Saturday, January 5, 2013 0 Comments:

These are the biggest numbers in the universe

There are numbers out there that are so enormously, impossibly vast that to even write them down would require the entire universe. But here's the really crazy thing...some of these incomprehensibly huge numbers are crucial for understanding the world.
When I say "the biggest number in the universe", what I really mean is the biggest meaningful number, the largest possible number that is in some way useful. There are lots of contenders for this title, but I'll warn you now: there is a very real risk that trying to understand all this will blow your mind. But then, with extreme math, that's half the fun.

Googol and Googolplex

We might as well begin with what are quite probably the two largest numbers you've ever heard of, and are in fact the two largest numbers with commonly accepted definitions in the English language. (There's a fairly robust nomenclature available for naming numbers as high as you want to go, but you won't find these in dictionaries at the present time.) The googol, which has since become world famous (albeit misspelled) in the form of Google, began life in 1920 as a way to get children interested in large numbers.

These are the biggest numbers in the universeTo that end, mathematician Edward Kasner (pictured) took his two nephews, Milton and Edwin Sirotta, on a walk through the New Jersey Palisades. He asked them for any ideas they might have, and the then nine-year-old Milton proposed "googol." Where he got this particular word is unknown, but Kasner decided that 10^100 - or, the number one followed by a hundred zeroes - would henceforth be known as a googol.

But young Milton wasn't finished - he also proposed an even larger number, the googolplex. This number, according to Milton, was 1 followed by as many zeroes as you could write before you got tired. Though a charming idea, Kasner decided a more technical definition was needed. As he explained in his 1940 book Mathematics and the Imagination, Milton's definition left open the dicey possibility that a random buffoon could become a greater mathematician than Albert Einstein simply by possessing greater endurance.

So, Kasner decided a googolplex would be 10^googol, or 1 followed by a googol of zeroes. To put that another way - and in similar notation to how we'll be dealing with various other numbers we'll be talking about - a googolplex is 10^10^100. To put that in some mindbending perspective, Carl Sagan once pointed out that it would physically impossible to write down all the zeroes in a googolplex, because there simply isn't enough room in the universe. 

If you filled the entire volume of the observable universe with fine dust particles roughly 1.5 micrometers in size, then the number of different combinations in which you could arrange and number these particles would be about one googolplex.

Linguistically speaking, googol and googolplex are probably the two biggest meaningful numbers (at least in English), but as we're about to find out, there's no end of ways to define "meaningful."

The Real World

These are the biggest numbers in the universe
If we're going to talk about the largest meaningful number, there's a not terrible argument that that really means we need to find the largest number with any real world significance. We can start the bidding with the current human population, which is currently about 6.92 billion. The global economy in 2010 is estimated to have been about $61.96 trillion, but both of those are dwarfed by the roughly 100 quadrillion cells that make up the human body. Of course, none of these can compare to the total number of particles in the universe, which is generally thought to be around 10^80 - a number so large that our language doesn't have an agreed upon word for it.

We can play around a bit with measurements as we get larger and larger - for instance, the weight of the Sun in tons will produce a smaller value than if you measure it in pounds. The fairest way to do this is to use the Planck units, which are the smallest possible measurements for which the laws of physics still hold. For instance, the age of the universe in Planck time is about 8 * 10^60. If we go right back to the first unit of Planck time after the Big Bang, we find the density of the universe was 5.1 * 10^96. We're getting bigger, but we still haven't even reached a googol.

The largest number with any real world application - or, in this case, real worlds application - is probably 10^10^10^7, which is one recent estimate of the number of universes in the multiverse. That number is so huge that the human brain would be literally unable to perceive all those different universes, as the mind is only capable of roughly 10^10^16 configurations. 

Actually, that number is probably the biggest with any practical application, assuming you don't buy into the whole multiverse idea. But there are still far larger numbers lurking out there. But in order to find them, we're going to need to venture in the realm of pure mathematics, and there's no better place to start than with the prime numbers.

The Mersenne Primes

Part of the difficulty here is coming up with a good definition for what a "meaningful" number actually is. One way to think about is in terms of prime and composite numbers. A prime number, as you probably remember from high school math, is any number whose only divisors are 1 and itself. So, 2, 3, and 5 are all prime numbers, while 4 (2*2) and 6 (2*3) are both composite numbers. This means that any composite number can ultimately be reduced to its prime divisors. In a sense, a number like 5 is more important than, say, 4 because there's no way to express it in terms of smaller numbers.

Obviously, we can extend this quite a bit further. 100, for instance, is really just 2*2*5*5, which means that in a hypothetical world where our knowledge of numbers only went up to 5, mathematicians could still express the number 100. But the very next number101 is prime, which means the only way to express it is to have direct knowledge of its existence. 

These are the biggest numbers in the universeThis means that the largest known prime numbers are important in a way that, say, a googol - which is ultimately just a bunch of 2's and 5's multiplied together - really isn't. And, because prime numbers are essentially random, there's no known way to predict what impossibly large number will actually be prime. To this day, a discovery of a new prime number is a big deal.

Ancient Greek mathematicians understood the concept of prime numbers at least as far back as 500 BCE, but 2000 years later people still only knew which numbers were prime up to about 750. Thinkers as far back as Euclid saw a potential shortcut, but it wouldn't be until the Renaissance that mathematicians could really put this in practice. These are known as the Mersenne numbers, named after 17th century French scholar Marin Mersenne. The idea is simple enough: a Mersenne prime is any number of the form 2^n-1. So, for instance, 2^2 - 1 = 4 - 1 = 3, which is prime, and the same is true for 2^5 - 1 = 32 - 1 = 31.

It's much quicker and easier to identify Mersenne primes than any other type of prime, and computers have been hard at work looking for them for the past six decades. Until 1952, the largest known prime number was 2^127 - 1, a number with 39 digits. That year, computers determined that 2^521 - 1 is prime, and that number has 157 digits, which already makes it far bigger than a googol.
Computers have been on the hunt ever since, and currently the 47th Mersenne prime is the largest known to humanity. 

Discovered in 2008, it is 2^43,112,609 - 1, which is a number with nearly 13 million digits. That's the largest known number that can't be expressed in terms of any smaller numbers - although if you want to help find an even bigger Mersenne prime, you (and your computer) are always welcome to join the search.

Skewes' Number

Let's stay with the prime numbers for a second. As I said before, the primes are fundamentally irregular, which means there's no way to predict what the next prime will be. Mathematicians have had to go to some pretty fantastic lengths to come up with any way to predict future primes in even the vaguest of senses. The most successful of these attempts is probably the prime-counting function, which the legendary mathematician Carl Friedrich Gauss came up with in the late 1700s.

I'll spare you the more complex math - we've got plenty still to come anyway - but the gist of the function is this: for any given integer x, it's possible to estimate how many prime numbers there are that are smaller than x. For instance, if x = 1,000, the function predicts that there should be 178 prime numbers; if x = 10,000, there are 1,246 primes smaller than it; and if x = 1,000,000, then there are 78,628 smaller numbers that are prime.

Here's the thing though - prime numbers really are irregular, and so this is just a close approximation of the actual number of primes. In reality, we know that there are 168 primes smaller than 1,000, 1,229 primes smaller than 10,000, and 78,498 primes smaller than 1,000,000. It's an excellent estimate, to be sure, but it's always just an estimate...and, more specifically, an overestimate.
In all known cases up to about 10^22, the prime-counting function slightly overestimates the actual number of primes smaller than x. 

These are the biggest numbers in the universe
Mathematicians once thought that this would be the case all the way up to infinity - it certainly holds true for some unimaginably huge quantities - but in 1914 John Edensor Littlewood proved that, at some unknown, incomprehensibly vast figure, the prime-counting function would start providing an underestimate of the number of primes, and then the function would switch back between over- and underestimates an infinite number of times.

The hunt was on for the crossover point, and that's where Stanley Skewes (pictured) makes his entrance. In 1933, he proved that the upper bound for when the prime-counting function first becomes an understimate is 10^10^10^34. It's hard to really comprehend in even the most abstract sense what a number like that actually is, and to that point it was easily the largest number ever used in a serious mathematical proof. Since then, mathematicians have been able to reduce the upper limit to the relatively tiny figure of about 10^316, but the original figure remains known as Skewes' number.

So just how big is 10^10^10^34, a number that dwarfs even the mighty googolplex? In The Penguin Dictionary of Curious and Interesting Numbers, David Wells relates one way in which mathematician G.H. Hardy managed to conceptualize the size of Skewes' Number:
Hardy thought it 'the largest number which has ever served any definite purpose in mathematics', and suggested that if a game of chess was played with all the particles in the universe as pieces, one move being the interchange of a pair of particles, and the game terminating when the same position recurred for the 3rd time, the number of possible games would be about Skewes' number.
One last thing before we move on: the Skewes' number we've been talking about is the smaller of the two. There's another Skewes' number that the mathematician demonstrated in 1955. The first number relies on something called the Riemann hypothesis being true - it's a particularly complex bit of mathematics that remains unproven but is massively helpful when it comes to prime numbers. Still, if the Riemann hypothesis is false, Skewes found that the crossover point jumps all the way up to 10^10^10^963.

A Matter of Magnitude

Before we get to the number that makes even Skewes' number look tiny, we need to talk a little bit about scale, because otherwise there's really no way to appreciate where we're about to go. Let's first look at the number 3 - it's a tiny number, so small that humans can actually have an intuitive understanding of what it means. There are very few numbers that fit that description, as anything beyond about six stops being a distinct number and starts being "several", "many", and so on.

Now, let's look at 3^3, which is 27. While we can't really intuitively understand what 27 is in the same way that we can for 3, it's perfectly easy to visualize what 27 of something is. So far, so good. But what if we go on to 3^3^3? That's equal to 3^27, or 7,625,597,484,987. We're well past the point of being able to visualize that amount as anything other than a generically large number - we lose the ability to comprehend the individual parts somewhere around a million. (Admittedly, it would take an insanely long amount of time to actually count a million of anything, but the point is that we're still able to perceive it.)

Even so, while we can't visualize what 3^3^3 is, we're at least able to understand in broad terms what 7.6 trillion is, perhaps by comparing it to something like the US's GDP. We've moved from intuition to visualization to mere understanding, but at least we still have some grasp on what the number is. That's about to change, as we move another rung up the ladder.

For this, we'll need to switch to a notation invented by Donald E. Knuth, known as up-arrow notation. In this notation, 3^3^3 can be rewritten as 3^^3. When we then move to 3^^^3, the value that we're talking about is equal to 3^^(3^^3). This is equal to 3^3^3^...^3^3^3, where there are a total of 7,625,597,484,987 terms. We've now well and truly blasted past all the other numbers we've discussed. After all, even the biggest of those had just three or four terms in the exponential series. For instance, even the super-Skewes' number was "just" 10^10^10^963 - even adjusting for the fact that those are all much larger numbers than 3, it's still absolutely nothing compared to an exponent tower with 7.6 trillion terms.

Obviously, there's no way to even begin to comprehend a number so huge...and yet, the process by which it's created can still be understood. We might not be able to grasp the actual number that is produced by an exponent tower with 7.6 trillion 3's in it, but we can basically visualize an exponent tower with that many terms in it, and indeed a decent supercomputer would be able to store the tower, even if it couldn't begin to calculate its actual value.

This is getting increasingly abstract, but it's only going to get worse. You might think that 3^^^^3 is an exponent tower of 3's that is 3^^^3 long (indeed, in a previous version of this post, I made precisely that mistake), but that is just 3^^^4. In other words, imagine you had the ability to calculate the precise value of an exponential tower of 3's that was 7,625,597,484,987 terms long, and then you took that value and created a new tower with that many 3's in it...that gets you 3^^^4.

Repeating that process with each successive number until you've done it 3^^^3 times will, at last, get you to 3^^^^3. This is a number that is just incomprehensibly vast, but at least the steps involved can still sort of be grasped, if we take things very slowly. We can no longer understand the number or visualize the procedure that would create it, but at least we can understand the basic procedure, if only in the vaguest possible terms.
Now prepare for your mind to be really blown.

Graham's Number

These are the biggest numbers in the universeHere's how you get to Graham's number, which holds a place in the Guinness Book of World Records as the largest number ever used in a mathematical proof. It's utterly impossible to imagine how vast Graham's number is, and it honestly isn't much easier to explain exactly what it is. Basically, Graham's number comes into play when dealing with hypercubes, which is a theoretical geometric shape with more than three dimensions. 

Mathematician Ronald Graham (pictured, awesomely) wanted to figure out what would be the smallest number of dimensions needed for certain properties of the hypercube to remain stable. (Sorry to be so vague in explaining this, but I'm pretty sure we'd all need to get at least two graduate degrees in mathematics before we get any more specific.)

In any event, Graham's number is the upper limit for this minimum number of dimensions. And just how big is this particular upper bound? Well, let's go back to 3^^^^3, a number so larger that we can only understand the procedure behind it in the vaguest of senses. Now, instead of simply jumping up one more level to 3^^^^^3, we're going to consider the number 3^^....^^3, in which there are 3^^^^3 arrows between those two threes. At this point, we're far beyond even the tiniest possible comprehension of what a number like this is, or even how you would go about calculating it.

Now repeat that process 62 more times.
That, ladies and gentlemen, is Graham's number, a number that is about 64 orders of magnitude past the point of human comprehension. This is a number that is so much greater than any number you could possibly imagine - hell, it's much larger than any infinity that you could ever hope to imagine - that it simply defies even the most abstract of descriptions.

But here's the weird thing. Because Graham's number is basically just a bunch of 3's multiplied together, that means that we can know some of its properties without actually calculating the whole thing. We can't represent Graham's number with any familiar notation - even if we used the entire universe to write it down - but I can tell you right now what the last twelve digits of Graham's Number are: 262,464,195,387. And that's nothing - we know at least the last 500 digits of Graham's number.

Of course, it's worth remembering that this number is just an upper limit for Graham's original problem. It's possible that the actual number of dimensions that you need for the properties to hold is much, much smaller. In fact, back in the 1980s, the considered opinion of most experts in this area was that the actual answer was just six - a number so tiny that we can understand it on an intuitive level. Since then, the lower limit has been raised to 13, but there's still a very good chance that the actual solution to Graham's problem isn't anywhere near as big as Graham's number.

Towards Infinity

So, are there numbers even bigger than Graham's number? Well, of course there are - there's Graham's number + 1, for a start. As for meaningful numbers...well, there are some fiendishly complicated areas of math (particularly an area known as combinatorics) and computer science that do feature numbers even bigger than Graham's number. But we've pretty much reached the limit of what I could ever hope to sensibly explain. For those foolhardy enough to delve still further, you can check out some of the additional reading at your own peril.

And yet, there's still something even bigger out there, something so big that the term ceases to have all meaning: infinity. So join us next week for part two of our odyssey into the largest numbers imaginable, as we examine all the many flavors of infinity. Until then, I leave you with this amazing quote attributed to Douglas Reay:
I have this vision of hoards of shadowy numbers lurking out there in the dark, beyond the small sphere of light cast by the candle of reason. They are whispering to each other; plotting who knows what. Perhaps they don't like us very much for capturing their smaller brethren with our minds. Or perhaps they just live uniquely numberish lifestyles, out there beyond our ken.

Thanks for reading :)


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