So say you had a computer that didn't treat base 10 as special. Then an apples-to-apples comparison would be that 355/113 (9/7 = 16 binary digits) has the same precision as 3141592/1000000 (22/20 = 40 binary digits). 16 is a lot fewer than 40.
That's to say, we know which base we are using... which is a requirement for writing down numbers in the first place.
>So say you had a computer that didn't treat base 10 as special
Your computer treats base 2 as special. The apples-to-apples comparison would be comparing 355/113 to 11.0010010000111₂ (or a 16-bit float representation of Pi).
355/113 is a better approximation, but not as drastically as you wrote.
Writing in binary, 355/113 = 101100011₂/1110001₂, and the equivalent one is 11.0010010000111111011₂ - hardly that big of a difference.
This is a compression problem, a little more breakage here or there is to be expected. If e and every other irrational number showed similar breakage, then I might get interested.
Eg area of a half circle is:
A = pi * r^2/2
22 * 4^2
---------
7 * 2
You can't 'just' memorize pi. You have to pick a number system first.
And yes, if you work a lot with decimal numbers, then memorising an approximation for pi in that system will come comparatively easy.
Obviously bits is the relevant concept here from a modern information theory perspective. But “digits in fraction” is more interesting. 22/7 vs 314/100. 355/113 vs 3141592/1000000. Of course you don’t need to “remember” the denominator when it is a power of 10. But the error in the approximation is unusually low, compared to the size of the denominator. This is true when comparing to decimal approximations, or other “nearby” rational approximations.
In that sense, these approximations should be “surprising” because they work much better than the decimal ones of similar size (or, much smaller than the decimal approximations of similar accuracy). But they also work better for pi than for many other common irrational numbers.
This is also historically meaningful, as some calculation was done in fractions, e.g. in ancient Egypt. It is an interesting mathematical fact that 22/7 is an approximation to pi that is small enough to be easily discoverable, easy to compute with, and still practically useful.
Edit: changed 7 to 8, my mistake