The mark that won't say how close

Twenty-two divided by seven gives 3.142857142857…, those six digits repeating for as long as you care to write them out. π opens 3.14159265… The two hold hands for three digits and then let go. The fraction sits about 0.00126 above the constant, an overshoot of four hundredths of one per cent, and in day-first date order it is written 22/7 — which is how the twenty-second of July ended up with an observance of its own.

That the two π holidays fall out of opposite date conventions is the quietly funny part. March 14 reads as 3.14 only where the month comes first; July 22 reads as 22/7 only where the day does. One constant, two calendars, two celebrations — and the July one is the more honest of the pair, because it says in its own name that it is settling.

22/7 did not start life as a rough guess, though. In Measurement of a Circle, Archimedes worked inward and outward from a 96-sided polygon and proved that the ratio of a circle's circumference to its diameter is less than 3 1/7 and greater than 3 10/71 — that π is trapped between 223/71 and 22/7. The famous fraction arrives in mathematics as a proved ceiling, not an estimate. Nothing in that argument means "about": there are two inequalities and a proof, which is a far stronger thing to be able to say.

It is also no accident that the fraction is as good as it is for its size. Expand π as a continued fraction and it opens 3; 7, 15, 1, 292, and stopping after each term in turn gives 3, then 22/7, then 333/106, then 355/113. That fourth one matches π to six decimal places out of three digits over three, and it is that good precisely because the next term is 292 — a large term means the fraction before it had already done nearly all the available work. Zu Chongzhi had 355/113 in fifth-century China, roughly a millennium before it surfaced again in Europe.

The mark itself sits at U+2248, in the Mathematical Operators block, assigned in Unicode 1.1 in 1993. Its name is ALMOST EQUAL TO, and the standard's names list files exactly one formal alias under it: asymptotic to. Which is odd, because U+2243 is already named ASYMPTOTICALLY EQUAL TO, and U+2245 — the one you would bet on for everyday approximation — is named APPROXIMATELY EQUAL TO and spends its working life on geometric congruence instead. The names and the usage ended up in a different order, and ≈ took the ordinary-writing slot regardless of what it had been called.

What makes it the right character for the day is what it declines to do. ≠ is a flat denial. ≡ claims identity, or definition, depending on which field you are standing in. The equals sign claims exactness and means it. ≈ asserts a relationship and then refuses to quantify it: it says near enough, and leaves near enough entirely to the reader. 22/7 is fine for the circumference of a dinner plate and hopeless for anything that has to hold together over a few million iterations, and the character you write between the two numbers is the same character either way.

So the tolerance never travels with the mark. It lives in the head of whoever typed it, and gets rediscovered by whoever didn't. Pi Approximation Day celebrates a number that is wrong on purpose by a known amount — and puts in the headline the one character that will not tell you what that amount is.