From 1,000,000 to Graham’s Number

Welcome to numbers post #2.

Last week, we started at 1 and slowly and steadily worked our way up to 1,000,000. We used dots. It was cute.

Well fun time’s over. Today, shit gets real.

Before things get totally out of hand, let’s start by working our way up the still-fathomable powers of 10—

Powers of 10

When we went from 1 to 1,000,000, we didn’t need powers—we could just use a short string of digits to represent the numbers we were talking about. If we wanted to multiply a number by 10, we just added a zero.

But as you advance past a million, zeros start to become plentiful and you need a different notation. That’s why we use powers. When people talk about exponential growth, they’re referring to the craziness that can happen when you start using powers. For example:

If you multiply 9,845,625,675,438 by 8,372,745,993,275, the result is still smaller than 829.

As we get bigger and bigger today, we’ll stick with powers of 10, because when you start talking about really big numbers, what becomes relevant is the number of digits, not the digits themselves—i.e. every 70-digit number is somewhere between 1070 and 1071, which is really all you need to know. So for at least the first part of this post, the powers of 10 can serve nicely as orders-of-magnitude “checkpoints”.

Each time we up the power by one, we multiply the world we’re in by ten, changing things significantly. Let’s start off where we left off last time—

106 (1 million – 1,000,000) – The amount of dots in that huge image we finished up with last week. On my computer screen, that image was about 18cm x 450cm = .81 m2 in area.

107 (10 million) – This brings us to a range that includes the number of steps it would take to walk around the Earth (40 million steps). If each of your steps around the Earth were represented by a dot like those from the grids in the last post, the dots would fill a 6m x 6m square.

108 (100 million) – Now we’re at the number of books ever published in human history (130 million), and at the top of this range, the estimated number of words a human being speaks in a lifetime (860 million). Also in this range are the odds of winning the really big lotteries. A recent Mega Millions lottery had 1-in-175,711,536 odds of winning. To put those chances in perspective, that’s about the number of seconds in six years. So it’s like knowing a hedgehog will sneeze once and only once in the next six years and putting your hard-earned money down on one particular second—say, the 36th second of 2:52am on March 19th, 2017—and only winning if the one sneeze happens exactly at that second. Don’t buy a Mega Millions ticket.

109 (1 billion1 – 1,000,000,000) – Here we have the number of seconds in a century (about 3 billion), the number of living humans (7.125 billion), and to fit a billion dots, our dot image would cover two basketball courts.

1010 (10 billion) – Now we’re up to the years since the Big Bang (13.7 billion) and the number of seconds since Jesus Christ lived (60 billion).

1011 (100 billion) – This is about the number of stars in the Milky Way and the number of galaxies in the observable universe (100-400 billion)—so if a computer listed one observable galaxy every second since Christ, it wouldn’t be anywhere close to finished currently.

1012 (1 trillion – 1,000,000,000,000) – A million millions. The amount of pounds the scale would show if you put the whole human race on it (~1 trillion), the number of seconds humans have been around (~100,000 years = ~3 trillion seconds), and larger than both of those totals combined, the number of miles in one light year (6 trillion). A trillion is so big that you’d only need 4 trillion millimeters of ribbon to tie a bow around the sun.

1013 (10 trillion) – This is about as big as we can get for numbers we hear discussed in the real world, and it’s almost always related to nations and dollars—the US nominal GDP in 2013 was just under $17 trillion, and its debt is currently just under $18 trillion. Both of those are dwarfed by the number of cells in the human body (37 trillion).

1014 (100 trillion) – 100 trillion is about the number of letters in every published book in human history, as well as the number of bacteria in your body.2 Also in this range is the total wealth of the world ($241 trillion, which we discussed at great length in a previous post).

1015 (1 quadrillion) – Okay goodbye normal words. People say the words million, billion, and trillion a lot. No one says quadrillion. It’s really uncool to say the word quadrillion.3 Most people opt for “a million billion” instead. Either way, there are about a quadrillion ants on Earth. Comparing this to the bacteria fact, it’s like you have 1/10th of the world’s ants crawling around inside your body.

1016 (10 quadrillion) – It’s in this range that we get to the number of playing cards you’d have to accidentally knock off the table to cover the entire Earth (89 quadrillion). People would be mad at you.

1017 (100 quadrillion) – The number of seconds since the Big Bang. Also the number of references to Kim Kardashian that entered my soundscape in the last week. Please stop.

1018 (1 quintillion) – Also known as a billion billion, the word quintillion manages to be even less cool than a quadrillion. No one who has social skills ever says the word quintillion. Anyway, it’s the number of cubic meters of water in all the Earth’s oceans and the number of atoms in a grain of salt (1.2 quintillion). The number of grains of sand on every beach on Earth is about 7.5 quintillion—the same number of atoms in six grains of salt.

1019 (10 quintillion) – The number of millimeters from here to the closest next star (38 quintillion millimeters).

1020 (100 quintillion) – The number of meter-long steps it would take you to walk across the whole Milky Way. So many podcasts. And heard of a Planck volume? It’s the smallest volume scientists talk about, so small you could fit 100 quintillion of them in a proton. More on Planck volumes later. Oh, and our dot image? By the time we get to 600 quintillion dots, the image would cover the surface of the Earth.

1021 (1 sextillion) – Now we’re even beyond the vocabulary of the weirdos. I don’t think I’ve ever heard someone say “sextillion” out loud, and I hope to keep it that way.

1023 (100 sextillion) – A rough estimate for the number of stars in the observable universe. You also had to deal with this number in high school—602 sextillion, or 6.02 x 1023—is a mole, or Avogadro’s Number, and the number of hydrogen atoms in a gram of hydrogen.

1024 (1 septillion) – A trillion trillions. The Earth weighs about six septillion kilograms.

1025 (10 septillion) – The number of drops of water in all the world’s oceans.

1027 (1 octillion) – If the Earth were hollow, it would take 1 octillion peas to pack it full. And I think we’ve heard just about enough from octillion.

Okay so now let’s take a huge leap forward into a whole different territory—somewhere where the Earth’s volume is too tiny and the Big Bang too recent to use in examples. In this new arena of number, only the observable universe—a sphere about 92 billion light years across—can handle the magnitude we’re dealing with.4

1080 –  To get to 1080, you take trillion and you multiply it by a trillion, by a trillion, by a trillion, by a trillion, by a trillion, by a hundred million. No dot posters being sold for this number. So why did I stop here at this number? Because it’s a common estimate for the number of atoms in the universe.

1086 – And what if you wanted to pack the entire observable universe sphere with peas? You’d need 1086 peas to make it happen.

1090 – This is how many medium size grains of sand (.5mm in diameter) it would take to pack the universe full.

A Googol – 10100

The name googol came about when American mathematician Edward Kasner got cute one day in 1938 and asked his 9-year-old nephew Milton to come up with a name for 10100—1 with 100 zeros. Milton, being an inane 9-year-old, suggested “googol.” Kasner apparently decided this was a reasonable answer, ran with it, and that was that.5

So how big is a googol?

It’s the number of grains of sand that could fit in the universe, times 10 billion. So picture the universe jam-packed with small grains of sand—for tens of billions of light years above the Earth, below it, in front of it, behind it, just sand. Endless sand. You could fly a plane for trillions of years in any direction at full speed through it, and you’d never get to the end of the sand. Lots and lots and lots of sand.

Now imagine that you stop the plane at some point, reach out the window, and grab one grain of sand to look at under a powerful microscope—and what you see is that it’s actually not a single grain, but 10 billion microscopic grains wrapped in a membrane, all of which together is the size of a normal grain of sand. If that were the case for every single grain of sand in this hypothetical—if each were actually a bundle of 10 billion tinier grains—the total number of those microscopic grains would be a googol.

We’re running out of room here on both the small and big end of things to fit these numbers into the physical world, but three more for you:

10113 – The number of hydrogen atoms it would take to pack the universe full of them.

10122 – The number of protons you could fit in the universe.

10185 – Back to the Planck volume (the smallest volume I’ve ever heard discussed in science). How many of these smallest things could you fit in the very biggest thing, the observable universe? 10185. Without being able to go smaller or bigger on either end, we’ve reached the largest number where the physical world can be used to visualize it.

A Googolplex – 10googol

After popularizing the newly-named googol, Krasner could barely keep his pants on with this adorable new schtick and asked his nephew to coin another term. He could barely finish the question before Milton opened his un-nuanced mouth and declared the number googolplex, which he, in typical Milton form, described as “one, followed by writing zeroes until you get tired.”6 At this, Krasner showed some uncharacteristic restraint, ignoring Milton and giving the number an actual definition: 10googol or 1 with a googol zeros written after it. With its full written-out exponent, a googolplex looks like this:

1010,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

So a googol is 1 with just 100 zeros after it, which is a number 10 billion times bigger than the grains of sand that would fill the universe. Can you possibly imagine what kind of number is produced when you put a googol zeros after the 1?

There’s no possible way to wrap your head around that number—the best we can do is try to understand how long it would take to write the number. What I wrote above is just the exponent—actually writing a googolplex out involves writing a googol zeros. First, let’s figure out where we’d write these zeros.

As we’ve discussed, filling the universe with sand only gets you a ten billionth of the way to a googol, so what we’d have to do is fill the universe to the brim with sand, get a very tiny pen, and write 10 billion zeros on each grain of sand. If you did this and then looked at a completed grain under a microscope, you’d see it covered with 10 billion microscopic zeros. If you did that on every single grain of sand filling the universe, you’d have successfully written down the number googolplex.

And just how long would it take to do that?

Well I just tested how fast a human can reasonably write zeros, and I wrote 36 zeros in 10 seconds.7 At that rate, if from the age of 5 to the age of 85, all I did for 16 hours a day, every single day, was write zeros at that rate, I’d finish one half of a grain of sand in my lifetime. You’d need to dedicate two full human lives to finish one grain of sand. About 107 billion human beings have ever lived in the history of the species. If every single human dedicated every waking moment of their lives to writing zeros on grains of sand, as a species we’d have by now filled a cube with a side of 1.7m—about the height of a human—with completed sand grains. That’s it.

Now to get a glimpse at how big the actual number is—as the Numberphilers explain, the total possible quantum states that could occur in the space occupied by a human (i.e. every possible arrangement of atoms that could happen in that space) is far less than a googolplex. What this means is that if there were a universe with a volume of a googolplex cubic meters (an extraordinarily large space), random probability suggests that there would be exact copies of you in that universe. Why? Because every possible arrangement of matter in a human-sized space would likely occur many, many times in a space that vast, meaning everything that could possibly exist would exist—including you. Including you with cat whiskers but normal otherwise. Including you but a one-foot tall version. Including you exactly how you are except instead of a pinky finger on your left hand you have Napoleon’s penis there as your fifth finger. What I’m saying isn’t science fiction—it’s the reality of a space that large.

Graham’s Number

You know how sometimes you go through life, and you’re lost but you don’t even know it, and then one day, the right person comes along and you realize what you had been looking for this whole time?

That’s how I feel about Graham’s number.

Huge numbers have always both tantalized me and given me nightmares, and until I learned about Graham’s number, I thought the biggest numbers a human could ever conceive of were things like “A googolplex to the googolplexth power,” which would blow my mind when I thought about it. But when I learned about Graham’s number, I realized that not only had I not scratched the surface of a truly huge number, I had been incapable of doing so—I didn’t have the tools. And now that I’ve gained those tools (and you will too today), a googolplex to the googolplexth power sounds like a kid saying “100 plus 100!” when asked to say the biggest number he could think of.

Before we dive in, why is Graham’s number even a number people talk about?

I’m not gonna really explain this because the explanation is really boring and confusing—here’s the official problem Ronald Graham (a living American mathematician) was working on when he came up with it:

Connect each pair of geometric vertices of an n-dimensional hypercube to obtain a complete graph on 2n vertices. Color each of the edges of this graph either red or blue. What is the smallest value of n for which every such coloring contains at least one single-colored complete subgraph on four coplanar vertices?

I told you it was boring and confusing. Anyway, there’s no single answer to the problem, but Graham’s proof includes a lower and upper bound, and Graham’s number was one version of an upper bound for that Graham came up with.

He came up with the number in 1977, and it gained recognition when a colleague wrote about it in Scientific American and called it “a bound so vast that it holds the record for the largest number ever used in a serious mathematical proof.” The number ended up in the Guinness Book of World Records in 1980 for the same reason, and though it has today been surpassed, it’s still renowned for being the biggest number most people ever hear about. That’s why Graham’s number is a thing—it’s not just an arbitrarily huge number, it’s actually relevant in the world of math.

So anyway, I said above that I had been limited in the kind of number I could even imagine because I lacked the tools—so what are the tools we need to do this?

It’s actually one key tool: the hyperoperation sequence.

The hyperoperation sequence is a series of mathematical operations (e.g. addition, multiplication, etc.), where each operation in the sequence is an iteration up from the previous operation. You’ll understand in a second. Let’s start with the first and simplest operation: counting.

Operation Level 0 – Counting 

If I have 3 and I want to go up from there, I go 3, 4, 5, 6, 7, and so on until I get where I want to be. Not a high-powered operation.

Operation Level 1 – Addition 

Addition is an iteration up from counting, which we can call “iterated counting”—so instead of doing 3, 4, 5, 6, 7, I can just say 3 + 4 and skip straight to 7. Addition being “iterated counting” means that addition is like a counting shortcut—a way to bundle all the counting steps into one, more concise step.

Operation Level 2 – Multiplication

One level up, multiplication is iterated addition—an addition shortcut. Instead of saying 3 + 3 + 3 + 3, multiplication allows us to bundle all of those addition steps into one higher-operation step and say 3 x 4. Multiplication is a more powerful operation than addition and you can create way bigger numbers with it. If I add two eight-digit numbers together, I’ll end up with either an eight or nine-digit number. But if I multiply two eight-digit numbers together, I end up with either a 15 or 16-digit number—much bigger.

Operation Level 3 – Exponentiation (↑)

Moving up another level, exponentiation is iterated multiplication. Instead of saying 3 x 3 x 3 x 3, exponentiation allows me to bundle that string into the more concise 34.

Now, the thing is, this is where most people stop. In the real world, exponentiation is the highest operation we tend to ever use in the hyperoperation sequence. And when I was envisioning my huge googolplexgoogolplex number, I was doing the very best I could using the highest level I knew—exponentiation. On Level 3, the way to go as huge as possible is to make the base number massive and the exponent number massive. Once I had done that, I had maxed out.

The key to breaking through the ceiling to the really big numbers is understanding that you can go up more levels of operations—you can keep iterating up infinitely. That’s the way numbers get truly huge.

And to do this, we need a different kind of notation. So far, we’ve worked with a different symbol on each level (+, x, and a superscript)—but we don’t want to have to remember a ton of different symbols if we’re gonna be working with a bunch of different operations levels. So we’ll use Knuth’s up-arrow notation, which is one symbol that can be used on any level.

Knuth’s up-arrow notation starts on Operation Level 3, replacing exponentiation with a single up arrow: ↑. So to use up-arrow notation, instead of saying 34, we say 3 ↑ 4, but they mean the same thing.

3 ↑ 4 = 81
2 ↑ 3 = 8
5 ↑ 5 = 3,125
1 ↑ 38 = 1

Got it? Good.

Now let’s move up a level and start seeing the insane power of the hyperoperation sequence:

Operation Level 4 – Tetration (↑↑)

Tetration is iterated exponentiation. Before we can understand how to bundle a string of exponentiation the way exponentiation bundles a string of multiplication, we need to understand what a “string of exponentiation” even is.

So far, all we’ve done with exponentiation is one computation—a base number and a power it’s raised to. But what if we put two of these computations together, like:

222

We get a power tower. Power towers are incredibly powerful, because they start at the top and work their way down. So 222 = 2(22) = 24 = 16. Nothing that impressive yet, but check out:

3333

Using parentheses to emphasize the top down order: 3333 =  33(33) = 3327 =3(327) = 37,625,597,484,987 = a 3.6 trillion-digit number

Remember, a googol and its universe-filling microscopic mini-sand is only a 100-digit number. So all it takes is a power tower of 3s stacked 4 high to dwarf a googol, as well as 10185, the number of Planck volumes to fill the universe and our physical world maximum. It’s not as big as a googolplex, but we can take care of that easily by just adding one more 3 to the stack:

33333 = 3(3333) = 3(3.6 trillion-digit number) = way bigger than a googolplex, which is 10(100-digit number). As for a googolplex itself, power towers allow us to immediately humiliate it by writing it as:

1010100 or, more typically, 1010102. So you can imagine what kind of number you get when you start making tall power towers. Tetration is intense.

Now those towers are Level 3, exponential strings, the same way 3 x 3 x 3 x 3 is a Level 2, multiplication string. We use Level 3 to bundle that Level 2 string into 34, or 3 ↑ 4. So how do we use Level 4 to bundle an exponential string? Double arrows.

3333 is the same as saying 3 ↑ (3 ↑ (3 ↑ 3)). We bundle those 4 one-arrow 3s into 3 ↑↑ 4.

Likewise, 3 ↑↑ 5 = 3 ↑ (3 ↑ (3 ↑ (3 ↑ 3))) = 33333

4 ↑↑ 7 = 4 ↑ (4 ↑ (4 ↑ (4 ↑ (4 ↑ (4 ↑ 4))))) = a power tower of 4s 7 high.

Here’s the general rule:

tetration generally

We’re about to move up another level, and this is about to become more complex, so before we move on, make sure you really understand Level 4 and what ↑↑ means—just remember that a ↑↑ b is a power tower of a’s, b high.

Operation Level 5 – Pentation (↑↑↑)

Pentation, or iterated tetration, bundles double arrow strings together into a single operation.

The pattern we’ve seen is each new level bundles a string of the previous level together by using a term as the length of the string. For example:

string bundle examples

In each case, is the base number and is the length of the string being bundled.

So what does pentation bundle together? How can you have a string of power towers?

The answer is what I call a “power tower feeding frenzy”. Here’s how it works:

You have a string of power towers standing next to each other, in a particular order, all using the same base number. The thing that differs between them is the height of each tower. The first tower’s height is the same number as the base number. You process that tower down to its full expanded outcome, and that outcome becomes the height of the next towerYou then process that tower, and the outcome becomes the height of the next tower. And so on. Each tower’s outcome “feeds” into the next tower and becomes its height—hence the feeding frenzy. Here’s why this happens:

3 ↑↑↑ 4 means a string of (3 ↑↑ 3) operations, 4 long. So:

3 ↑↑↑ 4 = 3 ↑↑ (3 ↑↑ (3 ↑↑ 3))

Remember, when you see ↑↑ it means a single power tower that’s high, so:

3 ↑↑↑ 4 = 3 ↑↑ (3 ↑↑ (3 ↑↑ 3)) = 3 ↑↑ (3 ↑↑ 333)

Now, you might remember from before that 333 = 327 = 7,625,597,484,987. So:

3 ↑↑↑ 4 = 3 ↑↑ (3 ↑↑ (3 ↑↑ 3)) = 3 ↑↑ (3 ↑↑ 333) = 3 ↑↑ (3 ↑↑ 7,625,597,484,987)

So the first tower of height 3 processed down into 7 trillion-ish. Now the next parentheses we’re dealing with is (3 ↑↑ 7,625,597,484,987), where the outcome of the first tower is the height of this second tower. And how high would that tower of 7 trillion-ish 3s be? 

Well if each 3 is two centimeters high, which is about how big my written 3’s are, the tower would rise about 150 million kilometers high, which would touch the sun. Even if we used tiny, typed 2mm 3’s, our tower would reach the moon and back to the Earth and back to the moon forty times before finishing. If we wrote those tiny 3’s on the ground instead, the tower would wrap around the earth 400 times. Let’s call this tower the “sun tower,” because it stretches all the way to the sun. So what we have is:

3 ↑↑↑ 4 = 3 ↑↑ (3 ↑↑ (3 ↑↑ 3)) = 3 ↑↑ (3 ↑↑ 333) = 3 ↑↑ (3 ↑↑ 7,625,597,484,987) = 3 ↑↑ (sun tower)

This final 3 ↑↑ (sun tower) operation is a power tower of 3’s whose height is the number you get when you multiply out the entire sun tower (and this final tower we’re building won’t even come close to fitting in the observable universe). And we don’t get to our final value of 3 ↑↑↑ 4 until we multiply out this final tower.

So using ↑↑↑, or pentation, creates a power tower feeding frenzy, where as you go, each tower’s height begins to become incomprehensible, let alone the actual final value. Written generally:

pentation generally

We’re gonna go up one more level—

Operation Level 6 – Hexation (↑↑↑)

So on Level 4, we’re dealing with a string of Level 3 exponents—a power tower. On Level 5, we’re dealing with a string of Level 4 power towers—a power tower feeding frenzy. On Level 6, aka hexation or iterated pentation, we’re dealing with a string of power tower feeding frenzies—what we’ll call a “power tower feeding frenzy psycho festival.” Here’s the basic idea:

A power tower feeding frenzy happens. The final number the frenzy produces becomes the number of towers in the next feeding frenzy. Then that frenzy happens and produces an even more ridiculous number, which then becomes the number of towers for the next frenzy. And so on.

3 ↑↑↑↑ 4 is a power tower feeding frenzy psycho festival, during which there are 3 total ↑↑↑ feeding frenzies, each one dictating the number of towers in the next one. So:

3 ↑↑↑↑ 4 = 3 ↑↑↑ (3 ↑↑↑ (3 ↑↑↑ 3))

Now remember from before that 3 ↑↑↑ 3 is what turns into the sun tower. So:

3 ↑↑↑↑ 4 = 3 ↑↑↑ (3 ↑↑↑ (3 ↑↑↑ 3)) = 3 ↑↑↑ (3 ↑↑↑ (sun tower))

Since ↑↑↑ means a power tower feeding frenzy, what we have here with 3 ↑↑↑ (sun tower) is a feeding frenzy with a multiplied-out-sun-tower number of towers. When that feeding finally finishes, the outcome becomes the number of towers in the final feeding frenzy. The psycho festival ends when that final feeding frenzy produces it’s final number. Here’s hexation explained generally: 

hexation generally

And that’s how the hyperoperation sequence works. You can keep increasing the arrows, and each arrow you add dramatically explodes the scope you’re dealing with. So far, we’ve gone through the first seven operations in the sequence, including the first four arrow levels:

↑ = power
↑↑ = power tower
↑↑↑ = power tower feeding frenzy
↑↑↑↑ = power tower feeding frenzy psycho festival

So now that we have the toolkit, let’s go through Graham’s number:

Graham’s number is going to be equal to a term called g64. We’ll get there. First, we need to start back with a number called g1, and then we’ll work our way up. So what’s g1?

g1 = 3 ↑↑↑↑ 3

Hexation. You get it. Kind of. So let’s go through it.

Since there are four arrows, it looks like we have a power tower feeding frenzy psycho festival on our hands. Here’s how it looks visually:

grahams festival

So g1 = 3 ↑↑↑↑ 3 = 3 ↑↑↑ (3 ↑↑↑ 3), and we have two feeding frenzies to worry about. Let’s deal with the first one (in red) first:

g1 = 3 ↑↑↑↑ 3 = 3 ↑↑↑ (3 ↑↑↑ 3) = 3 ↑↑↑ (3 ↑↑ (3 ↑↑ 3))

So this first feeding frenzy has two ↑↑ power towers. The first tower (in blue) is a straightforward little one because the value of b is only 3:

g1 = 3 ↑↑↑↑ 3 = 3 ↑↑↑ (3 ↑↑↑ 3) = 3 ↑↑↑ (3 ↑↑ (3 ↑↑ 3)= 3 ↑↑↑ (3 ↑↑ 333)

And we’ve learned that 33= 7,625,597,484,987, so:

g1 = 3 ↑↑↑↑ 3 = 3 ↑↑↑ (3 ↑↑↑ 3) = 3 ↑↑↑ (3 ↑↑ (3 ↑↑ 3)= 3 ↑↑↑ (3 ↑↑ 333= 3 ↑↑↑ (3 ↑↑ 7,625,597,484,987

And we know that (3 ↑↑ 7,625,597,484,987) is our 150km-high sun tower:

g1 = 3 ↑↑↑↑ 3 = 3 ↑↑↑ (3 ↑↑↑ 3) = 3 ↑↑↑ (3 ↑↑ (3 ↑↑ 3)= 3 ↑↑↑ (3 ↑↑ 333= 3 ↑↑↑ (3 ↑↑ 7,625,597,484,987= 3 ↑↑↑ (sun tower) 

To clean it up:

g1 = 3 ↑↑↑↑ 3 = 3 ↑↑↑ (3 ↑↑↑ 3) = 3 ↑↑↑ (sun tower) 

So the first of our two feeding frenzies has left us with an epically tall sun tower of 3’s to multiply down. Remember how earlier we showed how quickly a power tower escalated:

3 = 3
33 = 27
333 = 7,625,597,484,987
3333 = a 3.6 trillion-digit number, way bigger than a googol, that would wrap around the Earth a couple hundred times if you wrote it out
33333 = a number with a 3.6 trillion-digit exponent, way way bigger than a googolplex and a number you couldn’t come close to writing in the observable universe, let alone multiplying out

Pretty insane growth, right?

And that’s only the top few centimeters of the sun tower.

sun tower

Once we get a meter down, the number is truly far, far, far bigger than we could ever fathom. And that’s a meter down.

The tower goes down 150 million kilometers.

Let’s call the final outcome of this multiplied-out sun tower INSANITY in all caps. We can’t comprehend even a few centimeters multiplied out, so 150 million km is gonna be called INSANITY and we’ll just live with it.

So back to where we were:

g1 = 3 ↑↑↑↑ 3 = 3 ↑↑↑ (3 ↑↑↑ 3) = 3 ↑↑↑ (sun tower) 

And now we can replace the sun tower with the final number that it produces:

g1 = 3 ↑↑↑↑ 3 = 3 ↑↑↑ (3 ↑↑↑ 3) = 3 ↑↑↑ (sun tower) = 3 ↑↑↑ INSANITY

Alright, we’re ready for the second of our two feeding frenzies. And here’s the thing about this second feeding frenzy—

So you know how upset I just got about this whole INSANITY thing?

That was the outcome of a feeding frenzy with only two towers. The first little one multiplied out and fed into the second one and the outcome was INSANITY.

Now for this second feeding frenzy…

There are an INSANITY number of towers.

We’ll move on in a minute, and I’ll stop doing these dramatic one sentence paragraphs, I promise—but just absorb that for a second. INSANITY was so big there was no way to talk about it. Planck volumes in the universe is a joke. A googolplex is laughable. It’s too big to be part of my life. And that’s the number of towers in the second feeding frenzy.

insanity

So we have an INSANITY number of towers, each one being multiplied allllllllll the way down to determine the height of the next one, until somehow, somewhere, at some point in a future universe, we multiply our final tower of this second feeding frenzy out…and that number—let’s call it NO I CAN’T EVEN—is the final outcome of the 3 ↑↑↑↑ 3 power tower feeding frenzy psycho festival.

That number—NO I CAN’T EVEN—is g1.

Now…

I want you to look at me, and I want you to listen to me.

We’re about to enter a whole new realm of craziness, and I’m gonna say some shit that’s not okay. Are you ready?

So g1 is 3 ↑↑↑↑ 3, aka NO I CAN’T EVEN.

The next step is we need to get to g2. Here’s how we get there:

g2

Look closely at that drawing until you realize how not okay it is. Then let’s continue.

So yeah. We spent all day clawing our way up from one arrow to four, coping with the hardships each new operation level presented us with, absorbing the outrageous effect of adding each new arrow in. We went slowly and steadily and we ended up at NO I CAN’T EVEN.

Then Graham decides that for g2, he’ll just do the same thing as he did in g1, except instead of four arrows, there would be NO I CAN’T EVEN arrows. 

Arrows. The entire g1 now feeds into g2 as its number of arrows.

Just going to a fifth arrow would have made my head explode, but the number of arrows in g2 isn’t five—it’s far, far more than the number of Planck volumes that could fit in the universe, far, far more than a googolplex, and far, far more than INSANITY. And that’s the number of arrows. That’s the level of operation g2 uses. Graham’s number iterates on the concept of iterations. It bundles the hyperoperation sequence itself.

Of course, we won’t even pretend to do anything with that information other than laugh at it, stare at it, and be aroused by it. There’s nothing we could possibly say about g2, so we won’t.

And how about g3?

You guessed it—once the laughable gis all multiplied out, that becomes the number of arrows in g3.

And then this happens again for g4. And again for g5. And again and again and again, all the way up to g64.

g64 is Graham’s number.

All together, it looks like this:

grahams number

So there you go. A new thing to have nightmares about.

________

P.S. Writing this post made me much less likely to pick “infinity” as my answer to this week’s dinner table question. Imagine living a Graham’s number amount of years.8 Even if hypothetically, conditions stayed the same in the universe, in the solar system, and on Earth forever, there is no way the human brain is built to withstand spans of time like that. I’m horrified thinking about it. I think it would be the gravest of grave errors to punch infinity into the calculator—and this is from someone who’s openly terrified of death. Weirdly, thinking about Graham’s number has actually made me feel a little bit calmer about death, because it’s a reminder that I don’t actually want to live forever—I do want to die at some point, because remaining conscious for eternity is even scarier. Yes, death comes way, way too quickly, but the thought “I do want to die at some point” is a very novel concept to me and actually makes me more relaxed than usual about our mortality.

P.P.S If you must, another Wait But Why post on large numbers.

If you liked this, you’ll probably also like:

Fitting 7.3 billion people into one building

What makes you you?

What could you buy with $241 trillion?


  1. I’m using the American short scale system—in the British long scale system, you don’t get to a billion until 1012.

  2. Upsetting.

  3. Luckily, I’m not cool.

  4. I’m going to use the term “universe” to refer to the observable universe so I don’t have to type observable 49 times in this post.

  5. 59 years later, Sergey Brin and Larry Page named their new search engine after this number because they wanted to emphasize the large quantities of information the engine could provide. They spelled it wrong by accident.

  6. Fucking Milton.

  7. When my father was my age, he had children.

  8. Or a g65 number of years, which would be (3 [Graham’s number of arrows] 3)…or a gg64 number of years…I could go on.

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  • Andrew James Stevens

    Maths – removing deaths’ sting.

  • slovoflud

    I’ve watched and read a lot about Graham’s number and I think I kind of get it (not comprehend it), but I’m always stumped by “what about Graham’s number + 1” friend comment. I just loose it and go watch Numberfile on youtube again…

    • Andrew James Stevens

      Graham’s number is still finite despite being incomprehensibly big so you can arbitrarily create a larger number by adding one to it.

      • Tim Urban

        You can always make any number bigger. With Graham’s number, you could just square it or do Graham’s number to its own power, or like I mentioned in a footnote, you could make the number of Graham’s number “layers” Graham’s number instead of 64. The reason Graham’s number in particular is worth discussing more than any of those fabricated larger numbers is that it actually has some relevance in the world of math. So it’s not that it’s the biggest possible number—it’s just the biggest possible relevant one (or at least it was in 1980).

        • Andrew James Stevens

          Yes, got that. You got alot further than me with the visualisation of it, although it always breaks down quickly with our current perception of reality. Disturbingly, even g(Grahams number) tends to zero when compared to the mathematical concept of infinity (not a number, but can be expressed as x/0), which parallels eternity, which in turn makes death look like a pretty cushty option. A perspective that I didn’t expect Grahams number to give me, so thank you.

        • Truliner

          That’s actually what I was hoping this article to elaborate more, that how the Graham’s number is relevant in math. But I know other websites too!

          • yeah, graham’s problem is in fact not something insanely complicated like so many people think! it’s actually quite simple, and there are several pages on the internet that do a good job of explaining it.

  • .. (I don’t know why it’s here twice)…

  • KIC

    One day I’m going to be a googolionaire

  • Phil Orme

    Wow i will re read this when my head stops hurting.

  • Michael

    Another thought I just had Re: living to infinity, and somewhat relevant to this post is: if you live forever, EVERY possible thing that can happen within the universe WILL happen to you will happen an infinite number of times. You will have years, decades, centuries that will be identical in every way to years, decades, and centuries that you’ve had before. You will get bored of literally every quantum state, and will eventually be begging for death.

    • Mattchenzo

      Michael, I am certainly no expert but to the best of my knowledge that isn’t the way it works. There are infinite numbers (decimals) between 1 and 2, but none of them are 3. Living forever would not guarantee that every possible thing happens to you an infinite number of times, or even once. I’m on my phone or I would get you a reference, but the reasoning is the same reason pi can have infinite digits, be none repeating, but still not contain every finite combination of numbers.

      • Michael

        I totally understand what you’re talking about. I think my main point still stands, however, in that you will eventually hit a point where every single life event is a duplicate of one that came before it (though some theoretically possible things may never happen). While 1 and 2 contain an infinite number of numbers, all different, I don’t think the same is true of quantum states. Whether or not everything theoretically possible will eventually happen has something to do, I think, with the inherent randomness of the universe which is a physics can of worms I am not qualified to open.

        My basic point though was that there will eventually be no more novelty, except perhaps in the exact sequence in which things happen. But the length of sequences that are copies of previous sequences will probably get longer and longer as you live longer expanses of life i.e. first you will have a day that is an exact copy of a day you’ve had before, and eventually you will have entire billion-year spans that are identical to ones you’ve experienced before. I have no mathematical proof to back this up with, just intuition, so I’m hoping someone can bring in some math to either back me up or prove me wrong.

        • Mattchenzo

          I certainly agree with your point. Eventually, there is no more novelty no matter what you did. There are several great sci-fi short stories that deal with that concept. Good point!

    • (!) BEAUDISM

      ‘..you will get bored….’ (?) huh….You can NEVER experience the same ‘sequence’ as you bring to it the consciousness gleaned from the previous one…and therefore the perception will be different……..any repeated action is experienced DIFFERENTLY because consciousness has been (however slightly) altered…….so don’t panic about ‘living’ too long……or(?)

  • Bill Warren

    Wow.

  • 1 little 2 little 3 little Graham, 4 little 5 little 6 little Graham, 7 little 8 little 9 little Graham, 10 little Graham’s Number…said nobody ever x(

  • Unqlefungus

    Math classes have made many a student more comfortable with the concept of death. Depending on the instructor, one can actually long for it.

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  • Utens

    I walked out of the room in denial when I looked at the passage from g1 to g2

    • Tim Urban

      It’s not okay.

  • Michael

    “I do want to die at some point”

    Note: no post next week

    Tim, are you okay?

  • Anonymous

    Excellent post, however, I am here representing the spelling police to report on three possible mispellings of “googol” as “google.” The first instance is in superscript and is located in the paragraph immediatley proceeding the heading “A Googolplex – 10googol.” The second and third instances are located in the second paragraph proceeding the heading “Operation Level 3 – Exponentiation (↑)”; one is in superscript and one is in normal text. If you meant to do this then, by all means, disregard me completely.

    • Tim Urban

      Incredible how hard it is to type googol and not revert subconsciously to google. Thanks and fixed.

  • samuel

    Amazing, thanks for this post!

  • jasvisp

    I can’t handle this…..numbers are like a foreign language to me! When I scrolled down to see how long this post was my eyes landed on the word ‘Insanity’…… think I’ll stop there.

  • Guest

    What about different sizes of infinities?

  • wobster109

    Re: footnote 7 – When my parents were my age, they packed up a suitcase and started over in a new country where they barely spoke the language. And here I am surfing the internet at my desk job.

  • SJ P

    For others like me who can’t possibly comprehend the silliness of typed numbers like that – this is basically the gist of this post illustrated visually, to scale, from a Planck Length to the distance to the Hubble Deep Field. Also, the existence of a Japanese Spider Crab is pretty much reason #1 why I wouldn’t ever possibly pick infinity as a life span.

    http://htwins.net/scale2/

  • M1zzu

    Let’s just think about a few additional crazy things:
    1. There are debates in science whether the universe (the whole thing) might actually be or will become infinitely big.
    2. 1 and Graham’s Number and any other number are equally insignificant to infinity.

    • Truliner

      Those are truly crazy… One level of craziness after Graham’s number. I think just now my concept of infinity evolved a little when I was thinking about an universe that is g64 light-years wide… And still it’s “lot” less than infinity (like you pointed out). Mind = blown.

  • Vinay Kapadia

    Love big numbers, love this post! I’ve heard of Graham’s Number before, but only explained as a ridiculously large number. This provides a really good explanation to wrap my brain around it.

  • chingareke kuuraya

    This English major bowed out at googleplex 🙁

    • Ditto for this psych major turned software engineer, now retired!

    • Krattz

      I *just* made it (partly because i saw Numberphile’s explanation ages ago) I feel ready for university now…

  • Tom Miller

    To anyone interested about Huge numbers and infinity, I’d recommend this BBC documentary that will unravel your mind! “BBC Horizon – To Infinity and Beyond” http://www.dailymotion.com/video/x1kab09_bbc-horizon-2010-to-infinity-and-beyond-pdtv-xvid_tech

  • Christine

    Wow! I’ve read it twice now and am amazed at how well you are able to make comprehensible just how incomprehensible the incomprehensible is. And I’m also feeling a little regret to have spent the past fifteen years actively avoiding anything mathy. Numbers are… so cool…

  • Karson

    I was in almost in hysterics of laughter after finishing this post. Do you know why he stopped at g64? Not that we need more, Jesus. I’m not sure I even want to know the answer as I am now afraid of math in a very real way.

  • Dijana

    This post is the FIRST time in my Whole Entire Life that I felt grateful for that stalwart mathematical practice of ignoring a variable because it’s considered irrelevant. I don’t pretend to know why you enjoy this sort of thing, I freely confess it’s completely beyond me, but why worry that if it takes several Graham’s worth of ants to make a string from here to the edge of the Universe? Surely, nobody is every going to measure warp speeds according to how many sour patch kids takes from here to z8 GND 5296.
    I just don’t know why you have to come over all faint over something that essentially just means ‘Lots’.
    *shakes head in bewilderment*

  • Jon Lizarraga Diaz

    I like how every post usually takes you to the limit of a certain matter, and then a step further.Maybe two steps further. This post took me FAR beyond what my brain is able to process, so as I was being taken way to many steps further than I feel comfortable with, my brain just shut down and I couldn’t do anything but laugh.

    I don’t think I can go back to work now. everything seems so futile.

    • Michael

      Yeah, at pentation I just skimmed to the end to see where the crazy would end, without trying to follow it. Whoosh.

  • You are much, much smarter than me. My brain is aching, and I couldn’t even finish this post. But thank you for being there for those who can understand it all!

  • DeeDee Massey

    When dealing with math, the word INSANITY is appropriate. I look forward to numbers post #3 and the realm of LUDICROUS numbers. I wish my college math professors had been a fraction of this brilliant at presenting concepts. Tim, just don’t start striving to predict your own death date or writing manifestos. Hey, wait….. 🙂

    http://www.lomont.org/Math/Talks/Mathematics%20and%20Insanity.pdf

  • Regine

    This is giving me palpitations.

  • Thomas

    Really good work as always:) I wonder what was the amount of work to prepare and write this. Was that only a week or did you have a part of it ready?
    Tiny mistake at the definition of Graham’s problem, you obtain a complete graph on 2^n and not 2n vertices.
    Keep up the good work.

  • Great Pierre

    My mind was blown, setting flames to the rest of my skull, and sending splatters of my squishy brain all over the walls.

  • DK

    SCREW YOU TIM
    I HAD AN EXAM THIS MORNING AND I WANTED TO READ YOUR POST TO CALM ME DOWN AND GET MY MIND OFF IT, NOT GIVE ME AN EXISTENTIAL MELTDOWN

  • Neal Smith

    So in my state, teachers now have to have us do a “getting to know you” sheet, where you answer questions about you so teachers can document that you know them. So, being who I am, I put really sarcastic answers. One of the questions was “How many M&Ms can you fit in your mouth?” So I put “Graham’s number m&ms.” But here’s the kicker. To pose something so mind-numbingly large that your brain instantly creates Francium and sticks it in water, couldn’t you go with the lowercase g syntax to, say, g65? or g66? those are values, no matter how useless they are. So the other number question was “how many pieces of pizza have you eaten at once?” So I wrote down all the graham’s number definitions, and then I said “g(googolplex)” (note: I’m using parentheses because I have no easy access to subscript). That means (I did no math of this, but I’m sure) if you had the 64th g number for every planck volume in the observable universe, it’s not even close. In fact, there is insufficient data for a meaningful answer (bonus points).

    • Krattz

      I had to do those too but we had to share it with the class as well… the only information that would have been of any use to my teachers was none of my classmate’s business… I just left them blank but your idea is way better

    • Rodrigo Gomes

      sponsored post -> Graham’s number m&ms in one’s mouth would be a delicious thing to do.

  • Soundarya

    Thank you for blowing my mind

  • Ron Burgundy

    You can throw your big numbers around all you like but I’m still kind of a big deal.

  • marisheba

    I definitely had the same reaction to the infinity possibility in the dinner party conversation: “Oh, so I actually DO want to die someday. That’s…comforting.”

  • Kate

    I stopped when i got to Graham’s number and felt a headache coming on. I can’t decide if you’re a genius or a really (to the power of googol) weird person!!

  • Bogdan Voicu

    Cool! I’ve enjoyed reading the whole post and it was indeed mindblowing. Just a sidenote: numbers are fun to play with, interesting to observe, but one should keep in mind that numbers are just inside our brain, not real. They help us grasp the reality (or whatever we think is real) but at some point they diverge from our own reality. What I mean is that understanding the reality with the help of numbers is quite straightforward, while the reverse (what Tim in the last two posts has tried to do) can prove tricky. And again mindblowing.

  • Mihai

    Another pretty amazing fact. The explosion of a supernova is just about as energetic as a few octillion nuclear warheads detonating all at once.

  • Felipe Lisbôa

    Interesting article, but I still don’t speak numbers… (・・。)ゞ

  • Rodrigo Gomes

    This time you really entered a Psycho Festival. I love math, but could not keep up from the moment when hexation entered the stage.

    Note: no post next week because I need to recover in a mental health clinic. *FTFW*

  • Joey

    I am now sure Tim Urban is INSANE to be able to think about this things! Truly mind blowing! Thanks for sharing anyway. I like how you connected this insanity numbers with how insane it is to live forever

    • Roberto Lorenzo

      These*

  • Gabriel Santos

    It’s official, Math is scarier than Death !

  • jamaicanworm

    Among the powers of 10 under a googol, 10^67 holds a special place in my heart. Why? The number of ways you can arrange a standard deck of 52 cards is 52!, or about 8 x 10^67.

    So in your example of playing cards covering the earth, imagine each card actually being a differently-arranged deck of cards. Then we’d need something like 10^50 earths to fit them all.

    What does this mean? Whenever you’re shuffling a deck of cards, the chances that this precise shuffle (or any shuffles along the way) has ever existed in human history is almost too small to comprehend.

    So remember, your shuffles are one-of-a-kind. Just like you.

  • wobster109

    Tim, a word of advice. You’re excellent at giving names and personalities to abstract concepts. It’s an excellent mental shorthand that helps us understand. However I found “power tower feeding frenzy psycho festival” very hard to think about. It’s too many syllables, hard to say, sounds too similar to the just-plain “power tower feeding frenzy”, and that made it very hard to think about. I had to try very hard to keep those concepts clear in my mind. “Sun tower” was excellent: short and with vivid imagery. I would’ve stuck with “frenzy” (or “feeding frenzy”) and perhaps “psycho party”.

  • meregoround

    When I realised that Maths was a form of philosophy, the world made more sense. It also made this post easier to swallow. Thanks for distracting me from my real work for a bit, also fun reading your posts!

  • Jon Tretten

    So, I watched this video: https://www.youtube.com/watch?v=HX8bihEe3nA. I’m not sure how to explain this, so I would recommend you watch it yourself, if you haven’t so already.

    My thought was: Graham’s number (GN) is the number of dimensions where the configuration you want to avoid HAS to happen. But then, a cube in not 2, 3 or 4 dimensions (the latter which has 2^120 different pairings of “points”), but GN dimensions – how many different pairings of points can you have?

    I have a feeling that number just dwarfs GN… I think I’m done for today.

  • V

    curious fact, last digit of Graham’s number (in base ten) is 7

  • Thomas Peyrin

    Actually, Graham’s sequence is not the fastest growing one. Nothing can beat busy beavers 🙂 http://www.scottaaronson.com/writings/bignumbers.html

  • Eli Peter

    I never thought I’d be so unsettled by a number.

    I just bought a 2TB hard drive which feels like oceans of space, yet the biggest number I could store would be 2^16,000,000,000,000, or <2^^4 if I understand tetration right.

    That's laughably small on the graham scale. Even if you mined all of the matter in the universe for HD-making material you still couldn't even write down anything bigger than g1.

    I need to lay down.

    • PleaseMathsStopYou’reHurtingMe

      I believe that you couldn’t even come close to the googolplex, so g1 seems really far beyond reach.

  • math but not english

    no need for almost all of the apostrophes in this article.

    • math but not english

      by which i mean a select few, of course.

    • Tim Urban

      I know. But writing “3s” is just kind of upsetting?

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  • Roberto Lorenzo

    ¨Weirdly, thinking about Graham’s number has actually made me feel a little bit calmer about death, because it’s a reminder that I don’t actually want to live forever—I do want to die at some point, because remaining conscious for eternity is even scarier¨

    I like this quote very much. it is very appealing to my way of thinking, it made me calmer as well. Also I invite those loving christians to read this article so they get a glimpse of what eternity means, which hopefully makes them realize how absurd the eternal life they hope for or eternal suffering they wish upon non believers is…

  • Peter Piper

    I had a thought sequence that went like this:

    1. “Big numbers are big!”
    2. “Yea, big numbers are big, but the difference between big numbers is also big. Millionaires and Billionaires are two TOTALLY different classes of people and levels of wealth. I think most people lump them all together – even in the same breath of our national dept in the trillions”.
    3. “I want $1Googol
    4. “WTF do we worry about counting our petty dollars. My head hurts.”

    Fun times, cannot wait to forget I read this article and go back to thinking I am good with numbers.

  • mofo harry

    Excellent post Tim. In the domain of geeks and nerds who love the numbers but are normally incapable of human emotions and humour, you make the subject both interesting and enjoyable to read.
    You deserve a Honorary Doctorate from an Ivy League school.

  • Frank

    Wow, sorry I had to give up on understanding Graham’s Number… my mind’s definitely too small to hold it.

  • Brian

    Why did he stop at G64?

    • it was used as a serious upper-bound in a mathematical proof.

  • Phill

    Eternal Life? – if there’s even a hint of a chance that we live on for infinity after death – I’d at least want to be on God’s side – Couldn’t imagine the hell of being against him for that long! I’m checking out this Jesus Bible thing!

  • Wow, great blog post! I have a lot of experience with very large numbers, including those that make Graham’s number look tiny, and I have to say this is quite an impressive coverage of really big numbers. The best part is that you specifically clear up that Graham’s number is no longer the record holder for largest number in a mathematical proof 🙂

  • GC

    To a mathematiciaon, Graham’s Number (or any other number you can name) is “small”. Why? However large it is, there is only a finite set of integers that are smaller than it, but an infinite set of integers that are larger. There are no “large” numbers!

  • GC

    (That’s “mathematician”. Why can’t I spot those errors BEFORE hitting “submit”??)

  • dreamfeed

    “I’m not gonna really explain this because the explanation is really boring and confusing”

    Just because you don’t understand it doesn’t mean it’s boring. If you watched a movie in a language you don’t understand, you would probably be bored, but you wouldn’t say it was necessarily a boring movie.

  • Eliza Qwghlm

    Wonderful post, Tim. Would you be able to calculate / predict how far into a single power tower the world’s largest supercomputer could calculate and multiply all the way down?

  • Eliza Qwghlm

    I saw someone’s post that said your brain wouldn’t explode, it would collapse into a black hole if it contained this much information. I did a quick calculation, and it would take a lot less information than a googol before the collapse.

    Quick research estimates a black hole forms at the mass of 40 suns. The mass of our sun is 2 x 10^30 kg, so a black hole requires 8 x 10^31 kg of mass to form. The human brain has 1.36 kg of mass. It would require 8 / 1.36 = 5.9 x 10^31 brains to form a black hole.

    The human brain contains an estimated 100 billion neurons, each potentially holding an estimated 1,000 bits of data. The total data held by a single brain is estimated at 100 trillion, or 10^14, bits of data.

    So by my calculation, 5.9 x 10^31 brains would hold 5.9 x 10^45 bits of data before the collapse.

    So don’t feel bad if you can’t comprehend Graham’s number. It wouldn’t be healthy for you if you did.

    • Isaac Churchill

      Not sure what you meant by “a black hole forms at the mass of 40 suns”, the earth would form a black hole if it was crushed to the size of a pea

      • Eliza Qwghlm

        Fair point. I read the European Southern Observatory’s website to mean it requires approximately 40 solar masses to form a black hole: http://www.eso.org/public/usa/news/eso1034/

        You are correct, a micro black hole can theoretically appear at much smaller sizes, as long as the volume is sufficiently small. Thanks for the correction. That makes the question more interesting: assuming 1,200 cubic cm for a human brain, what is the mass required before it forms a black hole? I am assuming it is less than 8 x 10^31 kg (and therefore, able to hold a lot less data than 5.9 x 10^45), but my personal neurons don’t contain the math skills to figure it out.

  • Habfast

    Having done set theory, this is still by far smaller than the cardinality of the continuum (number of real numbers): http://en.wikipedia.org/wiki/Cardinality_of_the_continuum

    Which itself is far far smaller than other sets (see http://en.wikipedia.org/wiki/Beth_number)

    Sincere apologies for being pedant and annoying, I generally don’t do that.

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  • Nat Williams

    You’re ignoring probably the best part about Graham’s Number. From the original paper:

    “Is it possible to improve significantly the estimates of these numbers? For example, in Corollary 12, the
    upper bound on N(l, 2, 2) given by the theorem is truly enormous, where, in fact,
    the exact bound is probably < 10."

    It is the worst upper bound ever.

    • Yeah, I find it funny how graham’s number is an EXTREMELY huge upper-bound – even the current bound is 2^^(2^^(2^^9)), which is still quite big.

      Then again, some numbers bigger than Graham’s number have been used in mathematics not just as an upper-bound. for example, TREE(3), a number discovered by Harvey Friedman, is the answer to the problem:

      “What is the longest possible length of a sequence of 3-labeled trees such that no tree is homeomorphically embeddable into a later tree?”

      It’s a fairly simply problem that leads to a surprisingly big solution, TREE(3). It’s not known exactly how big TREE(3) is, but we know that it’s far far far far far far larger than Graham’s number.

      And there are still bigger numbers like that – for example SCG(13) or the numbers that we can make from the finite promise games, and the busy beaver function as well.

  • foobie

    Goodstein’s function grows _much_ faster than Graham’s number (function).

    You want to see a big number, you have to ask a proof theorist, not a combinatorialist.

    • Eliza Qwghlm

      Tim wrote a 6,000 word essay on large numbers, and you ONE UPPED him in only 26!!

      God, I admire you.

      • well, of course he didn’t describe goodstein’s function in detail

        • jeffhre

          And he didn’t show if why and how Goodstein’s number could be of any importance equal to or greater than…John Doe’s number.

  • arpit maheshwari

    Most mind blowing post that i have ever read. When You reached gogolplex(or something like that) I wanted this post to never end. But hey, I was too early and in the end I was begging for death.

  • V.

    This post made me love math.

  • Dr. Strangelover’s Number: Whatever the Greatest Number Is + 1

    • this “number” is cheating in the large number discussion, trying to make the new largest number. well guess what: there is NO greatest number. if you want to make a large number, actually make your own large number and not some bullshit like “the biggest number + 1”.

  • odput

    When I read what g2 was and figured out where we were going to get to g64, I had one of those 3rd level of consciousness “whoa” moments. It was the first one I had since reading about Tim’s levels of consciousness. It was pretty awesome.

  • Pelishka

    What about the busy beaver function? The fastest growing sequence that one can conceive (without going meta). It makes the series (g1,g2,g3,…) look like a really slow one!!!! and actually grows faster than any computable sequence.

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  • ravensperch

    This post has a lot of similarities to a virus. As a result, it hacked the limits of my comprehension. Installing a new hard drive. Loading…..

  • QuanTim Leap

    g-googolplex to the googolplexh?

  • Mogumbo Gono

    A trillion trillion up arrows isn’t even a smidgen of infinity.

    And there are different levels of infinity…

    But fortunately, infinity isn’t a number at all.

  • garthpool

    About living for eternity, suppose you could forget everything that had happened more than two hours before, or some time period of your choice. And if you could change that time period at will, that would be even better. Assuming, of course, that all experiences would be good. Maybe they would all be bad. That seems to be a possibility in an infinite universe.

  • Wile E. Coyote

    Where does TREE3 fall in at? I thought that was a biggy.

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  • alterreg

    is g64-1 a prime?

    • no, since it’s a difference of two perfect cubes (graham’s number (it’s a cube since it’s just a huge power tower of 3’s) and 1), it can be factored with the formula for factoring differences of cube, which means it is composite.

      • and to add on, the largest known prime number isn’t even close to graham’s number or even a googolplex – it has only 17 million digits.

    • the bang!

      No becuase its even!Grahams number is odd so subtract 1 and you’ll get an even number.

      • alterreg

        i really meant to subtract 1

    • Aximili

      Not a chance. It’s a power of three, therefore it must be an odd number. And any odd number minus 1 is an even number, none of which (except for 2) are prime.

  • Tacoplex

    For those who’d like to imagine living LONG time here’s weird japanese vid.
    http://www.youtube.com/watch?v=CN_eL5ukCbw

    • Degu

      That went very differently than I thought it would. It was actually kind of rewarding. I’d do that.

  • Peter Tibbles

    I’ve just invented Pete’s number. Graham’s number is G64 (in his notation), so
    I’d like to lay claim to G(Graham’s number).

    Get your head around that one.

    • although this is a tricky number to get your head around, rest assured you aren’t the first to come up with G(Graham’s number). it’s a common retort to graham’s number alright.

      • Peter Tibbles

        Oh dear, foiled again.

        • Alec Rhea

          and you could then recursively define G(G(64)) as H(64), then keep going until we had H(H(64)) and define that as I(64) so on and so forth. You’re iterating the iteration of iterating iteration, so on and so forth.

          • Aximili

            If you wanna see some some truly vast numbers, many of which absolutely dwarf g64 and everything you just described along with it, go have a look at Jonathan Bowers’ homepage: http://www.polytope.net/hedrondude/scrapers.htm

            You will need the notation described here to even begin to understand any of this: http://www.polytope.net/hedrondude/array.htm

            You’ll find Graham’s number in the sixth-smallest group he defined, and then it just keeps going. I don’t know how or why, but it would appear that coming up with huge numbers is all that guy does for a living.

    • Meh

      The whole reason Graham’s number is significant is because it’s an insanely large number that actually serves a practical purpose. Arbitrarily constructing large numbers isn’t really impressive in itself.

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  • Bassam Abdul-Baki

    Good read. I recently learned about Busy Beavers (http://en.wikipedia.org/wiki/Busy_beaver). Talk about a crazy sequence.

  • Guest

    I don’t really get why Graham’s number is G64. 64 seems randomly chosen but it probably isn’t, can someone explain?

    • Matthus Gougeus

      g64 is an upper bound to the solution of the problem about n-dimensional hypercubes.

  • Sean

    but what about Graham’s number + 1?

    • Grotoff

      It’s not comprehendable by our meat brains, but there is no limit of the real numbers. When we say a number continues to infinity, we’re serious. If you want to know about the more or less practical use of the number, it’s an upper bound in the solution to a geometry problem. Look Numberphile’s videos about it.

  • Phill

    You always make me laugh in these posts. You can make even some of the most dull topics hilarious! Think about what an actual feeding frenzy psycho festival would look like! Lol! Insane people walking around eating tonnes of things…

  • Socrates

    I’m a nuclear physicist and I must say we don’t think much about numbers larger than the age of the universe, that’s mostly it for us. This is quite interesting. Gave me more of a headache than reading about infinity and Georg Cantor, because I just accepted this is beyond comprehension. With this I tried… unfortunately.

    • Matthus Gougeus

      What ? Even Avogadro’s number is bigger than the age of the universe in seconds.

      • Penisman

        assume he meant the “size” of the universe

  • godsmotive

    I wrote a book called Infinity Squared…but it wasn’t supposed to actually mean anything…just a catchy title and looked neat using the symbols on the cover.

  • stcoleridge

    I was reading this, but had to take a break. When I picked up my iPhone again there was an ant running around in it … sort of an analogy to me vs the G64 concept.

  • Matthus Gougeus
    • i am a regular contributor to that wiki and i approve this comment

  • Darkness3827

    This could go on infinitely where you have g(g (g… g (64)…) where the number of g’s is grahams number. and then you could do the same thing where whatever that new number is is the amount of g’s, then do that again with this 2nd new number. Thinking that if you could some how calculate this number instantly and if you kept plugging this in once a second and getting a new number and so on for your entire life (or for all of time) and you would still be 0% of the way to infinity is a pretty terrifying thought.

  • Hennie Randolph

    who is this graham character anyhow

  • Ishraq Quayyum

    Here’s a video about Graham’s number. It features the man himself: Ron Graham. https://www.youtube.com/watch?v=HX8bihEe3nA

  • qusdis

    What’s truly amazing about Graham’s Number is the “pound for pound” insanity you get out of moving from exponentiation to tetration. I am easily able to construct larger numbers by, say, doing G64 factorial, etc., but it was genius to come up with tetration in the first place, whereby 3^3^3^3 is already bigger than a googol.

  • Red Sage

    W. O. W.

    This still makes more sense to me than irrational quadratic equations.

  • Impirren RyRy

    I got very, very worried once I saw Sextillion was nowhere near as far down the page as I thought it would be.

  • Augustine Esterhammer-Fic

    One problem with the segment on the Googolplex (I hope I’m not the guy repeating someone else’s idea):

    “random probability suggests that there would be exact copies of you in that universe. Why? Because every possible arrangement of matter in a human-sized space would likely occur many, many times in a space that vast, meaning everything that could possibly exist would exist—including you.”

    But the universe doesn’t tend toward random arrangements. Maybe there are [10 to the 10 to the 70] possible arrangements of atoms in a human-sized space. That wouldn’t mean that you’d find that random space with any frequency. Every day of our 3.5 billion year evolution was another day of chance events influencing whether or not YOUR specific genes are passed down. For that random human-sized space to be arranged into a Tim Urban, it would have to follow an equally contingent series of events. If you calculate the odds of yourself existing, not as a single probability of random particles, but as a probability of events beginning with a mostly hydrogen universe, I think the number would be many times higher.

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  • Orang Yousefian

    Now instead of each 3 put the graham number itself !!! And then calculate it!!! And repeat the process graham number times ! How about that !!and repeat what i said graham number tmes again .yoooohooooooo hooooo

  • bobthelob

    “10^80 – To get to 10^80, you take trillion and you multiply it by a trillion, by a trillion, by a trillion, by a trillion, by a trillion, by a hundred million. No dot posters being sold for this number. So why did I stop here at this number? Because it’s a common estimate for the number of atoms in the universe.

    10^86 – And what if you wanted to pack the entire observable universe sphere with peas? You’d need 1086peas to make it happen.”

    So more peas would fit in the observable universe than atoms in the universe? Or am I an idiot?

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