1.03. Sizes of the -illions
Now we know pretty well how the -illions came to be, and how they are used. But giving their definitions isn't enough to understand how big they really are. These numbers' sizes are often underestimated because they're so often used, and here I'll give a tour of the -illions and use examples to show how ridiculously huge they really are.
Much of the information is taken from this article by Sbiis Saibian, which has the same general purpose as this page.
Note that if you would like a less American-biased view of some of the samples on this list, you can swap Ohio with South Korea and get the same effect. It never fails to blow my mind that Ohio and South Korea are about the same land area.
Let's start off with a million, a number which is "so small" that it's very familiar in our world. The name, as we saw in an earlier article, literally means "great thousand", and it's the number that all the -illions are based upon. We hear of millions in a variety of contexts wide, whether it be science, economics, human statistics, and so on. A million is also very often used as hyperbole for any large number, and sometimes to talk about millions people will even say "literally millions" instead of just "millions".
But how much would a million of something really be?
Let's start off with a million words: how much would that be? Let's use a novel as a medium for calculating those words. We'll say here that an average novel has 300 words per page and 300 pages. That would amount to 90,000 words in a novel. If you do the math, to read a million words would require reading eleven novels! That's quite a lot of reading, but still something humanly attainable.
How much would a million gallons of water be? A faucet running at full force would take 347 days to waste a million gallons of water. That's almost a year of water. How much space would that take up? It would be a sphere about 19 meters wide, or about 11 times the height of the average person. That means a sphere of a million gallons of water would already look quite intimidating. If it was placed next to your house that sphere would probably tower above it.
Now, imagine a million people formed a tower, with each standing on top of the previous. How big would that tower be? A lot bigger than you probably think. The tower of a million people would be about 1700 kilometers high. That stack would reach waaaaaaaay past the Empire State Building, the height of Mount Everest, zooming past even the height of the International Space Station. Hell, the height of that stack would be about the diameter of the Moon, and 1/7 of the diameter of Earth! Yikes! And that's only a million people, and not even close to the population of the whole world!
How hard a task would it be to count to a million? If you did it nonstop, counting and pronouncing every number (154,236 154,237 154,238 etc.) even if you ran out of breath, would take 29 days, or about a month. But that isn't actually realistic at all. Something more attainable would be counting a few hours each day, adding an average of 2700 numbers per day. That would be realistic and take about a year. This means that there are definitely people who have counted to a million, though not without a lot of dedication.
What would a million be like in time terms? A million seconds is exactly 11 days, 13 hours, 46 minutes, 40 seconds. This may seem like not a long time at all, until you consider something like waiting this long for something. It's just long enough to make someone get impatient when waiting for, say, a game which you really want that has been pushed back. A million minutes is about 1 year, 10 months. This means that something a million minutes ago is long enough ago to feel like a fairly distant event. A million hours would be about 114 years, meaning that the longest-lived people in the world's history barely surpassed a lifespan of a million hours.
Continuing, a million days would be about 2738 years, meaning that a million days ago was the later part of the ancient Egyptian times. A million weeks would be 19,165 years, meaning that a million weeks ago was before even the beginnings of agriculture. A million months is about 83,000 years, so a million months ago was a time of early human civilization, and at that time the Sahara Desert wasn't a desert at all! And finally, a million years is, well, a million years; a million years ago was around the time of Homo erectus, a not-fully-developed ancestor of modern humans.
A million is clearly a pretty big number, and one that isn't easy to dismiss. But it isn't completely out of reach either. Sbiis Saibian has described a million as on the tenuous line between human attainability and utterly out of reach[1A]. That feel of "utterly out of reach" is where a billion comes in...
If you thought a million was pretty huge, it probably didn't quite hit the "wow mark" for you. So we'll naturally want to examine a billion and see if THAT is more impressive. Well guess what: a billion is very incredible and hard to wrap your mind around!
Does what I just said sound like an exaggeration? Well, that's because in our modern world, we hear of billions very often with things like billions of dollars and computers performing billions of operations per second. Therefore a billion just doesn't seem as big as it did a hundred years ago.
But that doesn't change that a billion is very big. To imagine a billion objects you need to imagine a thousand millions, which is difficult because already a million objects is difficult to get straight. Therefore, how are we to comprehend a billion at all? With the examples for a million, but using a billion instead of a million.
A billion words would take up about 11,000 novels, which is pretty massive. How much would 11,000 novels take up? A typical home bookshelf can hold about 150 books, and therefore a billion words would be about 73 bookshelves. That means that in your house it's very unlikely that you have a billion words worth of text, unless you're an EXTREMELY bookish person—ditto for reading a billion words.
How about a billion gallons of water? A faucet would take 950 years to waste a billion gallons of water. As Sbiis Saibian said[1A], if a faucet were to run at full force from the birth of Jesus to the present day it would waste only two billion gallons of water! So a billion really is very huge! In a sphere those billion gallons of water would be 190 meters wide, which means that sphere would be taller than the Great Pyramid of Giza, even how tall it was in ancient times! That means in ancient times a sphere of only a billion gallons would have been taller than any man-made structure!! Now a billion seems like a scary number!
Now imagine a tower of a billion people. That would be 1.7 million kilometers tall, and would reach past the moon and be not too much bigger than the diameter of the Sun!!! O_o; For good measure, that same tower but with the whole world population of 7.1 billion people would be 12 million kilometers tall. That's 30% of the way from Earth to Venus at their closest! That's pretty mind-blowing alright.
Counting to a billion is a task that would take 125 years if you count 16 hours a day, i.e. nearly all of your life except sleep. As of 2015, although nobody has lived 125 years, we have gotten close (122 years is the record). This means that it's impossible to actually count to a billion, since if you spent all your life trying to count to a billion you wouldn't even be able to make it to 125 years of life. This means that counting to a billion is just barely something that a human can't hope to do.
And what would a billion be like in time terms? A billion seconds would be 31 years, 8 months, meaning that most people live between two and three billion seconds. Hopefully with those examples a human lifetime seems a lot longer now. A billion minutes, on the other hand, is 1901 years, meaning that it was about a billion minutes between World War I and the birth of Jesus. A billion hours is about 114,000 years, much longer than all of recorded history, and a billion days is about 2.7 million years.
A billion weeks is about 19 million years, meaning that a billion weeks ago was before the time of organisms people would call "humanoid". A billion months is about 83 million years, meaning that a billion months ago dinosaurs were still around. And a billion years ago was about twice as long ago as the Cambrian Explosion, the time where many new forms of animals evolved.
A billion is only the second member of the -illion series and we already have some mind-blowing comparisons! Now a trillion and even bigger -illions probably seem to be utterly scary, saying nothing of numbers like a vigintillion!!!
But now think back to how often we hear of billions in today's world. This goes to show just how advanced our human civilization has become, and how much we've evolved from a species that would only on a few occasions think of numbers like a hundred. Now that I've said what needs to be said about a billion, on to the next -illion, a trillion!
A trillion is the 3rd -illion, equal to 10 to the 12th power, one followed by twelve zeros, and a million millions. It's a number still small enough that we regularly hear of it in life, most often with countries having trillions of dollars in things like debt. As I said in the previous article, if you do a Google search for "trillion" almost all the results will have to do with trillions of dollars. For many people, a trillion is the largest number they regularly hear of. Larger -illions than this mostly appear in science. Therefore it's good to know just how gigantically big a trillion really is. Let's jump right in:
A trillion words takes up about 11 million books. How much would that be? It's been said that the average American city library has about 18,000 books. Dividing 18,000 into 11,000,000 gives 611 libraries needed to house a trillion words worth of books. That's a LOT of books, and far more than anyone could hope to read in their lifetime.
A faucet would take 950,000 years (that's almost a million years) to waste a trillion gallons of water, meaning that a trillion gallons of water is something we just can't comprehend ... or is it? Sbiis Saibian suggests[1A] that to imagine a trillion or more gallons of water, we should ditch the weeny little faucet and use the Niagara Falls instead.
So how long would it take a trillion gallons of water to go down Niagara Falls? 150,000 gallons of water goes down the falls every second, so doing the math, it takes 77 days for a trillion gallons of water to go down the falls. That's a very huge amount, considering just how much water goes down the falls.
What exactly would those trillion gallons look like? A trillion gallons in a sphere would be 1.9 kilometers wide, or about 1.2 miles. A casual walk around that sphere would take about an hour, and the sphere would look quite intimidating if it were to float above any city. If a trillion gallons of water were laid on the state of Ohio they'd cover Ohio in a layer 3.26 centimeters deep.
How big would a tower of a trillion people be? Even though there aren't even a trillion people in the world, the tower would be about 1.7 billion kilometers high, so if the tower were placed on Earth facing away from the sun, the tower would reach past Mars, the asteroid belt, Jupiter, and even Saturn! Now that's pretty insane.
Counting to a trillion nonstop would take about 200,000 years, so trying to count to a trillion is UTTERLY hopeless. This proves that precise physical quantities on the scale of trillions will always need to be estimated and cannot be known precisely.
How would you put a trillion into time terms? A trillion seconds is about 32,000 years, meaning that a trillion seconds ago was still well before recorded history or even the onset of agriculture. A trillion minutes is 1.9 million years, so a trillion minutes ago was well before "anatomically modern" humans. A trillion hours is 114 million years, so a trillion hours ago was around the time of the dinosaurs. A trillion days is 2.7 billion years, well before Earth had any multicellular life. And a trillion weeks or longer is beyond the age of the universe!
Here are some other examples of trillions:
The nearest star to us (not counting the sun), Proxima Centauri, is already 25 trillion miles away, though this says more about the insane scales of our universe than the number itself.
The human body contains about 20-50 trillion cells based on estimates. It is often stressed that cells are extremely small (for example, give your arm a quick scratch and you've gotten rid of a few hundred thousand of them), so it may come as a surprise that the human body is only made of trillions of cells. Once again this goes to show how often large numbers are underestimated, and how ridiculously huge a trillion is.
A trillion pennies in a cube would be bigger than a football field, and there are about a trillion fish in the world. For what a trillion dollars tightly packed in American $100 bills looks like click here.
So a trillion is a pretty insane number, but only the beginning of some super-insane-extreme-mega-large numbers ... next up is a quadrillion, equal to 1,000,000,000,000,000.
A quadrillion is the next -illion after a trillion. It's equal to one followed by 15 zeros, and equal to a million billions or a thousand trillions, which is pretty insane. Quadrillions are seen most often in science, but like I said in the previous article they occasionally show up in economics when working with extremely large worldwide scales or when the currency unit isn't very valuable (yens for instance).
Often "quadrillion" is replaced with "thousand trillion" or "million billion", which has always bothered me: it doesn't feel right to pretend names for numbers beyond a trillion don't exist, and it certainly doesn't do these numbers justice. When the name "quadrillion" is used, it is often specified what a quadrillion is, since many people are not familiar with the name. It is commonly asked "what comes after a trillion". On the other hand, the term "quadrillion" for 10 to the 15th power is not too obscure; if you Google search "what comes after" the first suggestion is "what comes after trillion", but the second is "what comes after quadrillion" (the next few are "terabyte", "billion", and "quadruple"). So what are we waiting for? Let's get to some examples of how much a quadrillion is.
A quadrillion words would take up 11 billion books. A question you may have is, are there even that many books in the world? That's actually a good question, since it isn't easy to answer.
Google estimated in 2010 that there are about 130 million different books in the world, counting each edition or translation as a separate book. However, that figure isn't quite what we're looking for, because it counts how many different books have been published, not how many total books have been printed. But that figure can still help us estimate how many books there are.
Let's say that there are at least 100 physical copies of each of those 130 million books out there somewhere in the world. That's a huge underestimate since most of those books will have far more, but it allows us to say that there are at least 13 billion copies of those books in the world, amounting to AT LEAST a quadrillion words printed in books in the world. However, keep in mind that since they'd be in 11 billion books, we could forget about even skimming through every one of them!
How about a quadrillion gallons of water? It would take 210 years for a quadrillion gallons of water to go down Niagara Falls. To get an idea of that, Sbiis Saibian says since 210 years ago America was still a very young country, you can imagine the Niagara Falls from the Declaration of Independence to the present day and not much more than a quadrillion gallons would have gone down the falls[1A].
What would a quadrillion gallons of water look like? A sphere of this much water would be about 19 kilometers or 12 miles wide, so it would by now look quite ominous in almost any context, and just about visible from space. That many gallons would cover the state of Ohio 32 meters or 107 feet deep, and the continental United States 48 centimeters or one foot seven inches deep. That's a lot of water, and we're only at the fourth -illion number! For comparison, the Great Lakes themselves have 6 quadrillion gallons of water.
How about a tower of a quadrillion people? They'd tower 1.7 trillion km high, or about 1/5 of a light year, and they'd reach past the orbit of Pluto at its furthest point 232 times!!! That's pretty insane alright.
Counting to a quadrillion nonstop would be beyond hopeless, as it would take over 200 million years. With a quintillion and higher we can forget about the counting analogies in their entirety because higher time spans would dwarf the age of the universe.
Even just a quadrillion seconds would be about 32 million years, so a quadrillion seconds ago was well before any organisms that most people would consider "humanoid"! A quadrillion minutes would be about 1.9 billion years, and a quadrillion hours and longer would be more than the age of the universe!
Here's another interesting figure: according to Randall Munroe, the author of the popular webcomic xkcd, as of late 2011 the total economic production of the human race so far has been 2.397 quadrillion dollars.
What would a quadrillion dollars look like? With $100 bills put into the tight half-inch-thick $10,000 packs, a quadrillion dollars would take up a space about 5x as large as the Empire State Building. That's pretty fucking huge.
So as you can see a quadrillion is a really insane number, but only the fourth step on a journey towards madness! Up next is a quintillion.
A quintillion is equal to 1 followed by 18 zeros, or a million trillions or a billion billions, or a million million millions. Quintillions, unlike quadrillions, exhaust economic usage entirely, and therefore are rarely seen outside science.
Sbiis Saibian has described a quintillion as a cut-off point for large numbers[1A], as -illions after a quintillion just don't get used much and it's rare to find them outside of lists. For a while a quintillion was the largest -illion i knew of, other than a vigintillion, a trigintillion, and perhaps a decillion. Perhaps that may be because a quintillion was the largest number I saw often enough to remember the name. So let's get to it and try to get a feel of just how big a quintillion is.
A quintillion words would take up about 11 trillion books, so it's debatable whether a quintillion words actually exist on the world. It has been estimated that 100 quadrillion words have been printed in the first 500 years of the printing press (1456 to 1956), and since that figure is rather rough it only adds to the question. But with the vast amount of data on the Internet, it now is highly doubtful that a quintillion words do not exist within Earth in some way.
A better way to get an idea of a quintillion words would be, how long would it take for humanity to speak that many words? Studies have shown that men speak about 7,000 words per day and women speak 20,000. This averages to about 13,500 words per person. Then, 13,500 words per person per day * 7.1 billion people amounts to 96 trillion words spoken my humanity every day. With that, you get that it would take 28 years, 7 months for humanity to utter a quintillion words. This shows that a quintillion words really do exist in our world—but don't expect that to last much longer.
How about a quintillion gallons of water? It would take about 210,000 years for a quintillion gallons of water to go down the falls. That's about 30 times all of recorded history. The Niagara Falls are estimated to only last 60,000 years total before ceasing to exist, so in the falls' entire lifespan a quintillion gallons won't go down the falls! A sphere of a quintillion gallons of water would be 120 miles wide. It could hover over about 1/4 of the area of the state of Ohio, and it would be easy to see from space. For comparison, the entire Earth has 326 quintillion gallons of water in all its oceans.
A tower of a quintillion people, as you might have guessed, would be insanely huge. It would reach 180 light-years, or roughly the distance to the star cluster Omega Centauri.
A quintillion is too large to put in reasonable time terms at all. Even just a quintillion seconds is about 32 billion years, about twice the age of the universe. One could argue that we could quantify quintillions using units like nanoseconds, though they're too small for people to directly grasp and therefore don't have the appeal of using such tangible units as seconds.
After a quintillion we're after -illions we won't regularly encounter at all, but are numbers that we can't ignore at all. Up next we have an insanely huge number, known as a sextillion.
A sextillion is equal to 1 followed by 21 zeros. It's also equal to a billion trillions, which is mind-bogglingly massive, and only the sixth -illion. It's so big that it's almost never heard of, and I myself didn't know of the name until 2013, the year I discovered the amazing world of googology.
How big is a sextillion? We've exhausted many of the analogies we had access to with numbers like a million. The counting analogy is utterly hopeless with a sextillion, as is the Niagara Falls analogy. A population of 7.1 billion people would take 28,000 years to utter a sextillion words, and since the past population was so much less than 7.1 billion that we can safely say that humanity hasn't uttered that many words! Ditto for printing that many words. The time analogies aren't any better a way to get an idea of how big a sextillion is. So we'll try to get an idea of the magnitude of a sextillion with the analogies we do have.
First off, how big is a sextillion gallons of water? That's 3x the amount of gallons of water on Earth, but we can still imagine spheres this big. A sextillion gallons in a sphere would be about 1900 km wide, about half the length of the continental United States. That's pretty insane, and appreciably big when placed next to Earth itself.
Continuing with the tower-of-people idea, a tower of a sextillion people would be 180,000 light years tall. That tower would be bigger than the diameter of the Milky Way!!! o_O;;
Earth ways about 6 sextillion metric tons (a metric ton is 1000 kilograms, which can also be called a "megagram"). The Earth's oceans have about 6 sextillion cups of water, and the volume of Earth is about 1.085 sextillion cubic meters. As you can see, a sextillion can be considered an Earth-scale number.
Here are some other figures in the sextillions: the current estimate of the number of stars in the observable universe is 300 sextillion. Also, the entire observable universe may be about 547 sextillion miles in diameter.
Now is a good time to try and bring these numbers in terms of atoms. A sextillion carbon atoms in a cube would be 1.34 millimeters wide, about the size of an average flea. Therefore a sextillion can be seen as a number so big that this many atoms is large enough to see with the naked eye, though really this tells us more about how small an atom is than how big a sextillion is.
Nonetheless, hopefully it's clear now that a sextillion is a super-ultra-gigantic number. However, remember the first law of googology, it only gets worse, and as a corollary, you ain't seen nothing yet! Up next on the itinerary of ridiculously big numbers is a septillion!
You know the drill now. This is a septillion, equal to 1 followed by 24 zeros, or a trillion trillions. It's also the largest -illion to get an officially recognized SI prefix. An SI prefix is a prefix used to multiply a unit by a certain value. For example, mega- multiplies by a million, and therefore a megameter is a million meters. The SI prefix for a septillion is yotta-, and for further discussion on the prefixes and a proposal I made to extend upon them, look here.
Just how insanely huge is a septillion? Let's start with continuing the gallons-of-water idea. A septillion gallons in a sphere would be 19,000 km wide. That's 50% bigger than the Earth itself!! Pretty insane alright, and getting quite astronomical.
A tower of a septillion people would be 180 million light-years tall, which is more than the diameter of the Virgo supercluster, and also past the Centaurus galaxy cluster! That's pretty—no, VERY insane.
How about a septillion dollars? In tightly packed 100-dollar bills, that would cover the United States in a layer about a kilometer thick.
Here are a few other figures: Earth weighs about 6 septillion kilograms, and a liter of water contains about 33.4 septillion molecules. The observable universe is about 880 septillion meters wide. That's also equal to 93 billion light-years.
Also, a septillion carbon atoms in a cube would be 1.34 cm wide. That's about the width of your pinkie finger.
A septillion is insanely huge and really blasting into the stars, but still quite early in a journey to absolute madness. Next up is an octillion!
An octillion is equal to 1 followed by 27 zeros. That's equal to a billion billion billions, which is impossible to even comprehend! It's a number so large that most people haven't even heard of it, and there's a completely good reason why: it's ridiculously huge. It's far far more than things most people would care to know: more than the number of gallons of water on Earth, the monetary value of the entire Earth's crust, the number of cells in your body, the diameter of the observable universe in meters, and more and more. So how can we get an idea of how big an octillion is? Let's use some examples:
An octillion gallons of water in a sphere would be 190,000 km wide. That's somewhere between the size of Jupiter and the sun.
A tower of an octillion people would be 180 million light years high. That's more than the diameter of the observable universe! This clearly shows that from the next -illion onward, we can forget about even using the tower-of-people analogy because these numbers are so big!!
How would you put weight into terms of octillions? Earth weighs about 6 octillion grams. The sun weighs about 2 octillion metric tons, and it's about 1.4 octillion cubic meters.
An octillion dollars would cover the world in a layer 15 km thick. 15 km is about twice the height of Mount Everest!
An octillion carbon atoms would take up a cube 13.4 cm wide, or about 5.3 inches. That's as big as many household objects. Also, the average human body has about seven octillion atoms, which is pretty amazing. Given all these astronomically huge examples of octillions, hopefully this gives you a good idea of how tiny atoms are!
An octillion is unbelievably insane by any standards, yet it's not even close to the largest numbers we can compare to the universe ... next we have a nonillion.
A nonillion is equal to 10^30, a quadrillion quadrillions, and 1 followed by 30 zeros. That's enough zeros that it's pretty hard now to keep track of all of them. It has ten groups of three zeros, meaning that it's the tenth power of a thousand, so a way to imagine a nonillion is to imagine sphere of a thousand objects, then a thousand of those spheres, then a thousand of those spheres, then a thousand of those spheres, then a thousand of those spheres, then a thousand of those spheres, then a thousand of those spheres, then a thousand of those spheres, then a thousand of those spheres, then a thousand of those spheres. But that's too many repetitions to really comprehend, so we once again need to use examples to get a feel of the sheer massiveness of a nonillion.
First let's use the gallons-of-water idea: a nonillion gallons in a sphere would be 1.9 million km wide. That's 50% bigger than the sun itself! Among some of the more well-known stars, this nonillion-sphere would fall in size somwhere between Alpha Centauri A and Sirius.
A nonillion dollars would cover the Earth in a layer 15,000 km thick. This might not seem like a lot, but actually 15,000 km is more than the diameter of Earth! That means if the Earth had a layer of a nonillion dollars covering it, it would actually be more than 3 times as big in diameter!!! Now that's REALLY insanely huge.
A nonillion carbon atoms in a cube would have side-length 134 cm, or about 4 feet 5 inches. That's big enough that it would be very hard for a human to carry, especially since the atoms are quite tightly packed! And carbon is one of the lightest elements, so can you even begin to imagine lifting a nonillion atoms of iron or lead or gold?
Here's another figure in the nonillions: The sun weighs about 2 nonillion kilograms.
A nonillion is an absolutely insanely crazy huge number. Up next is a favorite -illion of mine, the decillion.
A decillion is equal to 1 followed by 33 zeros, or a billion trillion trillions. It's one of my own favorite illions, being the tenth -illion and also a cut-off point for using -illion names, somewhat akin to a quintillion. While the names sextillion through decillion show up now and then (albeit quite infrequently), any illions after decillion will almost never appear at all, expect in particularly googological circumstances (such as what you're reading right now) or in some lists. A decillion was also the largest -illion my mom taught me as a kid.
How big is a decillion? Let's use some examples just like previously.
A decillion gallons of water in a sphere would be 19 million km wide. That's so big that it would dwarf the sun!! A star that would be comparable to is the biggest star in the Albireo group, which is 22 million km wide.
A decillion dollars in a sphere would be roughly 200,000 km in diameter. That's a little bit bigger than Jupiter!
A cube of a decillion carbon atoms would be about 13.4 meters high, or about 44 feet. That's bigger than most people's houses!! Another way to put this is, a decillion is so big that even this many atoms looks intimidating to a person!!!
The sun weighs about 2 decillion grams, which is a really insane figure.
After a decillion, I'll start taking bigger jumps through the -illions, for the sake of cutting to the chase. Up next we have a jump by a factor of a quadrillion, a monstrously gigantic number known as a quindecillion.
Alright, here's a REALLY awesome number called a quindecillion. As its name may suggest, it's the fifteenth member of the -illion series, and it's equal to 1 followed by 48 zeros, or a trillion trillion trillion trillions. It's the logarithmic halfway point between a decillion and a vigintillion.
And yes, we just took a leap by a factor of a quadrillion, which is itself a huge number (though quite modest in comparison to the numbers we're looking at right now!), so prepare for some shocking figures about the massiveness of a quindecillion.
How big do you think a quindecillion gallons of water would be? It's certainly bigger than the sun, but would the sphere centered at the sun swallow any planets, Earth perhaps? Would it pass up the big stars like Rigel or Betelgeuse?
The sphere would be over 500 times bigger in diameter than any stars known to exist, even UY Scuti, the largest known star! And yes, it would pass the orbits of all the planets, even Uranus and Neptune, and it would exceed the orbit of Pluto at its farthest point from the sun 260 times!!! That's an INSANE jump from a decillion, and utterly and completely dizzying!!
How about a quindecillion atoms? Would that be able to cover up your neighborhood? A typical city? How much would it cover? A cube of a quindecillion atoms would be 1340 kilometers in side length, so that would cover almost 1/4 of the continental United States, and the cube would look appreciably big next to Earth itself! In fact, a sphere of a sexdecillion atoms (that's a thousand quindecillion) would be about as big as Earth!
Relating to that, according to Sbiis Saibian's site[1B] Earth is itself made of about 89 quindecillion molecules! This shows that a quindecillion is a tipping point for numbers in terms of atoms, ending the range of "dwarfed by Earth" and starting the range of "dwarfing Earth"!
As you can see, a quindecillion is SCARY and makes a decillion look adorably small. Up next is a vigintillion, another one of my favorite -illions.
The NEXT next number for us to cover is a vigintillion, equal to 1 followed by 63 zeros. It's a quadrillion times bigger than a quindecillion and a nonillion times bigger than a decillion. A vigintillion is one of my own favorite -illions, largely because it's the largest -illion to have a name that is considered part of the English language, other than the incomprehensible centillion. There are no official -illions between a vigintillion and a centillion. But another reason I like the vigintillion is because of how RIDICULOUSLY huge it is.
How big is a vigintillion? Imagine a cube made up of exactly one vigintillion atoms. How big do you think it would be? It would be 134 million km wide, which is so big that it EASILY dwarfs the earth. In fact, saying it dwarfs Earth isn't enough to get a feel of the magnitude of a vigintillion! That vigintillion-cube would be as big as a million suns!!! To get an idea of what that means, you probably have an idea of how big the sun is compared to the earth—if you don't, the sun is about 100 times bigger than Earth in diameter. Then, in size-terms, you can simply use the analogy: the vigintillion-cube is to the sun as the sun is to the Earth. Although there are stars which are that amazingly big, this still means that a vigintillion is far scarier than a quindecillion!!!
How about a vigintillion gallons of water? In a sphere that would be 190 quadrillion meters in diameter! That's about 20 light-years. That's about 10x past the outer boundary of the Sun's gravitational pull (aka the Oort Cloud), and an astronomical object this would be comparable to is the Orion Nebula.
Hmm, what else is there to say about a vigintillion's size? I think that's about all we can say about just how mind-bogglingly big a vigintillion is ... it's just a number that transcends most things in the universe. That's how amazingly big a vigintillion would be; numbers this big have little use in the everyday world.
After a vigintillion we have numbers so big that they don't even get a name in the English language! Up next is the 30th -illion, which is often known as a trigintillion.
Now we have a huge number equal to 10 to the 93rd power, or 1 followed by 93 zeros. It does not get its own name in English, though it could be called a "nonillion vigintillion" by combining names, just as less familiar -illions are often named in terms of more familiar -illions (e.g. "million trillion trillion" in place of "nonillion"). However, as I will discuss in a later article, many people have proposed extensions or even revamps for the -illion system, and among most systems that extend upon the existing -illions this number is known as a trigintillion, derived from Latin "triginta" meaning thirty. But just how big is a trigintillion?
If you thought a vigintillion transcends most uses, then a trigintillion is even worse! For one thing, a trigintillion is about 10 trillion times more than the number of atoms in the observable universe, a figure estimated as 10^80!
Does that mean that a trigintillion has zero real-world meaning? Absolutely not! For example, the volume of the observable universe is estimated as 3*1080 cubic meters. Put that into, say, cubic nanometers, and then we'd get 3*10107 as the volume of the observable universe in cubic nanometers. Things like DNA are measured in nanometers.
So then, to get a feel of the size of a trigintillion, we can imagine dividing the observable universe into a trigintillion equal-sized cubes. In that case, each cube would be 46 micrometers in length, which is about half the width of a human hair. So a trigintillion is a dizzyingly big number alright.
How about a trigintillion atoms? Even though there aren't that many in the observable universe, we can still consider this many atoms in a tightly packed cube. That sphere would be 142,000 light-years wide, and it would be a little bigger than the Milky Way galaxy!!! Now that's an insane sphere, epsecially since we're talking about atoms here.
Also, a trigintillion gallons of water would dwarf the observable universe itself, meaning that the gallons-of-water analogy is no longer useful here. Pretty insane.
Up next, with another jump by a factor of a nonillion, is 10^123 which is often called a quadragintillion.
This is 10^123, the 40th -illion, often known as a quadragintillion from Latin "quadraginta" meaning 40. You may notice that now we've surpassed the googol which is equal to 10100, but I'll discuss the googol in the next article.
How would you get a feel of how big a quadragintillion is? For one thing, we've now exhausted the atom-cube idea, since a cube of this many atoms would be 1.42 quadrillion light years wide, which is much bigger than the observable universe; 10110 atoms are enough to fill it up!!! So what can we still do? We've exhausted almost everything ...
We can still imagine dividing the observable universe into a quadragintillion portions. Each portion would be 4.6 femtometers (a femtometer is a quadrillionth of a meter) in size, which is insanely small: each portion would be able to hold no more than a hundred protons. As small as it is, it's big enough that it can contain things that we know well exist ... but barely. Soon we'll be to numbers where dividing the observable universe into that many portions is physically unmeasurable, which you'll see more about later.
That's about all I can say about a quadragintillion because it's so insanely huge. Up next is a quinquagintillion.
This is 10^153, the 50th -illion, often known as a quinquagintillion from Latin "quinquaginta" meaning 50. It's insanely huge, so huge that dividing the observable universe into this many portions would make each portion unimaginably small.
Just how small would that portion be? That would be 0.46 yoctometers in diameter. Over an octillion of those portions could fit inside a proton!! That's insanely small alright. In fact, almost nothing is known to be this small. One such thing that is estimated to be this small is a neutrino, but that's a very rough estimate that hasn't really been confirmed. But that unit is still larger than the Planck length, the smallest length that is considered physically meaningful. I'll further discuss that in the next entry.
Up next we have the truly insane sexagintillion, a major turning point in what we can measure, which we'll see why in just a little bit.
This is 10^183, the 60th -illion, often known as a sexagintillion from Latin "sexaginta" meaning 60. This one is part of an important figure: the number of Planck volumes in the observable universe, which is about 300 sexagintillion.
First off, what exactly is a Planck volume? A Planck volume is just a cubic Planck length, and a Planck length is about 1.6162*10-35 meters. That sounds like a pretty arbitrary value, but it's not. The significance involves some quantum mechanics which most people (myself included) don't really know about, so the best explanation for most people is that a Planck length is the smallest length that can actually be measured, and the smallest that is physically meaningful.
What exactly is important about that? It is theorized that when you're below the Planck length, the laws of physics just don't work, and all the time and position stuff can be completely scrambled if the theory is correct. This also relates to quantum foam, a theoretical Planck-unit-sized material that may forms the very base of the physically meaningful. In any case, we can't consider units below Planck length physically meaningful.
Therefor, the entire observable universe's measuring is limited as 3*10185 cubic Planck lengths ... or is it? We can still consider even larger numbers to measure the universe under the most theoretical scenarios. Up next we'll skip to the whoppingly huge centillion, where I'll discuss how to get that kind of stuff.
We finally reach a centillion, the 100th -illion, equal to 1 followed by 303 zeros. It's the largest -illion with an official name in English. Since it's about 10118 times the number of Planck times in the observable universe, surely a centillion is too big to represent anything in the real world, right?
Actually, there are several ways to get numbers as big as a centillion. One thing you could consider is that since the portion of the universe that is visible is always expanding, we can consider the future volume of the universe, which is also something Sbiis Saibian talks about in an article of his[1C]. In this case, the observable universe would have a volume of about a centillion Planck volumes roughly 1050 years in the future.
What will the universe be like in 1050 years? There are several theories. According to Wikipedia, there is a theory that after an EXTREMELY long time (believed to be between 1035 and 1043 years), protons will decay into a pion (which quickly decays into pure energy) and a positron, causing all the matter in the universe to eventually cease to exist. Even if protons do not decay, the universe will be very different from how it is now in 1050 years. Maybe all stars will be somehow thrown into black holes, and maybe even rigid matter will start rearranging atoms as if it was liquid on a very long scale. It's all pretty mind-boggling, and quite scary as well. Therefore this isn't the best way to get a feel of how much a centillion is.
Another not-as-scary way to consider numbers as big as a centillion is with something called quartic hypervolume. What exactly is that? It's what you get when considering time as a fourth dimension, along with the three spatial dimensions.
A simple example of quartic hypervolume would be as follows: Imagine a cube a light-year wide, and considering it only for a year. Then the quartic hypervolume of that cube for a year is one quartic light-year. Now, imagine a cube 2 light years wide, considering it for a year. Then the quartic hypervolume of that cube for a year would be 8 quartic light-years. But if you considered that same 2-light-year-cube for 2 years, then the quartic hypervolume would be 16 quartic light-years.
With that in mind, we can now consider quartic hypervolumes, but on a Planck scale. We'll use the Planck length, but what time unit will we use? The unit we can use is the Planck time. What is a Planck time? The Planck time is defined as the amount of time it takes for light to travel a Planck length, about 5.39*10-44 seconds. Planck times are very analogous to Planck lengths. When dealing with sub-Planck times it is theorized that the concepts of past and future get scrambled up. Therefore two events that occur less than a Planck time apart can be considered simultaneous.
How would you work with that for the observable universe? You need to do a bit of math. The real formula for quartic hypervolume of the entire observable universe is pretty complicated, but a decent approximation formula given by Robert Munafo is:
1/4 * a * 4/3 * π * r^3
where a is the age of the universe in Planck times and r is its radius in Planck lengths.
The current radius of the observable universe in Planck lengths is 2.75*1061, and the age in Planck times is 8.03*1060. The values are quite close, but relation between the values is complicated.
But for simplicity's sake, we can assume that at any time the Planck radius will generally be 3.42 times the Planck age. Though this is not strictly true, it's a decent estimation which allows us to simplify the formula to:
1/4 * a * 4/3 * π * (3.42*a)^3
Simplifying further gives us the formula for quartic hypervolume based on age:
Now, things just got a lot simpler. Now all we need to is solve the equation:
10^303 = 41.89*a^4
a = 2.2104e75
Keep in mind that a is in Planck times, so what we need to do ... all we need to is convert to years. That gives us 3.778 septillion years. Then we can say:
The quartic hypervolume of the observable universe from the Big Bang to 3.778 septillion years in the future is about a centillion quartic Planck units.
However, another way we can consider a centillion's size is by probability, but we'll get to that in a later article. For now, we'll push the idea of measuring things to its very limits by going hypothetical!
How would we get numbers beyond a centillion in the real world? By now it's difficult, but the best option we have goes as follows:
First off, it is theorized that the universe encompasses an area far greater than the observable universe. It is not known how big that area is. Some believe that the size of the whole universe is infinite, but that would be boring to us since that would not provide a real example of a large finite number. Something more interesting, however, is that the size of the entire universe, based on extrapolations from the theory of inflation in the Big Bang, may be something on the order of 1010^12 meters wide. That's quite big alright, since this estimate is one followed by a trillion zeros. This number is horrifying to even contemplate. To get the size of this entire universe, you need to dwarf the diameter of the observable universe by a factor of its diameter in meters, 37,000,000,000 times! And while If you did such a gigantic dwarfing factor once every second, then it would take 1190 years to get the size of what could be the entire universe!! This diameter is so big that it doesn't really matter what units we use (it could be Planck lengths or yottameters); the estimate is still around 1010^12.
However, chaotic inflation is a theory devised by Andrei Linde that allows for even bigger estimates of the size of the universe. With chaotic inflation, there can be something known as a Grand Universe, which can be imagined as not just one single universe, but a sort of "grand universe" with local Big Bangs constantly going off. Not a lot is known about how that theory would work in the real world, but it can lead to some even bigger numbers. Sbiis Saibian lower-bounded the chaotic inflationary size of the universe by assuming that it expanded by a factor of 1010^12 every Planck time and extending it to 13.7 billion years, to be on the order of 1010^64 . This is an interesting figure especially since it's a lower bound. With such a Grand Universe it could well be that our Big Bang was not even the first ... and yet, with that theory the universe still must be finite. It's impossible to really know how big the Grand Universe might be, or if it even exists ...
Is it possible to go further than this Grand Universe? Maybe theoretically, but anything past measuring the entire observable universe is quite theoretical, especially with the idea of chaotic inflation. For now let's review what we've seen.
I hope the analogies I gave showed that millions, billions, and trillions are way bigger numbers than you probably thought. And to get a scale for the more exotic numbers like a vigintillion, you have to go super-exotic and pit atoms against celestial objects! Our world is full of large numbers, but with the numbering systems we learned about, it's easy to generate numbers so big that our world can't even keep up with their size! But hold on, those are for later. For now let's continue with looking at the legendary numbers, the googol and googolplex, with their history, size, and cultural impact, and after that we'll return to the idea of large numbers in the real world with probability.