How many decimal digits of pi does NASA use

This week, we received a question from one of our Facebook subscribers on how many decimal places of the mathematical constant pi ( π ) NASA scientists and engineers use in their calculations.

Does JPL use a value of 3.14 for pi? Or you use more decimal places, like this:
3,1415926535897932384626433832795028841971693993751058209749445
923078164062862089986280348253421170679821480865132823066470938
446095505822317253594081284811174502841027019385211055596446229
489549303819644288109756659334461284756482337867831652712019091
456485669234603486104543266482133936072602491412737245870066063
15588174881520920962829254091715364367892590360

We directed this question to the head and chief engineer of the Dawn mission, Marc Rayman. Here is what he said.

“Thank you for your question! This is not the first time they have asked him. In fact, such a question was asked many years ago by a sixth grader, a lover of science and astronautics, who was later lucky to get a degree in physics and work on space exploration. His name is Mark Reiman.

To begin with, I will answer directly. In the Jet Propulsion Laboratory (JPL), for the highest accuracy of calculations in interplanetary navigation, we use the value 3.141592653589793, that is, 15 decimal places. Let's see why there are no more decimal places. I think that in general there are no physically realistic calculations for which scientists needed to include more decimal places than here. Let's consider such examples.

1. The farthest Voyager 1 spacecraft from Earth is located at a distance of about 20 billion km . Imagine a circle with a radius of this size, that is, 40 billion km in diameter, for which we want to calculate the length using the formula 2 π R. It turns out a little more than 125 billion km. We do not need to focus on whichthe exact circumference (you can multiply yourself if you want), we are interested in the measurement error due to rounding to 15 characters. So, it turns out that the length with a rounding constant to 15 characters is calculated with an error of less than 4 centimeters. Think about it. We have a length of 125 billion km, and the error is less than your little finger.

2. We can study the problem using the example of the Earth. The diameter at the equator is 12,756 km. The circumference of the equator is 40 075 km. You will have to overcome this distance if you want to go on a trip around the world (not counting mountains, valleys and obstacles like buildings, parking lots, ocean waves, etc.). How wrong is your odometer when using a rounded pi value? He will make a mistake about the size of the molecule. Of course, there are different types of molecules that differ in size, but you get the idea. The size of the error is approximately 10,000 less than the thickness of the hair.

3. Let's take the largest possible object: the visible Universe. Its radius is approximately 46 billion light years. And now the question is: how many decimal digits pi must be used to calculate the circumference of the Universe with an error of no more than the diameter of a hydrogen atom (the smallest atom)? Answer: you will need 39 or 40 decimal places. If you think about how huge the Universe is - really more than we can even understand - and what a tiny atom of hydrogen is, then you will realize that really accurate calculations do not require many decimals of pi. ”

Mark Reiman's articles can be found in the Dawn Journal , where he talks about the mission of the Dawn research apparatus, which is now exploring the dwarf planet Ceres, on a monthly basis.

Note. World Pi Day was celebrated on March 14 (3.14), but it can be noted for the second time on July 22, because 22/7 is even closer to the real pi value than 3.14.

Note 2 . The irrational number pi is an example of a source of random numbers. The randomness of decimal places is confirmed by all statistical and cryptographic tests. By last day, Pi, Stephen Wolfram launched the Find Your Pi Day website , which shows exactly where the number pi occurs on your birthday (or another number).