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Even Exacter Number

If you don't know how you got here, back up to the previous page and read about the likelihood 3 billion people will decide at some randomly chosen moment to jump off a chair.

A more nearly exact number for the denominator of that tiny number has been calculated for me by Dave L. Renfro:

18017021709787328669080622320043008147994628844676
19271816565805431774308335986840148669046334703107
91264200480218510549822340448202123366887578486051
05592266673923832915244729333630831950259021209282
84469018267821482812834413729638010489405666335838
66049142482107325856768164100544853209985487731646
65293820842522233063971775936321124266706511581306
92743447508571597762833889446486689352907136963602
90081821768447563546441112485651963326513577705429
92891866700174552072837831552212571660609210596501
80176016307804663134755255621757928891380955843621
31376651012934896668101680291737759484156056349329
03752307542751382876258414071164164975426901974370
05216429289575310412456919602187416733919811661283
48391682105223869865182817777463052123853399731395
26402032707920263389610728418433025076511836834668
68967072928418157421780841769714441900386520193839
50149145976198840462198528904195753925017483596761
80553163250057221000048705897609304633346345266701
78596814616820615705939621711771686303020388415131

x 10 ^ (22,497,310,902).

This is a big damn number.  Here are the instructions, with thanks to mensanator and Keith Kingsbury, for writing it out longhand:

1.  Write down the thousand-digit number above that starts with 1801 and ends with 5131.

2.  To the right of that number, write down 22,497,310,902 zeroes.

3.  Stop.

If someone hires you to complete that task, you will not have to worry about losing your job for quite a little while.  If you could write one digit per second for 40 hours a week -- no breaks to shake off writer's cramp -- it would take you nearly 30 centuries and a lot of pencils.

22,497,311,902 digits / 3600 seconds per hour = 6,249,253.30611 hours

6,249,253.30611 hours / 40 hours per week = 156,231.332653 40-hour weeks

(Number of weeks per year = 365.25 days / 7 days per week = 52.1785714286 weeks)

Number of years = 156,231.332653 40-hour weeks / 52.1785714286 weeks per year = 2,994.16653955 years.

And a lot of paper too.  Assuming a piece of paper is .003 inches thick, if you could write 5,000 legible digits on one sheet -- which you can't -- the stack of pages needed to write out this number would be 1,125 feet tall.

22,497,311,902 digits / 5,000 digits per page = 4,499,462.3804 pages

4,499,462.3804 pages X 0.003 inches per page = 13,498.3871412 inches of pages

13,498.3871412 inches / 12 inches per foot = 1,124.8655951 feet of pages.

If you could write 10 digits in an inch -- which you can't -- that number would stretch over 35,000 miles.

22,497,311,902 digits / 10 digits per inch = 2,249,731,190.2 inches

2,249,731,190.2 inches / 12 inches per foot = 187,477,599.183 feet

187,477,599.183 feet / 5,280 feet per mile = 35,507.1210574 miles.

All said, no matter how you look at it, it's a big damn number, yet it's one that you can apprehend if you imagine all 3 billion of us north of the Equator jumping off our chairs at the same randomly chosen moment for no particular reason.

And despite how extraordinarily unlikely it is that we would all do that, you have conceived it yet again, so I say again that it's not inconceivable.

It's fun to try to imagine that which is truly inconceivable and non-trivial.  Earlier I said I cannot conceive that time is not infinite in both directions, and I will add that I think it inconceivable that space is finite.

Please what you think is inconceivable. 

 

 

 

 

Even bigger numbers

On the previous page akprasad referred to some unusual names for counting numbers as well as to a special one, a googolplex.

Googol

To start with, a "mere" googol is 10 raised to the power of 100, or a 1 followed by 100 zeros.

In math terms it would be

10^100,

meaning 10 times 10 times 10 and so on for a hundred such multiplications.  The result is a number with 101 digits.

Written out in full a googol looks like this:

10,000,000,000,000,000,
000,000,000,000,000,000,
000,000,000,000,000,000,
000,000,000,000,000,000,
000,000,000,000,000,000,
000,000,000,000.

(In case you're wondering, the silly-sounding name came from the nine-year-old nephew of American mathematician Edward Kasner, who "invented" the number and coined it in 1938.)

By contrast, the number at left contains not just a hundred zeros but 22 billion of them.  A googol, with its mere hundred zeros, is as nothing compared to the number at left.  The number at left divided by a googol is, for all practical purposes, the same number.

Googolplex

But the number at left is as nothing compared to a googolplex.  A googolplex is 10 raised to the power of a googol, or a 1 followed by a googol of zeros!

A googolplex is so large a number that you (well, I, anyway) can't conceive it without referring to a googol, itself an impossibly large number.
  

And if you think a googolplex is about as big a number as one would ever need, it's not.  Much, much BIGGER numbers have been defined.  They have literally no practical purpose, but they're fun to think about.

I have convinced my eight-year-old there's an interesting difference between her brain and a vessel such as, say, a coffee cup.  The more you try to fill a coffee cup, the less room there is for more.  The more you try to fill a brain, the more room there is for more.

I've also convinced her, I think, that books make her brain bigger and TV shrinks it.

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