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How Many Decimals of Pi Do We Really Need?

jpl.nasa.gov

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Re: How Many Decimals of Pi Do We Really Need?

#21

Post by Etheryte »

notjpl wrote: post #14
WalterBright wrote: post #3

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.

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.

Re: How Many Decimals of Pi Do We Really Need?

#22

Post by julie1 »

WalterBright wrote: post #3

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 know about infinite.

3.1415926535897932384626433832795028841971 is precise.... it has a lot of digit and people loves that.

In ana num 3.15159 +- 0.00001 is exact. It bounds your result. Hence you can estimate your error and its propagation.

Because we thought humans were smart we thought that 3.14159 would be so meaningful people would understand that a constant should be considered to be exact with the implicit meaning that 9 was the last significant digit and people would be wise to use upper and lower bounds to estimates their results.

Then Computer Science was taught in university.

People not understanding why they had to study math and physics to simply program 2 + 2 and thought, stop bothering us. We just compute TVA we don't send a rocket to mars. Why learn boring math (integration, derivation, Newton's methods for approximation, Taylos's development, Cauchy Suites, condition of converging Suites, Integration in the complex field to compute generalized integrals, simplex, LU/RU matrices ....)

Yes people loves recurrence. They cannot apply the reasoning to simple maths series.

And that's how we have funny stuff like a lot of coder not understanding why :

   1.198 * 10.10
   12.099799999999998
Yes ... why are computers' maths so odd. What can we do about it?

Having a look at HP Saturn opcode makes you wonder if the lack of solution is because it does not exists or because people forgot. http://www.hpcalc.org/details.php?id=1693

Re: How Many Decimals of Pi Do We Really Need?

#24

Post by duaneb »

wtbob wrote: post #18
brador wrote: post #17

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).

Yea, but even in American high schools the metric system is taught for physics.

Re: How Many Decimals of Pi Do We Really Need?

#27

Post by todd8 »

mchahn wrote: post #2

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). I only knew one person that bought one; the rest of us continued carrying around our slide rules (they came in handy leather holsters with belt loops.) The HP that came out about a year later was a real scientific calculator.

Years before that, sometime between 1965 and 1968, on an episode of Lost in Space (a TV program with a family of early space explorers lost in outer space) the son, Will Robinson, was carrying around a large device about 3 inches thick and a foot tall that looked like a calculator. I thought the idea quite marvelous and went to bed thinking about it and how much better it would be than my slide rule for playing around with calculations. (I was a weird kid.)

Re: How Many Decimals of Pi Do We Really Need?

#28

Post by andrewflnr »

over wrote: post #20
wtbob wrote: post #18

Earlier quoted context omitted.

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).

Inch / foot / yard / mile is a really small range compared to what metric can handle (anything), so in practice most science is done with metric units.

Wide ranges of values are handled exactly the same way in imperial units as in metric: multipliers. You write "0.00023 inches" or "5.48e4 miles", etc. It's just not as pretty.
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