exploring the undesigned
intelligence of the numberverse
How big is BIG...?
Many people have heard the assertion that the number of particles in the universe is a quantifiable sum, variously estimated at between 1072 and 1087. But how many people have said to themselves: is that all?!
How can a number with fewer than a hundred zeros contain everything - the unimaginable hugeness of the universe?
It's hard enough to comprehend the billions of galaxies and the trillions of stars. Now try to imagine that all those galaxies and stars are composed of grains of sand. Now try to imagine how many atoms are in each grain of sand!
Here I can help you out. The number of atoms in a grain of sand is about 2.3 x 1019 or 23000000000000000000. This vast number appears to make the 1072 to 1087 estimate hopelessly wrong... but wait! We need to keep reminding ourselves that we're dealing with exponents....
You really can see the world in a grain of sand. The number of grains of sand on all the world's beaches has been calculated to be less than the number of atoms in a single grain of sand. Whether that is true or not, I don't know. The point is this - that between 1019 and 1087 there are more than enough orders of magnitude to account for every atom in the universe.
So let's consider how many grains of sand would be needed to fill up our star, the Sun? Let's start with the Earth. The volume of the Earth is about 1021 cubic meters. The number of grains of sand in 1 cubic meter is about 1012 - that's something we have a comprehensible name for: 1 trillion. So if we multiply 1021 x 1012, we get a result for the number of grains to fill the Earth: 1033.
Now for the Sun. You can fit about 1 million Earths in the volume of the Sun, so that means we need to add 6 more zeros or orders of magnitude (that is, multiply by 1 million) to calculate the number of grains of sand to fill the Sun:1039.
Let's now stop thinking about grains of sand, and consider atoms instead! For this we have to multiply our answers for grains of sand by the 19 orders of magnitude for the number of atoms in a single grain of sand. That produces the vast number,1057,for the number of atoms in the Sun.
We still have a very long way to get from 1 average star to all the billions in the Milky Way and from there to the billions of galaxies in the observable universe. But in orders of magnitude, it is not so very far.
There is a lot of uncertainty about the number of stars in the Milky Way galaxy, since the dense inner core and interstellar gas and dust obscure most of them. However, we can assume that even if the number were as high as 1 trillion - which is far higher than the usually stated range of 200 billion to 400 billion - the average solar mass is a third of our sun's. This means we should add another 11 zeros to the Sun's total to come up with a reasonable estimate for the galaxy. The number we have reached is now a colossal 1068 atoms.
To this I will generously - or foolishly - add one more 0, raising our number to 1069 atoms, to make sure we account for all the interstellar gas, dust clouds, and the relatively negligible contribution of those billions of unknown undiscovered planets we hope to find. Remember that just one zero multiplies the total atoms in our galaxy by 10, so I'm being positively careless here!
In the next 12 orders of magnitude we have to jump to our local galactic group and from there to our galactic supercluster and then to all the other superclusters and large structures - the walls and filaments - in the cosmos, until we reach the limits of the observable universe. But we can get there because each increase in magnitude contains 10 times the atoms of all the preceding magnitudes combined. So at 1070 we account for the atoms of 10 galaxies and at 1072 we account for the atoms in 1,000 galaxies.
How many galaxies are there estimated to be in the universe? The number could be as high as 500 billion. Even if that extremely high estimate were true, we would still only be adding another 11 zeros (and I'm adding yet one more 0, to account for rounding errors, which we've been ignoring.) This brings our total to 1084.
At the beginning I said particles, so now we have to go one big step further and convert our result to one that would account for all the known kinds of fundamental particles in the observable universe. This means electrons, photons, quarks, and even neutrinos.
To do this we need to add another 6 zeros. That's 1 million particles for every 1 atom in the universe! That means we've just increased our final answer 1 million times to 1090.. So I've arrived at a number a little bit larger - OK, a thousand times larger - than the accepted upper bound for particles in the observable universe!
Should we consider this a big number? It depends how you look at it :) . The number of particles in the observable universe is at least 10 orders of magnitude less than a Googol (10100) - 10 billion times smaller in fact. And this vast number is just a tiny speck compared with the unit that has taken hold in popular culture, the Googolplex: which is 10 raised to the power of a Googol.
The answer to that question depends on what you see when you look at a string of zeros.... It also rather depends on typography! Does this number look more manageable? 432043200000000000 Or this number.... 4.32*1017 Here the number is rounded to 3 significant digits with a base 10 exponent, the standard scientific notation. An 18-digit number may be modest from the vantage point of computational number theory. But let's consider for a moment how long it would take to count this number. Or better still, let's say this number represents something in the physical universe: a measurement of time. In the discussion to the left, where numbers represent grains of sand or atoms or particles in the observable universe, we examine one way to think about very large numbers. But here we consider a question that almost anyone can comfortably answer on a pocket calculator in only 3 easy steps: How many seconds has the universe existed? Imagine a clock that began keeping time at the moment of creation and has been faithfully counting the seconds till now. How many seconds would that be? We might reasonably expect that number to be unimaginably large. But with simple multiplication steps we can come up with an answer that is far less intimidating than we might imagine. Begin with a best estimate of the universe's age in years:
Step 1. How many seconds are there in one 24-hour day?
Step 2. How many seconds are there in 1 year?
Step 3. How many seconds are there in 13.7 billion years?
So the number we have calculated is the one we began with, and it's only 18 digits long: 4.32*1017 By the standards of computational number theory it is inconsequential. And yet it contains every second that the universal clock has ticked since time began. It's hard for human minds to get around that paradox even though the idea of counting huge numbers by orders of magnitude was discovered by Indian and Greek mathematicians thousands of years ago. How long would it take to factor such a number? - that is, find the unique prime numbers that multiplied together make this number - and let's assume it's an odd number so it's not too easy. Using only trial division and assuming the worst-case scenario of a symmetrical semiprime (a composite with just two factors within an order of magnitude of each other), it's unlikely to require more than a few minutes. Even if it took 1 hour, it demonstrates that in 86,400 seconds it's possible to factorize a number that would take 13.7 billion years to count at 1 digit per second. That says a lot about just how fast computers are. It also says something about the magic of square roots. Let's consider the square root of the age of the universe in years. The square root of 137,000,000,000 is a mere 370,135 (approximately). That is, the square root of 13.7 billion years is only 370,135 years - less than the period humans have been on the planet! How many orders of magnitude different is 1 year from 1 second? The answer is 6, because the number of seconds in a year is in the millions (31536000) versus the single unit of 1 second. The difference between 1 and 1000000 is 6 zeros. That difference holds true whether we consider 1 year or 13.7 billion years. There are only 6 orders of magnitude separating the age of the universe in years and the age of the universe in seconds. This conveys something of the difference between a 12-digit number and an 18-digit number. For factoring large numbers, the order of magnitude is critical in determining how long the process will take. It is trivial to factor a 12-digit number, even by trial division. Even in the worst case, it will take only a few minutes. It is much harder to factor an 18-digit number by trial division: worst case, it may take an hour or more. A 21-digit number (about a 1,000 times greater than the age of the universe in seconds!) may take days, in the worst case, using trial division. How many more numbers does a 21-digit integer contain compared with an 18-digit integer. Three orders of magnitude equals 1,000 times more numbers. The 18-digit number is 0.1% of the 21-digit integer. And the 21-digit integer comprises a billion 12-digit integers! That should give you a better conception of the exponential nature of growth by orders of magnitudes. To factor a number - or more precisely, to find its prime factors - we need only consider the odd numbers less than the square root. That means, the numbers to consider are just half its square root. If we could do it efficiently, we could eliminate approximately another 9 out of 10 remaining numbers because they are not prime. In fact, that is not usually a necessary or efficient step. The computational effort to eliminate composites is many times greater than the time required to test every odd number less than the square root if we are considering numbers small enough to be factored by trial division. The kind of semiprimes that are used for encryption purposes are numbers on a completely different scale - a scale that exceeds the number of particles in the physical universe. You might say they exist merely in the plain of thought on an infinite number line: the uncreated 'numberverse'. However, these numbers are actually one of the few products of number theory with practical importance. It is easy to make such numbers, but it's nearly impossible to factor them using current computational number theory. This property is the basis for cryptography keys that provide internet security. For example, RSA-200 comprises two primes of 100 digits each. The resulting number is many times larger than a Googol. 2799783391122132787082946763872260162107044678695542853 The square root of this number is approximately a Googol. The probability of finding one of the two prime factors by shear chance is analogous to finding the location of one of just two unique particles in the physical universe. With this in mind, the fact that factoring this number required the equivalent of just 75 years of computation (using a lattice sieving algorithm designed by four mathematicians) is quite impressive.
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First published July 8, 2008