At one point in the movie π our demented and twisted anti-hero uber-nerd says "surely you've already said every 216 digit number aloud?" to the Hassidic Jew conspiracy that is trying to bring him down. Upon hearing this, all of the math majors laughed, as did many of the computer science majors - but nobody else. Later when talking to people, they didn't immediately see why this number was so unimaginably and unacceptably huge that it made his statement ridiculous. So let's try to visualize a number this astronomically big; but let me tell you right now that your brain can't handle its biggitude - because it's REALLY big. In doing so, we'll explore lots of good techniques for allowing our brains to come to grips with numbers that are so freaking huge.
Firstly, how many 216 digit numbers are there? Well, to make a number, you can put 1 through 9 in the first spot, and 0 through 9 in any other spot. We have one first digit, and 215 "other" digits, so it there are 9*10215 numbers that are 216 digits long. That 9 is a little troublesome, so let's just round up to 10216, as you'll see soon, being inaccurate by 10% is peanuts in a situation like this. Let's try to figure out how big 10216 really is...
Method 1: 10216 as a quantity of stuff
One way of thinking about big numbers is as an amount of stuff. This will help, because the difference between macro and micro allows us to throw away lots and lots of orders of magnitude, and it's these powers of 10 that stand between us and understanding how big this number really is. First of all, let's recall some quantities that will be helpful. One mole of carbon masses 12 grams, there are approximately 6 billion people on earth, and the earth masses 6*1024 kilograms. So here is one method of visualization:
10216 is so big that, if you had that many atoms of carbon, you could give every person on the planet a lump of coal that weighs as much as the earth.
Now let's look at how big that described quantity actually is:
6*109 people * 6*1027 grams of coal/person * 1/12 moles/gram * 6*1023 atoms/mole = around 1060 atoms
Shit! That's way too low, and by way too low, I mean WAY too low. But it will serve to impress for now. Since I've already overly taxed my powers of visualizations with that "Earth's mass of coal" thing, let's give up on this and try method 2 for a bit.
Method 2: 10216 as a unit of man-hours
Okay, so we've given up on stuff. Let's try time. Time is good because, as long as you are not a young earth creationist, you'll agree that there is a lot of time to play in, and "a lot" becomes pretty essential. Also, because we are dealing with man-hours, we can give a task to every single person on the earth. So, let's try saying something like:
Assuming it only takes one second to say any given 216 digit number, then it would take everybody on earth saying numbers aloud 24 hours a day over 100 billion years (an approximate age for the universe - certainly this is high by a factor of 5 to 10, but that isn't really an issue as you'll see soon).
So how big is this number? Well, it's certainly quite large. Let's do some math:
6*109 people * 1011 years * 3.2*107 seconds/year = around 1027
Shit! That's even worse than the first method. It looks like we'll have to go to method 3.
Method 3: Something with a Probability of 1 in 10216
This method should only be used in the direst of circumstances because people are so notoriously bad at intuitively grappling with odds. But many things are so improbable that it is a good way of getting unimaginably huge numbers. You should never, however, use this method first. Instead, use one of the other methods, and then go to this one. As long as people still play the lottery, probability will never be our species' strong suit.
Because of the way probability works the probability two independent things both happening to you is simply the product of each one individually. This allows us to say things like:
This week, you have a 1 in 10216 chance of winning the Powerball lottery (1 in 80 million chance per drawing) in all 20 states that offer it in the United States this year with the same set of 6 lucky numbers as well as getting attacked by sharks 6 separate times. (Not either/or, but both!)
Which, while technically the most accurate, only provides a sliver of a glimpse into the mind-boggling hugeness of this number. I recommend just horribly underestimating and using the first method no matter what. People are always impressed by large amounts of stuff
Method 4: Zen
A king once asked his wisest advisor how long eternity was. The advisor responded:
Once every thousand years, a little bird sharpens his beak on a mountain made of diamond. When this activity has worn the mountain into a pebble, the first second of eternity will be over.
This writeup has been brought to you by the numbers G63 and Googol and the letters π and e.