Is super-accuracy matters?

[DA74254]

(10-20-2023 12:28 AM)Peter Klein Wrote:  I’m curious: How many digits are enough, in various real-world fields?

I notice that if I convert the distance halfway around the earth from km to miles, the difference between using 1.61 and 1.609344 is a whopping 5.25 km. Should I care?

How many digits did we need to go to the moon? To design a car engine, or the robotics in the car factory? To design a smartphone? The lenses in my glasses? The tomography machine that can measure the curvature of my retina? To navigate a ship or airplane? To measure spectroscopically the light from a distant star or planet and determine useful things about its composition? To determine that yes, you really did detect a Higgs Bosun or a gravity wave?

True story: My father was a theoretical physicist (classical physics, not nuclear). He used a slide rule from the 1930s through the early 70s. Then he got an early TI calculator that was, IIRC, only accurate to 4 or 5 decimal places when doing certain trig and log calculations. He was quite happy with for years. If he needed more accuracy, he punched Fortran programs on cards and sent them to the big computer at work.

My mother, OTOH, could not understand why, when she divided something by three on a four-function calculator, she got results that ended in .3333333 or .6666667. “That’s not good enough! I want the exact answer!” she would say.

(My highlighting).
I have asked myself the exact question.
Yes, it is fun to have something calculated to more than 1.000 decimal places or having my land plot calculated into the exact square Planck length.
Though, I was curious to how accurate one have to be. I found that NASA uses 16 decimal places of Pi when calculating orbits and space craft operations.

As a fun "experiment" for myself and obtaining the most accurate values for the distance earth/sun (1 AU, 149.597.870.800 metres) and knowing the exact length of a day (23h 56m 04,1s) and also the exact length of one tropical year (365,2422 days).
Now, the earth orbit is not exactly circular, but the AU is a median value, so for the purpose of finding the difference in accuracy, I assumed circular orbit based on the mean value of the AU.
Pi with 15 vs 16 decimal points yields a discrepancy of 0,0598mm or about the diameter of a human head hair (60µ).
12 vs 16 decimal digits is 237,322mm.

Given the exact time of one (circular) earth orbit (755.298.370,68048 seconds) one get the following:
Orbit time difference, 12 vs 16 decimal places of Pi: ~7,9µs.
Orbit time difference 15 vs 16 decimal places of Pi: ~2,00349 nanoseconds.

Worst case accuracy scenario should then give us a leap second every 160.000th year, while we have a leap second every 2-3 years.
Out of this, we can conclude that not even nature itself is that accurate