Is an infinitely small percentage equal to zero percent?

Question

Just came across a tweet with the following

0% of all integers are prime.

Thinking about that sentence as real numbers (0% of all reals are prime), in order to calculate that percentage, one ends up dividing by $\infty$ (even though smaller than the $\infty$ that one is multiplying above) and $\frac{\infty}{\infty}$ is undefined. The following links may be a good read:

• What exactly is infinity?

• Is infinity a number?

• What is infinity divided by infinity?

Knowing that primes exist, intuitively one would not consider them exactly zero, but infinitely small compared to the real numbers. However, I wonder if an infinitely small percentage is equal to zero percent?

The formulation is sloppy , but in some sense right. If we choose a random integer in the range $[1,x]$ , the probability that it is a prime number tends to $0$ , if $x$ tends to $\infty$. This is what is called the density. The prime numbers have density $0$ in the natural numbers. But something like a "random integer" (without a bound) does not exist , therefore as formulated, the claim is not true, stricly speaking. — Peter, Aug 23 '20 at 09:57
What that tweet means is that if $f(n)$ is the percentage of natural numbers less than or equal to $n$ that are prime, then $\lim_{n \to \infty} f(n) = 0$. This is can be made into a very rigorous definition that doesn't really have to deal with any kind of notion of "infinitesimally small". — Izaak van Dongen, Aug 23 '20 at 09:57
$\infty$ is not a number, it is a concept. I know that some users will again come with the extended real line, but this is also a concept.. And because $\infty$ is not a number, $\frac{\infty}{\infty}$ makes no sense either. It is only the notation for an expression $\frac{f(x)}{g(x)}$ , where both $f(x)$ and $g(x)$ tend to $\infty$. The limit can be anything, hence we cannot define $\frac{\infty}{\infty}$. — Peter, Aug 23 '20 at 10:00
Though I'd caution you to be careful of this tweet thread. The author says "Some respondents are claiming that there are the same numbers of primes and integers, since they can be put into a one-to-one correspondence, but what about 59? That's a prime number that can't be put into a one-to-one correspondence with any integer.". This is nonsense. There is a bijection between the set of integers and the set of primes. (This doesn't make the original tweet technically wrong) — Izaak van Dongen, Aug 23 '20 at 10:03
Also, the fact that 13% of numbers in criminal justice statistics are prime doesn't really have anything to do with how many of the integers are prime. If we make some very weak assumptions like "most numbers in criminal justice can be stored on a 1TB hard drive" then we'd immediately expect a nonzero percentage of these numbers to be prime. — Izaak van Dongen, Aug 23 '20 at 10:05
"numbers in criminal justice". What on earth is meant with that ? — Peter, Aug 23 '20 at 10:07
I do not think this question should be closed. It is about a maths fallacy found in the "real world" (that is, found in a tweet by someone who implies a certain level mathematical sophistication). As a community, we should encourage people to ask us to clarify such things rather than push them away. — user1729, Aug 24 '20 at 09:10

score 3 · Accepted Answer · edited Aug 23 '20 at 10:18

This is a very informal way of expressing the fact that the proportion of prime numbers less than a given maximum number $n$ tends to $0$ as we make $n$ larger and larger. More precisely:

$$\lim_{n \rightarrow \infty} \frac {\pi(n)}{n} =0$$

A mathematician would understand what was meant by "$0\%$ of all numbers are prime", but they would be very unlikely to express it this way, since it is (as you have found) confusing. A mathematician might say "almost all whole numbers are composite", because "almost all" has a precise mathematical meaning.

Is an infinitely small percentage equal to zero percent?

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