# Maddenation

## Caesar’s Last Breath

From p. 24 of John Allen Paulos’s book, *Innumeracy*:

Now for better news of a kind of immortal persistence. First, take a deep breath. Assume Shakespeare’s account is accurate and Julius Caesar gasped “You too, Brutus” before breathing his last. What are the chances you just inhaled a molecule which Caesar exhaled in his dying breath? The surprising answer is that, with probability better than 99 percent, you did just inhale such a molecule.

For those who don’t believe me: I’m assuming that after more than two thousand years the exhaled molecules are uniformly spread about the world and the vast majority are still free in the atmosphere. Given these reasonably valid assumptions, the problem of determining the relevant probability is straight-forward. If there are N molecules of air in the world and Caesar exhaled A of them, then the probability that any given molecule you inhale is from Caesar is A/N. The probability that any given molecule you inhale is not from Caesar is thus 1 - A/N. By the multiplication principle, if you inhale three molecules, the prob-ability that none of these three is from Caesar is [1 - A/N]

^{3}. Similarly, if you inhale B molecules, the probability that none of them is from Caesar is approximately [1 - A/N]^{B}. Hence, the probability of the complementary event, of your inhaling at least one of his exhaled molecules, is 1 - [1 - A/N]^{B}. A, B (each about 1/30th of a liter, or 2.2 × 10^{22}), and N (about 10^{44}molecules) are such that this probability is more than .99. It’s intriguing that we’re all, at least in this minimal sense, eventually part of one another.

**My Commentary**

First of all, Paulos’s estimate of breath volume can’t be right. Either he means 1/30th of a mole (which is 1/30th of Avogadro’s number, or about 2 × 10^{22} molecules) or 1 liter, which would be about 1/24th of a mole, or 2.5 × 10^{22} molecules. His estimate of molecules in the atmosphere is also wrong, by my reckoning. Look at it this way. Atmospheric pressure at sea level is 14.7 pounds/sq in. This means that the weight of a 1 sq in column of air extending up to the stratosphere (and beyond) is 14.7 pounds. The total pounds of air then is this times the surface area of the earth in inches. Pounds of air is converted to molecules of air by multiplying by Avogadro’s number divided by the molecular weight of air, about 29. Thus:

N = 14.7 psi x 4pi x (4000 mi. x 5208 ft/mi. x 12 in/ft)^{2} x 6.023 × 10^{23}/29 = 2.46 × 10^{41} molecules.

Computing with such astronomically large (or small) numbers is difficult, especially when taking powers as Paulos indicates. Neither my calculator nor Excel can do the math. However, the concentration of Caesar’s molecules in present day air can be calculated as: 2 × 10^{22}/2.46 × 10^{41} = 8.13 × 10^{-20}. On this basis, the average number of Caesar molecules contained in a single breath is:

8.13 × 10^{-20} x 2 × 10^{22} = 1620 molecules.

Even using Paulos’s values, the number of Caesar molecules in each breath would be:

2.2 × 10^{22} x 2.2 × 10^{22}/10^{44} = 4.84 molecules.

Thus the probability that you just breathed in one of the molecules Caesar exhaled in his last breath is very high, and probably even higher than the 99% Paulos quotes.

Of course, as David pointed out when I told him, all of this ignores the fact that oxygen could have been converted to CO2 and nitrogen might have gotten fixed by lightening and both might have been combined in plant or mineral matter in the past 2000 years.

Dad • Puzzles • 01/02/04 • 5 comments
## Comments

David • 01/07/04 • 3:17 PM:Excellent work Dad.

Some more info - most estimates of a breath are at 0.5 L. And 0.5 L is 1/44 of a mole of air. Still, not a big deal.

As for the air you breath. The 79% nitrogen doesn’t get used by us at all - it must be fixed by bacteria into usable forms, so I’d say it’s a fair assumption to say Caesar’s nitrogen is still in the air today. But, a good part of the 20% oxygen you breath is converted to most directly to water (the rest is exhaled back out) - and only later is some of that water and food changed to carbon dioxide.

Dad • 02/20/06 • 12:49 PM:Alas, my “last breath” calculations are wrong. In determining the number of molecules in the atmosphere, I failed to remember that Avogadro’s number refers to a “gram mole” of gas. I calculated numbers based on “pound moles” which are 454 times as big (a pound equals 454 g.). So Poulos’s estimate of molecules in the atmosphere is correct (If you convert my pound moles into gram moles, you increase the number of molecules by a factor of 454, resulting in 1.1 × 10^44. Paulos still misstated the breath volume, but I get close to his estimate using 1 liter. Using David’s lower estimate for breath volume does change the answer, however, because it not only reduces the number of molecules Caesar exhaled (A), but also reduces the number you inhale (B). I estimate it changes the probability of inhaling a Caesar molecule to about 75%, still ignoring the conversion of oxygen and nitrogen to other chemicals in the ensuing 2000 years.

Dad • 02/20/06 • 1:40 PM:To dispell the uncertainty, I just measured my lung capacity. I filled the basement wash tub with warm water (it’s cold down there!) and then prepared an inverted 4000 ml beaker full of water. I then blew the water out through a plastic tube and was able to almost empty the beaker. That’s almost 4 liters of air! So Paulos’s estimate of molecules in a single breath is conservative (i.e. low). My lung capacity may be a little larger than that of smaller folks, but anybody should be able to inhale a liter or so. Of course, Caesar’s last breath “Et tu, Brute,” may have come up a little short. Maybe we should be talking about his penultimate breath.

Patrick • 05/17/06 • 5:54 PM:As usual, Wikipedia has a page detailing lung capacities. I’m glad that Dad did the experiment, though. In any case, they point out that a normal breath does not use the lungs’ full capacity. They claim it’s about half a liter. Still quite a bit more than Paulos’s estimate. Maybe he was calculating Caesar’s last peep?

Patrick • 05/18/06 • 5:38 PM:I’ve been searching for the origin of this problem. It’s kind of hard to find, but here’s one site that traces it to James Jeans’s 1940 book “An Introduction to the Kinetic Theory of Gases” (p. 32). Other sites at least attribute it to the genre of Fermi problems, problems that require a heck of a lot of estimation. Anyway, it seems like the idea has probably escaped its originator to become anybody’s game. That same first site cites an article that takes Paulos to task for his incorrect assumptions and calculations. If you have JStor access, you can find it online, or you can try googling “Thoughts on Innumeracy: Mathematics Versus the World?” by Peter L. Renz from the October 1993

American Mathematical Monthly(and if that doesn’t work, email me, and I can send you the PDF).## Post a comment

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