1.04. Googol and Googolplex
Introduction
With the exception of the first few -illions, no numbers have captured the notion of "really big number" any more than the googol and googolplex. The googol has become a classic example of what non-googologists think of as a really big number, and many people see the googolplex as the largest number of all. Therefore, now is a good time to look at the googol and googolplex and talk about how they came to be, and their size and cultural impact.
NOTE: This article has a bit of a dual purpose. There's quite a lot you can say about the googol and googolplex, and instead of having long entries for them in my number list, both numbers have short entries in the list with a "main article" link to this page; the googol section and googolplex section can be thought of as full number list entries for these numbers. That's also why this article talks a lot about stuff covered in later parts of this website.
The Googol
10^100 = 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 = googol
Origin of the googol
The googol originated in 1938 when a mathematician named Edward Kasner wanted to introduce the world to some very large numbers that anyone can comprehend. He asked his 9-year-old nephew Milton Sirotta for a name for 1 followed by 100 zeros, and Sirotta suggested "googol". Two years later Kasner published his book Mathematics and the Imagination which had a passage that introduced the world to the googol and its big brother the googolplex (seen later). The googol quickly became a well-known large number, and gained much cultural impact as we'll see later.
Size of a googol
A googol is often regarded as so large that there are no real-world examples of the number, but as we saw in the last article this is not quite true. True, it's bigger than the number of atoms in the observable universe (10^80)—it's a common misconception that that's the largest "physically meaningful" number. But as we've seen in the previous article, we can actually use the physical world to beat a googol, such as the number of Planck volumes in the observable universe (3*10^185), the quartic hypervolume of the universe in Planck units (1.75*10^245), or the size of the inflationary universe (10^10^12). We examined those in detail previously, but let's look at figures specifically for a googol.
What if you were to divide the observable universe into a googol portions? Well, the volume of the observable universe is estimated at 3*1080 cubic meters. By dividing a googol into that number and then taking its cube root, we can estimate every portion to be a cube with a side length of 0.31 micrometers. That is far too small for us to perceive with our eyes. Let's try comparing such a portion to a grain of sand. Grains of sand of course vary in size, but an average size would be about 0.2 millimeters. That's just about big enough for us to see with the naked eye. But now imagine a tiny grain of sand about 0.31 micrometers wide. Let's call that grain a googol-grain since about a googol of those would fill up the observable universe. How does the googol-grain compare to the normal grain of sand?
The googol-grain would be 645 times smaller in side length than the normal grain! Here's a picture to visualize this comparison:
To put this in perspective, imagine expanding the googol-grain to the size of a normal grain of sand, and then expanding the normal grain by the same factor. The googol-grain would now be as big as a typical grain of sand, but the normal grain would now be 149 cm (4 feet 2 inches) tall! That's roughly as tall as an elementary school student, and it would be big enough that it's difficult for a human to carry! Put another way, the googol-grain is to a grain of sand what a grain of sand is to a human being!
That tiny googol-grain would be exactly how small something would be such that a googol of it could fit in the observable universe! It could compare to the some of the smallest known bacteria, which is pretty insanely small alright. And that's how big a googol is.
Googol in culture
A googol has much cultural significance from being a simple yet very large number. For example, it may be the largest number to appear in printed comics, in this Peanuts comic strip:
This shows that a googol is a number well-known enough to reach large audiences through comic strips. The thing is, how would you read what Schroeder said in the third panel? Ten thousand thousand thousand thousand ... ... thousand, where thousand is said 33 times? 1 followed by 100 zeros? Doesn't really matter, since the audience gets the point. A googol is probably the current record-setter in printed comics, but still larger numbers have appeared in webcomics. If you only count well-known online comics, the largest might be the so-called xkcd number, equal to A(G,G) in the Ackermann function where G is Graham's number (more on both of them in section 2).
The technology superpower company Google even named itself after a misspelling of this number because it wanted to emphasize that its goal was to be able to locate anything and everything on the seemingly endless Internet. Therefore the pronunciation of the word "googol", which sounds the same as the name of Google, has become a very everyday term!
Googol in googology
Since it's such a famed large number, it's no surprise that the googol has become the very namesake of the term googology for the study of large numbers, a name originally coined by Andre Joyce.
Many googolisms are based upon the googol. For example, analogous to googol = 10^100 and googolplex = 10^10^100, Jonathan Bowers defines numbers like giggol = 1010^10^...^10 with 100 tens and giggolplex = 1010^10^...^10 with a giggol tens. Sbiis Saibian also has an intricate system to name numbers extended from a googol, with numbers like a grangoldex, a graatagold, and a tetrougolthra. We'll learn about both Bowers' and Saibian's work in section 3.
A stranger extension of the googol is the googo- prefix coined by Andre Joyce, which extrapolates from the name "googol" using Roman numerals. We'll learn about it in section 2.
To review, a googol is probably the quintessential googolism, and in some sense the smallest. Now let's look at a googolplex.
The Googolplex
For many people, googolplex is the largest number with a name. It's equal to 1 followed by a googol zeros, or 10^10^100 or 10^10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 in exponent form.
Origin of the googolplex
If you remember Milton Sirotta from the story behind the googol, he came up with the name "googolplex" for a continuation of the googol, He suggested that a googolplex should be one followed by writing zeros until you get tired. However, his uncle Edward Kasner was not satisfied with this vague definition, as different people get tired at different rates and thus a googolplex cannot be precisely defined that way. So Kasner gave a better definition of a googolplex, and cleverly defined a googolplex as equal to one followed by not 100, but a googol zeros.
Size of a googolplex
How can you get an idea of the size of a googolplex? It's not going to be easy. For one thing, let's think back to the volume of the observable universe in Planck volumes, which is about 3*10^185. That's NOWHERE CLOSE to a googolplex! You would need 10^9,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,814 observable universes to get a googolplex Planck volumes. That seems barely less than a googolplex itself! AAAAAAHHHHHHH?????
As you can see, putting a googolplex Planck volumes into a number of observable universes didn't help us get an idea of how much a googolplex is at all. Let's try to get a handle of the difficulty of just writing a googolplex.
Now, there are huge numbers that we can write out quite easily. For example, you can write the already ridiculous googol out in full as:
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
and that would take only about 40 seconds! But how about writing a googolplex?! That would be A LOT harder. First consider that the decimal expansion of a googolplex is
1000000000000 ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 000000000
with a googol zeros total. Therefore the vast majority of the time one would spend writing a googolplex would be writing zeros. Let's say you could reasonably write 3 zeros each second, which is nothing unrealistic. Now imagine spending every waking moment of your life (that'd be 16 hours a day), from birth to death, writing zeros at this rate. You'd probably get drowsy pretty quickly, but let's forget about that for a second. If you lived for 122 years, the longest anyone has lived, you would only write
7,694,784,000
zeros! Compare this to
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
the number of zeros you'd have to write total. So writing zeros every waking moment of your life wouldn't take you even close to writing a googolplex!!
To put all this in perspective, let's go back to the googol-grains we encountered earlier. Now let's say each zero you wrote during your whole 122-year life would be the size of a googol-grain. All those 7.7 billion googol-grains, when packed together in a cube, would only be 0.612 millimeters wide! That's about as big as a tip of a pen, and obviously nowhere close to filling up the observable universe!
So let's say that you got everyone else in the world to help you write a googolplex. All 7.2 billion people alive today would write zeros in exactly the same way you do, writing zeros every waking moment of their 122-year-long lives. That would give you about
55,556,340,500,000,000,000
zeros. That's much better than before, but still not putting a dent on the
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
zeros you'd need to write. Once again, let's put the number of zeros the world would have wrote in terms of googol-grains. All those 55.556 quintillion googol-grains that represent all the zeros the world would have wrote, if packed in a cube, would be only 118 cm (3 feet 10 inches) wide. That's literally it.
Let's shove aside anything that could plausibly happen, since what we had before clearly isn't working. Imagine every person couldn't just write three zeros a second, but they could write a zero every Planck time! Calling this superhuman is an absurd understatement. If you could write a zero every Planck time in only a second, you'd write about 1.855*10^43 zeros. Sounds pretty insane, doesn't it? Not only that, but everyone would write nonstop continuously from the Big Bang to the present! With this ungodly ability we'll definitely be able to write a googolplex, right? Let's find out!
With 1.855*10^43 zeros each second, if every living person wrote zeros every waking moment of the 13.7 billion years the universe has been around, then we'd all write
58,098,600,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
zeros. That number has exactly 71 digits, but it's still a tiny fraction compared to the
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
zeros we'd need to write to get a googolplex! We would have only gotten roughly a nonillionth (a millionth of a trillionth of a trillionth) of the way to writing a googolplex. Writing a googolplex that way would take the whole world 2.38*10^39 (2.38 duodecillion) years! That is unbelievably long and insane, especially considering that we're writing a digit each Planck time here!
With the insane unbelievable size of a googolplex, surely there's no number that represents anything in the real world that comes anywhere near a googolplex, right? Surprisingly, it's possible to surpass a googolplex in several ways. A common example is considering the number of possible parallel universes.
Sbiis Saibian has calculated an estimate of how many different parallel universes there can be, assuming sub-Planck units aren't meaningful. He came up with the value 10^10^343, by taking the factorial of the volume of the observable universe in Planck volumes, to the power of the number of Planck times in 10^32 years. That is a number that, interestingly, has real-world meaning AND surpasses a googolplex (we'll examine more numbers like this in a later article). That shows that as vast as a googolplex is, it can have meaning in the real world!
Googolplex in culture
Because of its huge size and simple explanation, a lot of attention has been given to the googolplex, and it's become a classic benchmark for large numbers, and the largest number with a name for many people. I was no exception for quite a while until I learned of a googolduplex, 1 followed by a googolplex zeros (which for a little while I knew as a googolplexplex).
Above: A googolplex regularly appears in pop culture like this movie theater in The Simpsons. Other examples include Googolplex Mall in the kids' TV show Phineas and Ferb, and astronomer Carl Sagan has referenced a googolplex when discussing the vast scales of the universe. Also, the company Google named its headquarters the Googleplex in reference to the googolplex and also as a portmanteau of Google + complex.
Googolplex in googology
Probably the biggest innovation in googology to come out of the name "googolplex" is the -plex suffix. Since googolplex is 10^googol, x-plex can be thought of as 10^x. For example, a "millionplex" can be thought of as 10^1,000,000, which is one followed by a million zeros. Sbiis Saibian coined many other numbers with the -plex suffix like "googolplexigong", and weirder numbers with -plex have been coined like "piplex", ten to the power of pi.
However, Jonathan Bowers seems to interpret -plex a little differently. For example, he defines the following numbers (and many others) similarly to googol and googolplex:
Giggol = 10^^100, giggolplex = 10^^10^^100
Boogol = {10,10,100}, boogolplex = {10,10,{10,10,100}}
(Don't worry, in sections 2 and 3 we'll learn what all this fancy notation means.)
This shows that Bowers seems to interpret "plexing" a number as replacing the number's largest argument with the original number. Golapulusplex, however, is an exception.
In Sbiis Saibian's Hyper-E notation, a googolplex can be written as E100#2, or as E2#3. We'll examine that notation in section 3.
Some people have extended the familiar -illion sequence to name very large numbers in terms of -illions. The two best known such systems are Conway and Guy's illions and Jonathan Bowers' -illions (we'll discuss those later). While Conway and Guy's is simpler and has shorter names, Jonathan Bowers' is more extensible with a colorful naming system.
In terms of Conway and Guy's -illions (more on those in a later article), a googolplex can be named:
ten trillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentilliduotrigintatrecentillion
(hyphens were created automatically and are NOT part of the name)
And in terms of Bowers' -illions, a googolplex can be named:
ten tretriotriaconto-tretrigintitrecentiduetriaconto-tretrigintitrecentimetriaconto-tretrigintitrecentitriaconto-tretrigintitrecentienneicoso-tretrigintitrecentiocteicoso-tretrigintitrecentihepteicoso-tretrigintitrecentihexeicoso-tretrigintitrecentipenteicoso-tretrigintitrecentitetreicoso-tretrigintitrecentitrioicoso-tretrigintitrecentidueicoso-tretrigintitrecentiicoso-tretrigintitrecentienneco-tretrigintitrecentiocteco-tretrigintitrecentihepteco-tretrigintitrecentihexeco-tretrigintitrecentipenteco-tretrigintitrecentitetreco-tretrigintitrecentitreco-tretrigintitrecentidueco-tretrigintitrecentimeco-tretrigintitrecentiveco-tretrigintitrecentixono-tretrigintitrecentiyocto-tretrigintitrecentizepto-tretrigintitrecentiatto-tretrigintitrecentifemto-tretrigintitrecentipico-tretrigintitrecentinano-tretrigintitrecentimicro-tretrigintitrecentimilli-doetrigintitrecentillion.
(here, hyphens are part of the name)
What about one followed by a googolplex zeros or beyond?
Continuing the Googolplex
Googolduplex
One followed by a googolplex zeros has two widely recognized names: googolduplex and googolplexian. I prefer the name googolduplex because unlike "googolplexian" which is arbitrary, "duplex" in googolduplex literally means two plexes, and that's exactly what a googolduplex is. There are other names for a googolduplex, such as gargoogolplex or googolplusplex. It is unknown who coined any of those names.
A googolduplex is, well, unimaginably huge. It's so big that it's difficult to compare to real-world values, though Poincaré recurrence times can take us past it, as I'll discuss in the next article. There are also two well-known large numbers known as the Skewes' numbers (used as upper-bounds in a mathematical proof, we'll go into them in detail a little later), one of which surpasses a googolduplex. For more on a googolduplex see its entry on part 3 of my number list.
In my opinion, the series "googol, googolplex, googolduplex (or googolplexian if you prefer)" is a pretty cool trio all by itself, but the name googolduplex is just asking for a name for one followed by a googolduplex zeros.
Googoltriplex and beyond
If a googolduplex is one followed by a googolplex zeros, then what is one followed by a googolduplex zeros? Here are the generally accepted names:
10^googolduplex = 10^10^10^10^100 = googoltriplex (also has been called googolplexianite in analogy to "googolplexian")
10^googoltriplex = googolquadriplex
10^googolquadriplex = googolquintiplex
10^googolquintiplex = googolsextiplex
continue with googolseptiplex, -octiplex, -noniplex, -deciplex, etc. following the pattern of Latin roots like the -illions.
However, though they're perfectly valid names, none of them have the appeal of googol, googolplex, and googolduplex: googol can be compared to the number of small spaces in our universe, googolplex to the number of histories of the universe, but googolduplex just transcends our world. "Here be dragons", as you might say.
But dragons (and by dragons, I mean numbers that transcend even the largest stretch of the physical world) are for another time. Up next we'll stay grounded in reality a little more and go into more detail into the large numbers that probability can generate.