Alright. So we all know e (Euler’s number), we all know pi, phi, (the golden ratio, divine ratio, divine proportion etc.), and the square root of two in the lore of transcendental numbers; however, there is one such irrational number you may not know of: 1.13198824...
Where does this come from?
First you start with the Fibonacci Sequence. (1, 1, 2, 3, 5, 8, 13…) This sequence was originally derived from a problem about rabbits (I will not explain that here; that is a whole different node). What makes this sequence special is that each consecutive integer after the second one is equal to the sum of the two numbers prior and the ratio of any two numbers as you approach infinity approaches phi (or the golden ratio which is equal to 1.6180 33988 749894…. This is where it gets interesting-
50% chance of rain?
Take a coin. Flip it. The coin will land on heads 50% of the time and will land on tails 50% of the time. (This is not the actual distribution: The more tosses you do, the closer the distribution will get to this split) This process where there is no 100% certainty in the outcome of any one event converges at a given number- The number 1.13198824... helps to demonstrate this…
Adding and Subtracting
in 1999 a computer scientist Divakar Viswanath asked himself a question that went like this: Start with the first two numbers of the Fibonacci Sequence 1 and 1. Instead of just adding them, as you would in the standard Fibonacci sequence, suppose you toss a coin and add or subtract them based on the outcome of the toss. If the coin comes up heads-, the numbers are added. If the coin comes up tails-, the two numbers are subtracted. Starting with these two numbers there is a 50% chance that your third number will be 0, and a 50% chance that your third number will be two. Then do it again, and again and again- This process can continue as long as you would like; however, the trend will be established at about the hundredth number…
Viswanath found that upon examination of his sequence of his numbers he found that if he took the absolute value of each number obtained from a random series of results generated by coin tosses that the number would steadily increase in a well-defined and predictable manner! The 100th number in his sequence (or any random sequence generated following his method) would approach the hundredth power of the number 1.13198824... If you further conduct this experiment you will notice that after the nth flip of the coin, the value you have is approximately equal to the number 1.13198824 to the nth power!
Viswanath calculated this number using fractals and a theorem proposed by Furstenburg and Kesten. The method is difficult and involved and I cannot adequately explain it any further than that; however, the real value in this discovery of a new constant is that even a seemingly random process can lead to a deterministic conclusion.
The value of this is immense for physical sciences where probability runs free. While the location of an electron may be random and only explained through probability, the discovery of this constant is proof that definite conclusions can be drawn from random data…. Even scarier is what this could do for sociology- while the actions of a single human being cannot be predicted the end result will converge upon the same inevitable conclusion… Kind of makes you rethink the whole free-will thing- yeah?