘ 0 π₯3 π₯5 π₯7 =π₯β + β + β― 3 5 7 Substituting π₯ = 1 formally proves the wellknown Leibniz formula (1671-74): Ο 1 1 1 1 1 = 1β + β + β +β― 4 3 5 7 9 11 10 Newton discovered a different more effective formula, considering the area A of the left most Red region in the figure: Now, A is the integral 1/4 π΄=β« βπ₯ β π₯ 2 ππ₯ 0 Also, A is the area of the circular sector, π/24, less the area of the triangle, β3/32. 11 Newton used his binomial theorem, 1 4 π΄=β« 1 π₯ 2 (1 0 1/4 =β« 0 1/4 =β« 1 π₯2 β 1 π₯)2 ππ₯ π₯ π₯ 2 π₯ 3 5π₯ 4 (1 β β β β β β― ) ππ₯ 2 8 16 128 1 (π₯ 2 β 0 3 π₯2 2 β 5 π₯2 8 β 7 π₯2 16 β 9 5π₯ 2 128 β β― ) ππ₯ Integrate term-by-term and combining the above produces 3β3 1 1 1 π= + 24 ( β β β― ). 4 3 β 4 5 β 32 28 β 128 12 The Irrationality of π Let π = π/π, the quotient of positive integers. We define the polynomials π₯ π (π β ππ₯)π π (π₯) = , π! πΉ (π₯) = π(π₯) β π (2) (π₯) + π (4) (π₯) β β― + (β1)π π (2π) (π₯), the positive integer π being specified later. Since π! π(π₯) has integral coefficients and terms in π₯ of degree not less than π, π(π₯) and its derivatives π (π) (π₯) have integral values for π₯ = 0; also for π₯ = π = π/π, since π(π₯) = π(π/π β π₯). By elementary calculus we have π {πΉ β² (π₯) sin π₯ β πΉ (π₯) cos π₯} ππ₯ = πΉ β²β² (π₯) sin π₯ + πΉ (π₯) sin π₯ = π (π₯) sin π₯ 13 and (5 ) π β« π(π₯) sin π₯ ππ₯ 0 = [πΉ β² (π₯) sin π₯ β πΉ (π₯) cos π₯]π0 = πΉ (π) + πΉ(0) Now πΉ (π) + πΉ(0) is an integer, since π (π) (0) and π (π) (π) are integers. But for 0 < π₯ < π, π π ππ 0 < π (π₯) sin π₯ < , π! so that the integral above is positive, but arbitrarily small for n sufficiently large. Thus (5) is false, and so is our assumption that π is rational. οͺ In 1761 Lambert proved that π is irrational. This is Nivenβs 1947 short proof. οͺ In 1882, Lindemann proved that π is transcendental. 14 Why Pi? ο· One motivation is the raw challenge of harnessing the stupendous power of modern computer systems. ο· There have been substantial practical spinoffs. For example, some new techniques for performing the Fast Fourier transform (FFT), heavily used in modern science and engineering computing, had their roots in attempts to accelerate computations of π. 15 References ο· D. Bailey, J. M. Borwein, P. Borwein and S. Plouffe, The Quest for π ο· J. Borwein, The Life of π: From Archimedes to Eniac and Beyond 16