Whenever this comes up, I feel the urge to remind people that 40 digits is not a small amount at all. Each digit is exponentially smaller than the next.
Imagine you had 4 (integer) digits in your bank account. Now imagine 5. Now 6. Now 7. That’s unfathomable, right? The world’s richest men are worth 15 digits. But 40 digits? That’s bloody unbelievable.
Same way in reverse. 40 fractional digits is so incredibly small that mathematically, it’s enough to do everything at a universe scale.
Not small, that's true, but I think the 40-is-small argument often has something to do with the fact that we've[1] come up with computer programs and mathematical formulas that can churn out literally trillions[2] of provably valid digits of Pi in more than just decimal. Compared to that, 40 is peanuts. Barely even a short jog down the road to the chemist's [3][4].
[1] Where "we" means a specific very clever subset of humanity and so by adjacency, all of us humans and human approximants.
[2] Billions if your native language isn't English and know that words like "one milliard" will cause most English speakers' heads to implode.
[3] Or "pharmacy" if you're not British, though we often call them pharmacies now anyway.
[4] This is a Hitchhiker's Guide to the Galaxy reference.
[5] Unlinked footnote error. I like footnotes today for some reason.
Then you also consider the mind-boggling size of the observable universe, to go in context, to put on top of those forty digits of Л.
Just a week ago an astronomer friend gave me something to (barely) latch on to:
Imagine the Milky Way is the size of an American penny. How far away is the edge of the observable universe, in all directions? Fifteen kilometers, he said.
So, pi involves measuring circles. In fact, just think of a pie. It's a circle, right? There you go. The observable universe is the size of forty of those circles (or pies, shortened to "pi"). The size of it all just blows your mind. As for the hydrogen atom, single atoms are tiny, so you can safely ignore it.
The observable universe is the size of forty of those circles (or pies, shortened to “pi”).
No… Pi is a unitless number representing half a circles angular size.
Forty in the title is the number of digits, the title means that the relative size of the universe compared to a hydrogen atom is 1 followed by 40 zeros. Pi needs to be known to that accuracy to have a proper amount of significant figures.
Imagine trying to measure an ant with an unmarked foot long ruler. Not going to work super well. Your measurement uncertainty is +/- 6 in or 0.5ft. Well above the size of any ant.
Adding in inch marks improves that to +/- 0.5 in or +/- 0.042 ft. Closer to some ants, maybe about right, still not going to give you a measure of the ant but you’ll be able to say if it’s more or less than that. Now measure a circle with this ruler, to get the full accuracy of the ruler, you need only know pi to 4 digits., 3.142. Roughly. Actual uncertainty has some additional stuff going on, but without getting into it there you go.
The observable universe: The most of the Universe that we can see from Earth. There's more universe past what we can see (we think) but we just can't see it.
Circumference of the observable universe: We can only see so far out into space from Earth in all directions, so that makes a circle that has Earth at the center of it. That circle has a circumference, which is the length of that circle if you were to actually walk it.
Calculate circumference: We calculate circumference with the formula 2πr, where r is the radius, the distance from Earth to any one point along that circle we just talked about.
Pi: Pi is irrational and goes on forever. If you calculate the circumference with only five digits of pi, there's a bit of a "rounding" error in your calculation. If you want a more accurate value, you add in more digits of pi. The more digits you add, the more accurate you are.
Accuracy of 1 hydrogen atom: so the furthest we can see out is 46.508 billion light years (I'm not going to get into how we can see that far with a universe that is only 14 and some change billion years old, but we can). So 46.508 * pi * 2 = 292.218382266 billion light years. That's how long the path would be if you were to walk the edge of the observable universe. In meters that would 2.7645993537522 * 10^27.
If you used 5 digits of Pi you would be off by something like 100s of lightyears, or basically billions of meters. If you use 40 digits of pi you would be off by 120pm or 0.000000000012 meters. So for most things, 40 digits of pi is accurate enough for pinpointing any particular atom within the universe. Obviously we can use fewer digits of pi when we want to land a giant rocket on a massive planet like Mars, but knowing 40 digits is good enough for atom sized things, it gives us an upper bound of how accurate we actually need to be. So we can forgo using 50 digits of pi or whatever.
Well, this is still the most practical usability I’ve found for all the digits I learned in high school challenges. The first 110 has for some reason stuck on as almost permanent knowledge.
As a representative of these great United States of America, I will pledge the domestic everyday use of the metric system if those honorable representatives of other nations pledge to stop mixing up commas and periods in numerical notation.
About 17 bytes of data. For reference a common size for numbers is 8 bytes. I imagine Nasa has hardware for efficiently handling larger numbers such as 16 bytes and possibly more.
I mean, so do you. Any common device can handle computation with numbers a lot larger than 8 bytes, using appropriate software. Hell, even Python can handle that pretty routinely.
Very large numbers are used routinely in cryptography.
The article literally states that NASA uses at most 16 digits of pi, that’s around 53 bits.
IEEE double precision float also have 16 decimal digits / 53 significant bits. Dedicated hardware for that format exists since the 1980 launch of the Intel 8087.
Thanks Anton. Do you know anything more? I once made a physics simulation the scale of the solar system, including a space shuttle. Doubles weren’t good enough.
There is a trick to achieve double the precision of your hardware, it’s explained in this video. Many languages also come with arbitrary precision numbers which are probably a lot slower.
The main way of improving the code is however is probably understanding floats and keeping their limitations in mind.(Wikipedia article, interactive website) Floats are denser near zero and more imprecise the larger the absolute value is. This means subtracting numers that are close to each other or is a problematic operation.
I don’t know how your code is structured but lets say your origin point is at the star. If the shuttle is approaching a space station they are relatively close together (say 1km) in compared to their distance to the origin point (150 million kilometres). If you get their relative positions by subtracting their absolute position vectors, you probably have a high rounding error (with 32 bit floats they would literally have the same position). In that case it would probably be better to use a fixed width format or chunks with local origin points, as those approaches spread representable positions more evenly in space instead of having most of them inside the star.
I hope I gave you some useful tips on how to deal with floating point numbers, Anton.
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