| T.R | Title | User | Personal
Name | Date | Lines |
|---|
| 37.1 | | HARE::STAN | | Sat Feb 25 1984 17:34 | 19 |
| In [1], Frank Rubin asked for the polynomial of degree less than
or equal to 5 with integer coefficients in the range [-100,100]
that had a zero closest in value to pi.
The best approximation was found by Harry Nelson. It is
5 4 3 2
11x - 17x - 42x - 37x - 27x + 42
which has a root of 3.1415926535897 which differs from the correct
value of pi by approximately 10^(-13).
Lynn's approximation, x^4=2143/22, is not of the requested class,
but is very easy to remember.
Reference
[1] Frank Rubin, Problem 589, Journal of Recreational Mathematics,
11(1978-1979)45-46.
|
| 37.2 | | HARE::STAN | | Fri Apr 13 1984 05:08 | 6 |
| The reference to Lynn's article is:
Lynn Yarbrough, Problem 925, The Malfunctioning Calculator.
Journal of Recreational Mathematics 14(1981)66.
See also his filler article, loc. cit. 13(1980)262.
|
| 37.3 | | HARE::STAN | | Fri Jun 08 1984 04:11 | 4 |
| According to Philip J. Davis in The Lore of Large Numbers
(New Math Library), page 65, the approximation 22 pi^4 = 2143
was discovered in 1914 by the famous Indian mathematician,
Srinivasa Ramanujan.
|
| 37.4 | A good recent article | AKQJ10::YARBROUGH | Why is computing so labor intensive? | Wed Jan 13 1988 12:10 | 5 |
| For more on Ramanujan and Pi, see the Feb. 1988 issue of Scientific
American, p.112-115. How to calculate Pi to 2 billion places with your
'pocket' calculator!
Lynn
|
| 37.5 | | BEING::POSTPISCHIL | Always mount a scratch monkey. | Thu Feb 11 1988 13:47 | 18 |
| I was surprised the article did not mention the sequence:
pi/4 = 1/2 + 1/6 + 1/15 + 1/35 + 4/315 + . . .
It has the nice property of being fairly easy to write an algorithm
for; you can compute the n-th term (starting with zero) by multiplying
the previous term by n and dividing by 2n+1. Since 2n+1 remains small
for a long time, the multplication and division consist of multiplying
a many-digit number by a single-digit number (considering the words the
processor multiplies as single digits). Then you just add the term to
the sum, a simple algorithm even for many-digit numbers.
Each term is less than half the preceding term, so figuring the error
is simple; it is the last term plus (n+1) times the least significant
bit.
-- edp
|
| 37.6 | other approximations | ULYSSE::ZITTA | ULYSSE in wonderland | Tue Oct 11 1988 14:45 | 30 |
|
Here are other values (some better than others..)
( # will mean "approximately equals")
pi^2 1 1 1 1
---- = --- + --- + --- + --- + ..........
6 1^2 2^2 3^2 4^2
pi # sqrt(10)
pi # sqrt(2) + sqrt(3)
pi # 4 / sqrt(k) k= (1 + sqrt(5))/2 =1.61803399...
pi # (4/3)^4
pi # 2*sqrt(2*sqrt(5) - 2)
pi # 22/7
pi # 20*sqrt(2) /9
pi # 3 + 1/8 + 1/60
pi # sqrt[(40 - 6*sqrt(3))/3 ]
pi # (13/50)*sqrt(146)
pi # 5/4 + [sqrt(229)]/8
pi # 355/113
pi # 167/80 + [sqrt(10)]/3
|
| 37.7 | A new mathematical truth | EVTSG8::ESANU | Au temps pour moi | Fri Sep 22 1995 08:48 | 13 |
| From
Ebbinghaus et al., Numbers, Springer-Verlag, GTM 123, 1991, ISBN
0-387-97497-0 or 3-540-97497-0 :
> The House of Representatives of the State of Indiana in the U.S.A.
> unanimously passed in 1897 an "Act introducing a new mathematical truth",
> which proposed two values for PI, namely 4 and 3.2. The Senate of Indiana
> postponed "indefinitely" the adoption of this measure. Fortunately for the
> people of Indiana, the "indefinitely" still continues (see D. Singmaster,
> The legal values of PI, Math. Intelligencer, 7(2), 1985, 69-72).
Mihai.
|