This week’s episode of More or Less on the BBC World Service answered a question that involved estimating big numbers: Are there more stars than grains of beach sand?

This claim was famously made by Carl Sagan in the seminal programme Cosmos.

The cosmos is rich beyond measure. The number of stars in the universe is larger than all the grains of sand on all the beaches of the planet Earth.

More or Less come to a fairly standard answer, that Sagan was correct. This sort of problem, which involves approximating unknowable numbers based on a series of estimates, is called a Fermi problem. I’ve written about Fermi problems here before. The More or Less approach to answering this raised a question from a reader of this blog.

But that's less than a factor of 3 difference! For Fermi estimates of numbers of that size, those two answers are essentially the same. It wouldn't take much of an error in either estimate to push sand ahead of stars…

— Paul Taylor (@aPaulTaylor) July 8, 2018

Alright, actually Paul is one of the writers of this blog, rather than a reader. Even so, are his concerns warranted?

The numbers in the More or Less piece are

- for stars: “1 followed by 22 zeros” (approx. 4:18 in the podcast audio), described as 10 sextillion;
- for sand: “3.65 with 21 zeros after it” (approx. 8:30), described as about 4 sextillion.

Fermi estimation involves making order-of-magnitude estimates based on informed guesses about the inputs. (Listen to the More or Less piece if you want to hear this process in action.) Then, given the level of estimation, you aren’t supposed to take too seriously differences within the same order of magnitude – 3 million and 5 million can’t be taken to be seriously different – but if one number comes out at least an order of magnitude bigger than the other, you take this to be your conclusion. In that sense, what More or Less did was correct – 10 sextillion is one order of magnitude bigger than 4 sextillion and so, we conclude, bigger.

There is an uncomfortable feeling which comes from how similar these numbers are, given the amount of guessing involved. Something about comparing so-many sextillions and some-other-number-of sextillions seems like there isn’t that much difference. Actually, the difference between $4 \times 10^{21}$ and $1 \times 10^{22}$ is $6 \times 10^{21}$, so we’d need More or Less to have missed out more than half the sand on Earth in estimating to have come to the wrong conclusion.

There are people who’ll tell you that the only number to pay attention to in a Fermi estimate is the $10^n$ term, because the other is just noise. But this leads to awkwardness. In this case, $4 \times 10^{21}$ is four times bigger than $1 \times 10^{21}$ but only two-and-a-half times smaller than $10^{22}$. Can we really say $1 \times 10^{21}$ and $9 \times 10^{21}$ are effectively the same, while $10 \times 10^{21}$ is significantly bigger?

It’d certainly be a clearer conclusion if the numbers came out several orders of magnitude different, but then obviously very differently-size numbers wouldn’t make for such an interesting piece of rhetoric, would they? The real question you have to ask, then, is whether any doubts you have about the way each number was estimated amounts to a big enough difference in either number.

I guess it is also possible to think about this estimation method too seriously — it’s only really designed to give a rough-and-ready, back-of-the-envelope guesstimate. There’s something about the way the More or Less piece suddenly comes to a conclusion and ends, with a definitive-sounding “Carl Sagan was right” and “he nailed it”. I’d say a few more caveats would have been appropriate. Some discussion of upper and lower bounds, or just an acknowledgement of how much error would cause the conclusion to flip, might have been nice. Paul points out, and I think he’s right, that the fact these numbers are pretty close is remarkable, and perhaps this could have been remarked upon. So, in conclusion, I don’t think the More or Less conclusion is necessarily wrong, but it could have been framed better.

10^4*10^4*10^4 equals 10^12 NOT 10 billion grains of sand per cubic meter (if one assumes sand diameter is 0.1mm as proposed in the programme)