Mania de Mathematica
The Numbers Game 10^k
k American SI -33 revo -30 tredo -27 syto -24 fito -21 ento -18 quintillionth atto -15 Quadrillionth femto -12 trillionth pico -9 Billionth nano -6 Millionth micro -3 Thousandth milli -2 Hundredth centi -1 Tenth deci 1 Ten deca 2 Hundred hecto 3 Thousand kilo 4 Myriad 6 Million mega 9 Billion giga 12 Trillion tera 15 Quadrillion peta 18 Quintillion exa 21 Sextillion hepa 24 Septillion otta 27 Octillion nea 30 Nonillion dea 33 Decillion una 36 Undecillion 39 Duodecillion 42 tredecillion 45 quattuordecillion 48 quindecillion 51 sexdecillion 54 septendecillion 57 octodecillion 60 novemdecillion 63 VIGINTILLION 6*n (2n-1)-illion n-illion 6*n+3 (2n)-illion n-illiard 100 Googol Googol 303 CENTILLION 100^100 Googolplex Googolplex
‘pi’npointing pi(
p)You see pi in almost every subject that you learn, but do you actually know the value of pi (sorry! stupid question). But read on for ways to get near that answer. Computation software such as Maple or Mathematica can compute 10,000 digits of pi in a blink, and another 20,000-1,000,000 digits overnight. It is possible to retrieve 1.25+ million digits of pi via ftp from the site wuarchive.wustl.edu.The Chudnovsky brothers have computed 2 billion digits of pi on a home computer. The current record is held by Yasumasa Kanada and Daisuke Takahashi from the University of Tokyo with 51 billion digits of pi (51,539,600,000 decimal digits to be precise). Nick Johnson-Hill has an interesting page of pi trivia at: http://www.users.globalnet.co.uk/ nickjh/Pi.htm
There are essentially 3 different methods to calculate pi
One of the oldest is to use the power series expansion of atan(x) = x - x^3/3 + x^5/5 - ... together with formulas like pi = 16*atan(1/5) - 4*atan(1/239). This gives about 1.4 decimals per term.
A second is to use formulas coming from Arithmetic-Geometric mean computations. They have the advantage of converging quadratically, i.e. you double the number of decimals per iteration. For instance, to obtain 1 000 000 decimals, around 20 iterations are sufficient. The disadvantage is that you need FFT type multiplication to get a reasonable speed, and this is not so easy to program.
A third one comes from the theory of complex multiplication of elliptic curves, and was discovered by S. Ramanujan. This gives a number of formulas, but the most useful was missed by Ramanujan and discovered by the Chudnovsky's. It is the following (slightly modified for ease of programming): Set k_1 = 545140134; k_2 = 13591409; k_3 = 640320; k_4 = 00100025; k_5 = 327843840; k_6 = 53360; Then pi = (k_6 sqrt(k_3))/(S), where S = sum_(n = 0)^oo (-1)^n ((6n)!(k_2 + nk_1))/(n!^3(3n)!(8k_4k_5)^n) .
An interesting new method was recently proposed by David Bailey, Peter Borwein and Simon Plouffe. It can compute the Nth hexadecimal digit of Pi efficiently without the previous N-1 digits. The method is based on the formula: pi = sum_(i = 0)^oo (1 16^i) ((4 8i + 1) - (2 8i + 4) - (1 8i + 5) - (1 8i + 6)) in O(N) time and O(log N) space.
The following 160 character C program computes pi to 800 digits.
int a=10000,b,c=2800,d,e,f[2801],g;
main()
{
for(;b-c;)
f[b++]=a/5;
for(;d=0,g=c*2;c-=14,printf("%.4d",e+d/a),e=d%a)
for(b=c;d+=f[b]*a,f[b]=d%--g,d/=g--,--b;d*=b);
} — Saumitra
Math Mysteries
These are a compilation of mathematical mysteries yet unsolved or unproven. A sort of Math X-files ain’t it!
Twin primes conjecture : "There exist an infinite number of positive integers p with p and p+2 both prime. See the largest known twin prime section. There are some results on the estimated density of twin primes."
Goldbach's conjecture : "This conjecture claims that every even integer bigger or equal to 4 is expressible as the sum of two prime numbers. It has been tested for all values up to 4.10^(10) by Sinisalo."
Tips ‘n’ Tricks
a. Take your age, multiply it by 7, then multiply that product by 1443. What do you get? Your age repeats 3 times. Watch this: 28 (age) x 7 = 196. 196 x 1443 = 282828!
b. Pick a number, Multiply this number by 2, Add 5, Multiply it by 50,If you have already had your birthday this year (1998), add 1748 else add 1747, Subtract the 4 digit year that you were born.You get back your original number.
c. Multiply the last digit of any no. by 2, subtract this answer from the remaining digits, if this number is evenly divisible by 7 the whole number is divisible by 7.
