|
|
|
---
Pi is 3.14159265358979323846.... The decimal digits do not stop and they do not repeat. Pi is irrational. At present Pi is known to an incredible 206,158,430,000 decimal places after the point. The calculation was completed on 20th September 1999. In case you need to know, the 206,158,430,000th digit (not counting the initial 3) is 3. The computer used was a HITACHI SR8000 at the Information Technology Center, Computer Centre Division (old Computer Centre,) University of Tokyo. Congratulations should be sent to : kanada@pi.cc.u-tokyo.ac.jp Do each of the digits occur equally often? The following tables gives the number of each of the digits that occur in the first 6 billion decimal places:
1 is the most frequently occurring digit making up 10.00055433% of the total.In numerical order 0 599963005 1 600033260 2 599999169 3 600000243 4 599957439 5 600017176 6 600016588 7 600023761 8 599975659 9 600007998 In order of frequency 1 600033260 7 600023761 5 600017176 6 600016588 9 600007998 3 600000243 2 599999169 8 599975659 0 599963005 4 599957439 4 is the least frequent making up only 9.99929065% of the 6 billion digits. --- One over Primes
Notice that for some primes the size of the repeating block is 1 less than the prime.1/2 0.5 Terminates 1/3 0.33333... Repeating block: 1 digit 1/5 0.2 Terminates 1/7 0.1428571428... Repeating block: 6 digits 1/11 0.090909... Repeating block: 2 digits 1/13 0.0769230769... Repeating block: 6 digits 1/17 0.05882352941176470588... Repeating block: 16 digits 1/19 0.0526315789473684210526... Repeating block: 18 digits 1/23 0.04347826086956521739130434... Repeating block: 22 digits These are called Golden Primes or Long Primes. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 9 out of the 25 primes less than 100 are golden, that's 36%. Emil Artin made a conjecture that in the long run the proportion of primes that are golden is given by: 
Can you spot the pattern in the terms of this product? You might like to calculate the first few terms of this product and compare it with 36%. What are the next few golden primes? --- We all know and love the Positive Integers:
1 is the first positive integer. It is special since it divides all the positive integers.1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...
2 is the first prime number. Now find all the multiples of 2 (even numbers):1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...
Positive Integers other than 1 which are not prime numbers are called Composite Numbers.2 4 6 8 10 12 14 ...
This leaves 3 as the next prime number. Now find all the multiples of 3:1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...
We now have identified all the multiples of 2 and 3. This gives 5 as the next prime:3 6 9 12 ...
Continue this process by identifying the multiples of 5 then finding the next prime and then its multiples and so on:1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...
Try this process with the numbers 1 to 100 crossing out the unwanted multiples of each new prime.1 2 3 4 5 6 7 8 9 10 11 12 13 14 ... At what stage are there no more multiples to cross out? This sifting out of the composite numbers was first done by Eratosthenes (275-194BC). It is called the "Sieve of Eratosthenes". --- A Prime Rich Decade 1993 , 1997 and this year 1999. Since all primes other than 2 and 5 end in either 1, 3, 7 or 9 then this decade has been RICH in primes. The most prime years possible in a decade is four. The last such VERY RICH decade was 1800-1899: 1801, 1803, 1807 and 1809 are all prime. An amazing fact is that there has been no VERY RICH decade for one hundred years. In fact the current RICH decade is the first since 1780-1789: 1783, 1787 and 1789 are all prime. So how long do you have to wait until the next RICH or VERY RICH decade? Will it be in your lifetime? The good news is that if you are of school age just now you may just experience the next VERY RICH decade. However the bad news is that you will definitely not experience another RICH decade like the present one. So when are they? Search for yourself in the The Prime List.
In the decade 1990-1999 three of the years were prime:
Cabalistic Numerology If the letters of the alphabet are assigned numerical values: A=1, B=2, C=3 etc then words take on values. For example MADRAS COLLEGE has a value of 13+1+4+18+1+19+3+15+12+12+5+7+5 = 115 Also HAD GOOD MATHS has a value of 8+1+4+7+15+15+4+13+1+20+8+19 = 115 Is this a coincidence or is there some deeper message here?!! Can you produce similar deep messages using your own school name? ONE gives 15+14+5 = 34 which is certainly not ONE. Let's assign different values to the letters so that it does give ONE: O=23, N=-22 and E=0 so that now you get: ONE gives 23+(-22)+0 = 1 which is definitely ONE. Can we now get TWO correct? Make T=32 and W=-53: TWO gives 32+(-53)+23 = 2 which is definitely TWO. For THREE try H=3 and R=-32. Can you continue assigning values that make the sums for FOUR, FIVE, SIX, SEVEN etc work out correctly? It is very difficult to do this. Here is a table to help you on your way:
Eventually, of course, you will arrive at a number which is impossible to make the sum correct. What number is this?E = 0 L = work this out T = 32 F = -33 N = -22 U = 48 G = work this out O = 23 V = -13 H = 3 R = -32 W = -53 I = 53 S = 42 X = -89 It may be possible to get further by assigning values in a completely different way from this table and do even better. You might like to try this! --- The 29th February The probability of being born on 29th February is 1/1461 (Why?) Scotland has a population of approximately 5.1 million. This means approximately 3500 Scots were born on this date (check the calculation). How many of these are likely to die on 29th February? Check that the calculation gives approximately 2! What are the corresponding figures for Friday the 13th? --- Fibonacci's Brothers We all know and love the Fibonacci Numbers: 1 1 2 3 5 8 13 21 34 55 ... where each term is the sum of the preceding two terms (zeros precede the initial 1) Some of you may even know the wonders of the ratio of consecutive Fibonacci Numbers: 1/1 1.00000... 2/1 2.00000... 3/2 1.50000... 5/3 0.66666... 8/5 1.60000... 13/8 1.62500... 21/13 1.61538... 34/21 1.61904... 55/34 1.61764... 89/55 1.61818... 144/89 1.61797... 233/144 1.61805... 377/233 1.61802... and it certainly appears that these ratios are converging towards a particular number. This number is called The Golden Ratio and is the positive root of the quadratic equation: x2-x-1=0 Check that this appears to be true by calculating the root. It is also a root of the cubic equation: x3-2x2+1=0 Generalising these ideas produces the amazing Tribonacci Numbers: 1 1 2 4 7 13 24 44 81 149 ... where each term is the sum of the preceding three terms (again with initial zeros). The ratios of these numbers approach 1.83929.... which is the root of the cubic equation: x4-2x3+1=0 and then there are the Tetranacci Numbers ... ... the ratios approach the root of the equation ... Do the calculations and be amazed! --- Square Permutations What have 169, 196 and 961 in common? They are different PERMUTATIONS of the digits 1,6 and 9. They are also square numbers: 132, 142 and 312. Are there other examples like this? Try calculating 362, 542 and 962. Can you find more examples? Is it possible to find more than three squares with this property? Calculate: 1282, 1782, 1912, 1962 and 2092 and be amazed! Now, if you have recovered, then consider: 10242=1048576. There are another six square numbers with the same digits but in a different order. Can you find the six different permutations of 0,1,4,5,6,7 and 8 that give square numbers? A final thought... 101282=102576384. This square contains one each of the digits: 0, 1, 2, 3, 4, 5, 6, 7 and 8 Can you find any of the eighty seven square numbers that contain one each of the digits: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9? Happy hunting!! --- The Divided Sum Write down the first three positive integers: 1 2 3 Keeping the order the same it is possible to divide them into two groups with the same total: 1 + 2 = 3 This is not possible with 1 2 3 4 or with 1 2 3 4 5 or....and further examples are extremely rare. So what is the next example? Check the following calculation: 1+2+3+4+5+6+7+8+9+10+11+12+13+14=15+16+17+18+19+20 The crucial numbers are 2, 14 ... and 3, 20 ... What are the next two numbers in these sequences? Look at these calculations: 6 x 14 - 2 + 2 = 84 and 6 x 20 - 3 + 2 = 119 Now check the calculation: 1+2+3+ ... +84 = 85+86+87+ ... +119 Is the next example given by: 6 x 84 - 14 + 2 = 492 and 6 x 119 - 20 + 2 = 696 ? What have these amazing sums got to do with triangular numbers that are double other triangular numbers? And what have they to do with Pell's Equation?
Labels:
Amazing Number Facts No
Previous Article
