Infinite mathematicians walk into a bar. The first orders one drink. The second orders half a drink. The third orders a quarter of a drink. The fourth one orders an eighth of a drink.
The bartender pours two drinks and says “You mathematicians just don’t know your limits.
The bartenders in jokes are a strange and varied bunch; some of them have concentrated hydrogen peroxide on tap; others have an acute understanding of mathematical limits.
Since this is basically a joke about limits, let’s start there…but I do want to come back later and talk about some difficulties you might encounter when you try to serve progressively smaller drinks to infinite mathematicians.
When drinking in bars, know your limits.
Let’s look at the number of drinks each mathematician orders: 1, 1/2, 1/4, 1/8, and so on. This is a sequence. A sequence can be any set of numbers – it may be finite or endless, and it may or may not follow a pattern. Since there are infinite mathematicians, we may assume that this sequence is infinite. It also seems to follow a pattern; each mathematician asks for half of the amount requested by the fellow immediately prior to him.
Since you can continue the sequence ad infinitum, multiplying by the same number to get the next term in the sequence, this is called a geometric sequence. Here are some more examples of geometric sequences:
- Start with 1 and multiply each term by 5 to get the next term: 1, 5, 25, 125, 625, 3125, etc
- Start with 4 and multiply each term by -1/2 to get the next term: 4, -2, 1, -1/2, 1/4, -1/8, 1/16, etc
Geometric sequences can be further classified as convergent or divergent. Convergent sequences seem to hone in on some particular value. Divergent sequences, on the other hand, balloon out of control. The first sequence above is a divergent sequence; its terms will get farther and farther apart. The second sequence is convergent.
Now imagine that you decided to add the terms in a sequence. If the sequence is finite, that’s easy enough. To illustrate my point, let’s try adding the first ten orders made by the mathematicians:
1 drink + 1/2 drink + 1/4 drink + 1/8 drink + 1/16 drink + 1/32 drink + 1/64 drink + 1/128 drink + 1/256 drink + 1/512 drink = 1023/512 drinks
1023/512 is very close to two, but not quite there. What would happen if you could add infinitely many terms in the sequence?
The idea is that the sum gets forever closer to two. That’s what we mean when we say that the limit of this sum is two.
So our bartender, being well-versed in mathematics, quickly realized that the infinite mathematicians were ordering drinks in a convergent geometric series, and that the sum of the infinite orders would be two drinks. Rather than listen to drink orders for the rest of eternity, the bartender took a shortcut. Let’s hope that every mathematician after the first one doesn’t mind sharing a glass.
Now that we understand the bartender’s response, let’s examine the logistics of serving infinite mathematicians.
Is it even possible for there to be infinite mathematicians?
Maybe, but only if there is infinite mass in the universe. Nobody knows if that is the case or not. Even if the universe is infinitely massive, and there are infinite mathematicians, you wouldn’t necessarily have all of them together in one drinking establishment.
At some point the collected mass of the assembled mathematicians will pass a critical limit and they will collapse into a black hole, pulling the bartender, the pub, and perhaps Earth itself along with them. Even before that point it would get uncomfortably stuffy in the building.
Assuming the mathematicians used their advanced mathematical knowledge to avoid collapsing into a singularity (mathematicians can do that, right?), how many of them would actually get something to drink?
An excellent point! (Thank you!) Even though numbers may be divided infinitely, liquid refreshment cannot. Eventually one of the mathematicians’ orders will constitute only a single molecule, and then what? Presumably any mathematician after that is drinking in theory only.
Let’s assume that each drink is one liter, and that they’re all asking for water (just to make things easier for me). One liter of water contains something like 33,420,000,000,000,000,000,000,000 (about 33.4 septillion) water molecules, and that’s how many molecules the first mathematician would consume. The second mathematician would get half as many; the third mathematician would get a fourth as many, and so on. How many times do you have to divide a liter of water in half until there is only one molecule remaining?
Not as many as you might think. (WARNING: Math ahead!) We can write a little equation to calculate how many molecules the nth mathematician consumes.
Start with the assumption that 1 liter of water contains 3.3418×1025 water molecules. The mathematician in position n consumes (3.342×1025)(1/2)n-1 molecules. So which mathematician only gets one molecule?
(3.342×1025)(1/2)n-1 = 1
Divide both sides by 3.342×1025, which gives…
(1/2)n-1 = 2.992×10-26
Break out the natural logarithms.
ln(1/2)n-1 = ln(2.992×10-26)
(n-1)ln(1/2) = ln(2.992×10-26)
n-1 = ln(2.992×10-26) / ln(1/2)
n-1 = 84.79
Finish up by adding one to both sides…
n = 85.79
So the 85th mathematician’s order amounts to little more than a single water molecule. Everybody else will have to be thirsty.
But you forgot to account for the fact that in the short span of time between when the bartender pours the drinks and serves them, some of the water molecules will have evaporated from the surface, which means that the 85th mathematician won’t even get his order (and neither will the 84th, 83rd, 82nd, and a few others, depending on conditions like ambient temperature and humidity.
Yes, and I also neglected to account for the fact that the combined body heat of infinite mathematicians (mathematicians are warm-blooded, right?) in one place would probably flash boil the water anyway, making the whole thought experiment moot.
You may be thinking too much about this joke.
Indeed I am, but that’s the stated purpose of this blog. Good night!