The delusion of counting to a billion

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In math or science class, it was sometimes helpful to give my students some idea of the size of numbers. One of those numbers was 1 billion. We worked it out that if you counted one number per second from 1 to 1 billion, it would take you around 32 years to get there. If you subtract 8 hours a day for both eating and sleeping, it would take around 42.7 years. It turns out that this length of time is wildly optimistic. In reality, accounting for the same daily breaks, it takes way longer. How much longer?

Jeremy Harper

Back in September 2007, software developer and Alabama native Jeremy Harper held the Guiness World Record for counting up to 1 million (highest number on record, spoken out loud). It took him 89 days, accounting for 8 hours of sleeping and eating per day. At 1 number per second, it should have taken him 15.4 days with the same breaks. It could be that, going by his videos left online, that he was deliberately slow in counting. He would give himeself a second or so in between each number, and not rush the number he was speaking. It would also be better to say the number in one breath, so that 1 second breaks between numbers allowed for a good inhale. I think there was a recognition that he not waste his energy on speed, otherwise he might not make it.

Guiness would usually send observers for such world records, but he had the entire 89 days of counting, eating, and sleeping documented on a YouTube livestream, making it easier for his feat to be proven. I have seen this stated elsewhere as the “fastest” count to 1 million, but I suspect that it is because it is the “only” count to 1 million. But accounting for breath control, it is likely the fastest one can expect.

Getting back to our count to 1 billion, which is 1000 million. It would have taken Jeremy, if he had the time, 89,000 days to count to 1 billion accounting for meal/sleep breaks and breath control. That would be around 243.7 years. Not possible. But even if Jeremy were to double his efforts, by counting every number in half the time, it would still take around 121.8 years to count that high. This is about as optimistic as one can be with 1 billion. But as you can see, this is triple our original estimation of 42.7 years, based on our naive assumption of 1 number per second. We are not taking into account that with more digits, numbers beyond 1 million will take longer to speak out than numbers below 1 million. The fact that there are 999 million numbers above 1 million and below or equal to 1 billion would likely skew closer to 243.7 years than to 121.8 years, even if we sped up the counting in the manner just described. Again, this is all pie-in-the-sky, since neither of these targets are humanly possible, and are hopelessly beyond the “naive” target of 42.7 years.

But to proceed ad absurdum, we have read about numbers in the trillions, such as the American military budget or their American infrastructure package. A trillion is a thousand billion, or a million million; literally \left(10^6\right)^2=10^{12}. The British used to call a trillion a “billion”, in the sense of a “bi-million” or a million million. But that got confused with the North American use of “billion”, so the British adopted the American parlance of a billion, and have been using it for trade since 1975. So, a thousand million is a billion, and a million million is a trillion, pretty much worldwide.

A trillion is “this” big.

The 89 million days (or more) it would take to count to 1 trillion would take us past the next few ice ages, should humanity last that long. It is the equivalent of 243,669 years, rounded to the nearest year. If these were light years, this would be more than the diameter of the Milky Way galaxy. And speaking of the Milky Way, there are  on the order of 1 trillion stars in our galaxy. There are also on the order of 1 trillion galaxies of every conceivable size in the known universe. Indeed, one trillion is a truly astronomical number.