PI root t(w)o Irrationality...

Rationality and irrationality is not always attributed only to humans, there are some mathematical entities too whose behaviour seems irrational.

Most common and popular of it all is the π (PI) pronounced as Pie and square root of two. We have always thought that pi is nothing but 22/7 and as you would expect here, it is not. There is a very simple proof for that available in Wikipedia, if you are more into the equation, read here.

This mathematical beauty has been used by humans from time immemorial, the oldest known usage is in the construction of pyramid some 2500 B.C. The proportion of height to perimeter of the giza pyramid is about 3.142857 (3+ 1/7). In one of the Egyptian texts, Rhind papyrus, they have computed the value of Pi as (16/9)^2. Indians in the 9^th century BC(Shatapatha brahmana) used the value 339/108 as the value of Pi. Archimedes finally suggested that 223/71 < π < (22/7).

So what this Pi is ? Pi is the ratio of circumference of a circle to its diameter. What does it mean when we say that it is an irrational number, when we say that a number is irrational we mean that it is not possible to represent that number as a ratio of two rational numbers. Lets take this situation where we draw a circle, whose diameter we are sure is of 7 units, if u try to find the circumference of the circle, you will find it is approximately 22 units but a little less than that, but you cannot get the exact value. In other words it means there is no single unit which can be used to measure the diameter of the circle and the circumference of the circle accurately. Similary applying the same logic to square, we cannot determine the exact value of square root of 2.i.e., when you know that the length of the side of the square, you can find the diagonal of the square only approximately. These irrational numbers kind of create a singularity in mathematics, it is like saying that we know that this particular number lies between x and y but just cannot pinpoint the exact location. We can narrow down the value of X and Y but still will always give only approximate value for Pi and root two.

We have found the value of pi to a million decimal points which actually has no reason, for instance the value of pi to 12 decimals is enough to draw a circle with the diameter fitting the known universe to an error of 1mm.

One of the biggest obsession in maths world is to find the value of pi more accurately and the other one is to remember them and be able to recite them. People have recited it to thousand place of decimal values. This obsession led to a complete new method of writing poems named as PIEMS, in which the count of the letters in the word of the piem give the actual value of pi. Here is an example..

(3)

Pie

(1) (4) (1) (5) (9) (2)

I wish I could determine pi

(6) (5) (3) (5) (8)

Eureka, cried the great inventor

(9) (7) (9) (3)

Christmas pudding, Christmas pie

(2) (3) (8) (4) (6)

Is the problem's very center.

Now the value of Pi to 2000 decimal places

3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510
58209 74944 59230 78164 06286 20899 86280 34825 34211 70679
82148 08651 32823 06647 09384 46095 50582 23172 53594 08128
48111 74502 84102 70193 85211 05559 64462 29489 54930 38196
44288 10975 66593 34461 28475 64823 37867 83165 27120 19091

45648 56692 34603 48610 45432 66482 13393 60726 02491 41273
72458 70066 06315 58817 48815 20920 96282 92540 91715 36436
78925 90360 01133 05305 48820 46652 13841 46951 94151 16094
33057 27036 57595 91953 09218 61173 81932 61179 31051 18548
07446 23799 62749 56735 18857 52724 89122 79381 83011 94912

98336 73362 44065 66430 86021 39494 63952 24737 19070 21798
60943 70277 05392 17176 29317 67523 84674 81846 76694 05132
00056 81271 45263 56082 77857 71342 75778 96091 73637 17872
14684 40901 22495 34301 46549 58537 10507 92279 68925 89235
42019 95611 21290 21960 86403 44181 59813 62977 47713 09960

51870 72113 49999 99837 29780 49951 05973 17328 16096 31859
50244 59455 34690 83026 42522 30825 33446 85035 26193 11881
71010 00313 78387 52886 58753 32083 81420 61717 76691 47303
59825 34904 28755 46873 11595 62863 88235 37875 93751 95778
18577 80532 17122 68066 13001 92787 66111 95909 21642 01989


38095 25720 10654 85863 27886 59361 53381 82796 82303 01952
03530 18529 68995 77362 25994 13891 24972 17752 83479 13151
55748 57242 45415 06959 50829 53311 68617 27855 88907 50983
81754 63746 49393 19255 06040 09277 01671 13900 98488 24012
85836 16035 63707 66010 47101 81942 95559 61989 46767 83744

94482 55379 77472 68471 04047 53464 62080 46684 25906 94912
93313 67702 89891 52104 75216 20569 66024 05803 81501 93511
25338 24300 35587 64024 74964 73263 91419 92726 04269 92279
67823 54781 63600 93417 21641 21992 45863 15030 28618 29745
55706 74983 85054 94588 58692 69956 90927 21079 75093 02955

32116 53449 87202 75596 02364 80665 49911 98818 34797 75356
63698 07426 54252 78625 51818 41757 46728 90977 77279 38000
81647 06001 61452 49192 17321 72147 72350 14144 19735 68548
16136 11573 52552 13347 57418 49468 43852 33239 07394 14333
45477 62416 86251 89835 69485 56209 92192 22184 27255 02542

56887 67179 04946 01653 46680 49886 27232 79178 60857 84383
82796 79766 81454 10095 38837 86360 95068 00642 25125 20511
73929 84896 08412 84886 26945 60424 19652 85022 21066 11863
06744 27862 20391 94945 04712 37137 86960 95636 43719 17287
46776 46575 73962 41389 08658 32645 99581 33904 78027 59009...

And the best part is there is no repetition of any sort determined till date in this series...

The decimal marked in red is called as Feynman point, cos Feynman once stated during a lecture he would like to memorize the digits of π until that point, so he could recite them and quip "nine nine nine nine nine nine and so on", suggesting, ironically and incorrectly, that π is rational.

Think about this,

Lets imagine a situation, where we have built a computer which displays a particular digit at a given point of time. We now program the computer to display the digits in the pi where the time period of display of a digit should be half of the display period of the previous digit. That is 3 should be displayed for .5 minutes, 1 should be displayed for ¼ mins, 4 should be displayed for 1/8 mins, and so on. By the end of one minute, the last digit of Pi will be displayed on the screen. This is because 1/2 min + ¼ min+ 1/8 min + 1/16 min +... = 1 min, but this is an infinite series, so will actually never happen because it will never be possible to process and display anything as the series approaches the final value. This is a combination of the continuum problem and the irrational number.