Really, really, big numbers

Let’s start with a question. Imagine all the sodium chloride in all the world’s oceans crystallised into one enormous prismatic salt crystal with a 1cm x 1cm base. How long (or tall) would it be? 200 miles? 2000 miles? 20 000 miles maybe? Just think about that last one for a minute; if stood up on its base, it would reach almost a tenth of the way to the moon. So pretty big then. Unfortunately, while the 20 000 figure is correct, it’s not miles we’re actually talking about, it’s light years! In fact when stood on its base it would reach most of the way to the centre of our galaxy. 

Cue furious shakings of heads amongst Chemistry World  readers and the grabbing of calculators. But, taking the total ocean volume as 1340 million km³, the average salt content as 30kg per cubic metre, and converting to the 1cm x 1cm example, this is exactly what we get. That this should seem so gob-smackingly big is, perhaps, symptomatic of chemists tending to avoid the gratuitously macroscopic. Then again, we do deal with an awful lot of atoms at a time and Avogadro’s number is  impressively big. Shifting scales to the cosmological this makes for an awfully big number of atoms (as an aside, the total number of stars in the visible universe is quite similar to Avogadro’s number - do we live in a Molar Universe?), and some impressively big timescales as well. 

In their 1999 work The Five Ages of the Universe,  Fred Adams and Gregory Laughlin delve deep into one potential future of the cosmos. The picture they paint of the universe some 10¹⁰⁰ years from now is unremittingly bleak. All conventional matter has long since gone; the black holes it spawned have fizzled out as well, after hawking the last of their radiation to the pawn shop of eternity. All that is left is a supra-tenuous melange of positrons, electrons, neutrinos and hyper-extended photons - the latter’s unimaginably colossal wavelengths spanning vastly greater distances than the current diameter of creation. They indicate that by now the only ’chemistry’ left will be that of positronium ’atoms’, each consisting of just one positron and one electron, ’paired’ up in a stupendous volume dwarfing today’s entire observable cosmos. An even bigger time span now comes into play as they arc in on each other over the course of the next 10¹⁴⁵ years to an eventual, inconceivably lonely annihilation. Then there will be nothing left, neither big nor small, to measure bigness or smallness against. 

If chemistry and physics will have ceased, I suppose mathematics will still be there as an abstract ghost in the clapped out machine. Mind you, big numbers can get pretty abstract even now. Look into the maths of really, really, big numbers and you find people resorting to inventing their own strange notations to get by; one such is the Knuth notation, in which arrows replace towers of powers. This is for numbers that make even Avogadro’s look minuscule. Start with a sensible number and begin applying this notation and very soon you’re way beyond physical meaning. Keep going and eventually, after pushing more arrows than a volume of Perkin Transactions, you arrive at Graham’s number. This is commonly held to be the biggest number ever discussed in the context of a mathematical problem. It is the upper bound for the ’smallest dimension n of a hypercube such that if the lines joining all pairs of corners are two-coloured, a planar complete graph K4 of one colour will be forced’, which should come in handy. Yet the smart money in maths circles puts the lower bound for this answer at eleven! Whatever else you say about chemists, I don’t think we’d let ourselves get away with an error bar that big! 

Paul Kelly