Miles and inches? Please learn and use standard international (SI) units. It's important.
He's an American, writing for an American audience, and thus he's using the units Americans use. There's absolutely nothing more scientific about one set of units or another (although different sets of units may be more convenient in different situations).
It prevents mistakes when we all use the same units and the SI are agreed by an international committee of scientists and engineers. It's one less thing to go wrong.
I remember back in high school physics when we were calculating the volumes of a few stars and my teacher said "Just round out 4\pi/3 to 4". I completely understand why we'd do that -- the error terms in the radius of the star completely drown out that approximation -- but goddammit it still feels wrong. I guess I'm a mathematician and not a physicist for a reason.
15 digits is about the precision hand-held calculators provide, right? Many early NASA missions took HP calculators along in missions with trajectory routines in case the computer failed.
Hand-held calculators might have been carried on many NASA missions, but not the early ones. The first missions started in 1961 and hand-held HP calculators weren't invented until more than ten years later. By that time we had already been to the moon 6 or 7 times. The first actual hand-held calculator I every saw was a Bowman Brain (simple 4 function calculator) it was for sale in 1971 at the MIT COOP (the bookstore…
We must be about the same age. I went through college with a leather holster for my slide rule. I saw the first TI calculator after graduating from college in '71. I lusted after it and later I went to work at HP and got an HP. I was a super-fan of reverse-polish at that time. HP stayed with RPN for some time.
I agree, that's a poor answer by NASA director and chief engineer. Here is a better answer: The precision used for calculations is dependent on the number of "steps" required to get to the final result. Roughly, for N repeated calculations you lose somewhere between sqrt(N) * eps to N * eps of precision (eps=2e-16 for IEEE64). Here are some actual examples: IEEE64 (~16 decimal digits) is OK for interplanetary navigat…
Disagreed, the answer isn't aimed at engineers or actually at any science-oriented people, it's directed at the general public who don't even care what IEEE is.
As a software engineer, though, I find this answer way more interesting. That one part in 10^15 ends up being ~an inch on the scale of the solar system is thoroughly unsurprising.
I have heard, but never done the math the verify, that with 50-ish digits of pi, one's error on a circle the size of the Universe would be smaller than a plank length.
Although this really should have been posted 4 days ago,
2pi x 3x10^8 x 40x10^10 / 1.6x10^-35
is just over 10^54 so that's about all the digits you need to memorize. Unless you're looking for a really strong password, reciting 100,000 digits is probably more than necessary:
http://blogs.scientificamerican.com/observations/how-much-pi...
If you know the diameter of the observable Universe and you want to calculate its circumference with the accuracy of the diameter of a proton, the number of digits of pi that you need is 43.
This overlooks the issue that for repeated calculations, such as numerical integration, the trouble comes from accumulated roundoff errors. Even 16 digits of precision can become 0 digits pretty quickly if you're not very careful.
A branch of physics used to be taught a long time ago called "numerical analysis" to deal with this issue. We even used to be careful about the difference between 'precise and exact'. Pi = acos(0) is absolutely exact. But computer don't know about symbolic calculus. So to put the value in a register we used tricks. Pi as a the converging value at the infinite of the Taylor development is awesome. But computer don't k…
I can't speak for other nations, but they still teach numeric analysis in Chinese universities as an undergraduate course. In my university it is a required subject. Many of us have countless dreadful memories of Runge-Kutta method, Euler's method, Newton's method, rate of convergence, numerical stability and error margins, just to name a few of the dreads...
If you know the diameter of the observable Universe and you want to calculate its circumference with the accuracy of the diameter of a proton, the number of digits of pi that you need is 43.
I've always heard the number is around 50, but do you have any references to show that as being correct?
If you know the diameter of the observable Universe and you want to calculate its circumference with the accuracy of the diameter of a proton, the number of digits of pi that you need is 43.
Would have been so much better if that answer was 42.
15 digits is about the precision hand-held calculators provide, right? Many early NASA missions took HP calculators along in missions with trajectory routines in case the computer failed.
Hand-held calculators might have been carried on many NASA missions, but not the early ones. The first missions started in 1961 and hand-held HP calculators weren't invented until more than ten years later. By that time we had already been to the moon 6 or 7 times. The first actual hand-held calculator I every saw was a Bowman Brain (simple 4 function calculator) it was for sale in 1971 at the MIT COOP (the bookstore…
Did the astronauts use slide rules? I know the E6B is pretty common for pilots still.