Really Big Numbers

It’s common knowledge in the West how Roman numerals work.  Some Roman numerals get more use than others.  Smaller numerals, like I, V and X (meaning 1, 5 and 10) get the most use, while the higher ones like L, C, D and M (meaning 50, 100, 500 and 1,000) aren’t as common.  They’re more often found in official use.  Movie credits often give the date in Roman numerals, maybe because MMXVII looks more impressive than 2017 (or, more to the point, MCMLXXXIX looks more impressive than 1989).  The date of the founding of the United States appears on the Great Seal as MDCCLXXVI (have a look at the base of the pyramid on a $1, if you want to see), and writing it that way does seem a lot grander, a lot more appropriate for the founding of a nation than the simpler (and clearer) 1776.  Even the Super Bowl gets a Roman numeral for its title.  (In 2016, what might have been Super Bowl L was marketed as Super Bowl 50, since the NFL had difficulty designing a logo for the game around the L.  As a single, asymmetrical letter, this wasn’t easy, though their graphics team tried.  They put together 73 different logos, incorporating either a Roman numeral “L” or an Arabic numeral “50” before deciding to suspend the use of Roman numerals for that game.  It was only for that year, though.  The logo for the Super Bowl scheduled for February 7, 2017, promotes “Super Bowl LI”, so fret not, Roman numeral fans!)

The Romans weren’t the first to use letters to do double duty as numerals.  This was also done by the Hebrews and the Greeks.  Greek numerals didn’t just use a few letters; they employed the entire Greek alphabet, from alpha to omega (including three archaic Greek letters) to represent numbers from 1 to 800.  After they ran out of letters, the Greeks created a new character to represent 900 and 1,000, and marked each thousand with a superscript letter.  Things got simpler when they got to 10,000, which was simply the letter mu.  Well, not that simply.  Mu (M) already represented 40, so this new number was written with a bar over it, to make the difference clear.  This new number was given a name: myriad.

The word myriad still exists in modern Greek, and it’s also made its way into Russian, French, Spanish, and of course English.  In English, it’s rarely used to mean strictly 10,000 anymore; its more common use is to mean many. We might talk about “myriad possibilities” (or possibly use the corruptedform “a myriad of possibilities”), but to an English speaker, they’re talking about countless possibilities, definitely not a finite number.  To the Greeks, though, myriad was where numbers stopped.

Of course, that’s not really where the Greeks stopped counting.  Archimedes saw to that.  He set out to create a system of numbers to count well beyond what the system of the day allowed.  To do so, he figured he should get something to count, so he decided on counting the number of grains of sand it would take to fill up the universe.  He published his ideas in a work called The Sand Reckoner.

It was a blueprint for bigger numbers.  His first post-myriad number was a myriad myriads or 10⁸ (since one myriad, or 10,000, is 10⁴.  A myriad myriads works out to 100,000,000.)  He referred to numbers below 10⁸ as “the first order”, and numbers above 10⁸ as “the second order.  Above that were a myriad myriad myriads, or 10¹⁶.  These were numbers of “the third order”, capping out at what we call 1 trillion.  Above that, Archimedes decided that a myriad myriads times a myriad myriads (what we would call 10 quintillion) and that all the orders up to that point would be “orders of the first period”.  His “orders” were exponents, and they kept expanding like this.  His largest defined number was a myriad myriads to the myriad myriadth order to the myriad myriadth order.  Got that?  It’s hard to wrap your head around it.  In simpler terms (which still aren’t that simple), we would write hisnumber as a 1 followed by 80 quadrillion zeroes!

While the word myriad and the expression of the first order have symbolic meaning in modern English, we have come up with our own names for what Archimedes laid out for us.  The number googol was invented in 1920 by mathematician Edward Kasner and named by his nine-year-old nephew.  Googol is often thrown around as the largest name for a finite number, but that’s not true.  A googol is just 10¹⁰⁰, which is a lot, but we have bigger number names than that.  An alternative name for googol is 10 trigintillion, and believe it or not, there’s a method to where that name comes from. But let’s start from the beginning.

The word million is the word for 1 followed by six zeroes, and this number gets tossed around quite a bit.  Beyond that is billion, for 1 followed by nine zeroes, and trillion for 1 followed by twelve zeroes, and quadrillion is 1 followed by fifteen zeroes, and so on.  The system of prefixes, you’ve probably noticed, borrows from Latin.  Tri  comes from Latin for three, quad comes from Latin for four, etc.  These larger numbers don’t get a lot of use in everyday conversation (though I did use a couple earlier in this article), but in case we need to name them, we’re ready.  The numbers are easy enough to follow.  10¹⁵ is a quadrillion, like I stated above, then 10¹⁸ is a quintillion, 10²¹ is a sextillion, etc., to septillion, octillion, nonillion and decillion.  Dec- is the prefix meaning ten, of course, and since Latin numbers went on, this system goes on, too.  Next are Undecillion, duodecillion, tredecillion, etc.  10⁶³ is 1 vigintillion, 10⁹³ is one trigintillion, and so on.  10³⁰³ is 1 centillion, 10⁶⁰³ is 1 ducentillion, and so forth.  10³⁰⁰³ is 1 millinillion, 10⁶⁰⁰³ is 1 dumillinillion, and so on, until you run out of Latin.

These words, especially the words for the really high numbers, don’t get used much, even by scientists and mathematicians.  When they need to talk about numbers that large, they prefer to write them as exponents, which is a lot simpler and more practical.  

Edward Kasner proposed a longer version of the number googol, calling it googolplex.  When asked how much that was, he said googolplex is written by “writing one and following it by zeroes until you get tired.”  Since mathematicians prefer numbers to be more exact than that, Kasner settled on a pat definition, reasoning that “because different people get tired at different times, and because it would not do to have [world heavyweight boxing champion Primo] Carnera be a better mathematician than Einstein, simply because he had more endurance and could write longer,” the standard became 10 to the googol power or, if you wish, 10 to the 10 trigintillionth power, if you want to show off your new vocabulary.