Would you like to see, for yourself, a demonstration of the power of exponentiality?
This came from Bill Gates’ book “The Road Ahead,” which I am currently reading. It was written in 1995, fifteen years ago. The book is about his prediction for the future of the internet.
From his book, as a demonstration of the power of exponentiality (is there such a word??): If you took one grain of wheat and put it on the 1st square of a chess board, put two grains of wheat on the 2nd square, 4 on on the 3rd, 8 on the 4th, and so on, how many grains of wheat would you have total by the time you finished at the last, 64th square?
Also, assume you took 1 second to count each grain, how many seconds, minutes, hours etc. would you have been at it when you were done.
(Hint: According to the book, this story comes from ancient India. The prince was so tickled with his minister for inventing chess, that he told his minister that he could have anything he wanted as a reward. The minister told him he wanted the wheat thing on the chess board. The prince said, “No prob!” and got to it. Long before he was half-way done he got so pissed that he had the minister’s head cut off! So the story goes… : ) And, the moral of the story is, “Chess is evil.”)
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This is strange…I don’t remember typing in the word “mathematics” in the topics….I thought I just typed in “math…” Doggie? Auggie?
Only if it’s applied to my paycheck, lol!
Beam me up Dog…I can do better! I promise!
It’s not really an example of exponents, though, so much as it’s an example of compounding.
You’re right. (What was I thinking?)
Benjamin Franklin once called compound interest the 8th Wonder of the World, and demonstrated how a dime invested at a positive rate of compound interest would turn into a fortune if left alone. And that’s true as far as it goes… but he didn’t take into account “inflation”. (Since the British pound at the time was backed by gold, he couldn’t have known otherwise.)
Well, the first square contains 2^0 grains, the second 2^1, the third ^2, right up to 2^63 on the last square.
So the total amount of grain would be 2^0 + 2^1 + 2^2 +...+2^63 which I’m far too lazy to add up right now, but is a hell of a lot.
Here’s the answer (Bill Gates figured it out for me…wasn’t that nice?!)
18,446,744,073,709,551,615 grains of wheat. And at 1 second per grain it would take 584 billion years of counting (the universe is only 14 billion years old.) Like WOW Man!!!
sum { 2^k | k = 0,...,n } = 2^n – 1, so there would be 2^64 – 1 grains. Also, 2 ~ 10^0.3, so 2^64 ~ (10^.3)^64 = 10^(.3*64) ~ 10^19.2 = 10^19 * 10^.2 ~ 1.6 * 10^19.
Mine looks more better, @ratboy! See, 10 and 19 are numbers you think you can wrap your brain around, but 18,446,744,073,709,551,615???
If a continuous flow of nutrients were available, bacteria would fill the entire universe in only a few weeks.
Do you like to experience something even more powerful than standard exponentiality? This might give you a buzz:
The fifth Ackermann number would require more paper tightly packed to fill the entire universe to type out that number. Check this out and it will blow your mind:
http://en.wikipedia.org/wiki/Ackermann_function#Ackermann_numbers
It gets even better: What is the likelihood of @Dutchess_III standing in front of her house one moment and inside it the very next moment (all your particles being quantum tunneled at the same time)?
The likelihood is greater than zero. Yes, it’s perfectly possible. But let’s try to type this small number. Well, if we put every digit in a tiny Planck-length sized cube, the volume of our universe would be far too small.
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