NATURAL NUMBERS  
exploring the undesigned 

Is 432043200000000000 a big number?  
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 10^{72} and 10^{87}. 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 10^{19} or 23000000000000000000. This vast number appears to make the 10^{72} to 10^{87} 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 10^{19} and 10^{87 } 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 10^{21} cubic meters. The number of grains of sand in 1 cubic meter is about 10^{12}  that's something we have a comprehensible name for: 1 trillion. So if we multiply 10^{21} x 10^{12}, we get a result for the number of grains to fill the Earth: 10^{33}. 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:10^{39}. 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,10^{57},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 10^{68} atoms. To this I will generously  or foolishly  add one more 0, raising our number to 10^{69 }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 10^{70} we account for the atoms of 10 galaxies and at 10^{72} 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 10^{84}. 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 10^{90.}. 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 (10^{100})  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.




© 2008 Michael M. Ross  First published July 8, 2008 
