How big is a trillion, anyway?

2020 was a sad year for lovers of large numbers. On 9th March, we lost Richard K. Guy at the age of 103; barely a month later, John Horton Conway is no longer with us. These two mathematicians, between them, in their Book of Numbers (1996) outlined a system for naming all numbers strictly less than \(1,000\)-illion. More specifically, for any \(n\) from \(21\) to \(999\), they gave a systematic way to name the number \(n\)-illion (the numbers from a million up to \(20\)-illion – or one vigintillion – already had names).

I say \(n\)-illion, because the precise meaning differs depending on whom you ask. There are two systems in use today: the short scale or American system, for which one \(n\)-illion is equal to \(1,000^{n+1}\), and the more sensible long scale, or European system, for which one \(n\)-illion is equal to \(1,000,000^{n}\).

I don’t find either of these systems particularly satisfying, as neither seems to follow the pattern that we use for smaller numbers (up to one million). Up to a million, we seem to be using the following rules.

  • Our basic unit is \(1,000\). That is to say, any \(n\)-illion should be some power of \(1,000\); to get the name for arbitrary \(10^n\), we add ten or a hundred to the start.
  • Repeating the same number twice – e.g., ‘a thousand thousand’ or ‘seventeen billion billion billion billion billion’ is ugly and inefficient, so we have to come up with new names to avoid doing so.

Just based on these axioms, then, both the short and the long scales seem to add new names exponentially more often than necessary. As any long scale adherent will tell you, there is no need to call \(10^9\) one billion when one thousand million will do. But the same devotee will, in the same breath, refer to \(10^{18}\) as one trillion, when the perfectly congenial one million billion is available. If we choose the principle that we should add new illions as sparingly as possible — after all, we have only \(999\) of them to cover all the numbers we might conceivably want to use — then the correct definition of \(n\)-illion is \(1,000^{2^n}\).

This gives us a nice way to convert \(1,000^n\) into words: it amounts to writing \(n\) in binary. Let’s see how the other systems compare.

The Googol Test

The first test of any system of naming large numbers should be: does it give a name to a googol? Recall that one googol is a one followed by a hundred noughts; i.e., \(10^{100} = 10 \times 1,000^{33}\).

  • The short scale calls this number ten duotrigintillion (\(10 \times 1,000^{33} = 10 \times 1,000^{32 + 1}\)).
  • The long scale calls it ten thousand sedecillion (\(10 \times 1,000^{33} = 10,000 \times 1,000^{32} = 10,000 \times 1,000,000^{16}\)).
  • Our scale calls it ten thousand quintillion (\(10 \times 1,000^{33} = 10 \times 1,000^{2^0 + 2^5} = 10 \times 1,000^{2^0} \times 1,000^{2^5}\)).

All the systems passed the test. However, while the long and short scales used the rather fantastical sounding duotrigintillion and sedecillion, we are still in the realm of the relatively pedestrian sounding quintillion.

The Googolplex Test

Not convinced? Let’s move on to the next test: does the system give a name to a googolplex? A googolplex is one with a googol noughts after it; i.e., \(10^{10^{100}}\).

Unfortunately, neither the long nor the short system is able to give a name to a googolplex: the largest power of ten that the short system can name is \(10^{3,000}\), while the long system can go only as far as \(10^{5,994}\), and one googol is far larger than either \(3,000\) or \(5,994\).

Our system, however, is perfectly suited for describing numbers of the form \(10^{10^n}\). One googolplex is equal to \(10,000\times 10^{3\cdots3}\), where there are \(100\) \(3\)s in \(3\cdots 3\). \(3\cdots 3\) is the number we have to convert into binary. At this point, I’ll hand over to the computer.

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Doesn’t exactly trip off the tongue, does it? But at least it’s possible to write down.

Some History

Our new number system might seem strange and unfamiliar, but it is in fact closer than both the long and the short scales to one of our very earliest methods for naming large numbers, namely that described by Archimedes in his The Sand Reckoner. As with our system, Archimedes’ displayed double exponential growth. The largest number that he could name was
\left((10^8)^{(10^8)}\right)^{(10^8)} = 10^{80000000000000000}\,.
Again, this is far larger than anything that either the long or the short system can name. But we can give it a name – and in our system it isn’t even as large as a centillion.

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