Edu News | March 16, 2016
How Many Decimals of Pi Do We Really Need?
Earlier this week, we received this question from a fan on Facebook who wondered how many decimals of the mathematical constant pi (π) NASA-JPL scientists and engineers use when making calculations:
Does JPL only use 3.14 for its pi calculations? Or do you use more decimals like say: 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360
We posed this question to the director and chief engineer for NASA's Dawn mission, Marc Rayman. Here's what he said:
Thank you for your question! This isn't the first time I've heard a question like this. In fact, it was posed many years ago by a sixth-grade science and space enthusiast who was later fortunate enough to earn a doctorate in physics and become involved in space exploration. His name was Marc Rayman.
To start, let me answer your question directly. For JPL's highest accuracy calculations, which are for interplanetary navigation, we use 3.141592653589793. Let's look at this a little more closely to understand why we don't use more decimal places. I think we can even see that there are no physically realistic calculations scientists ever perform for which it is necessary to include nearly as many decimal points as you present. Consider these examples:
- The most distant spacecraft from Earth is Voyager 1. It is about 12.5 billion miles away. Let's say we have a circle with a radius of exactly that size (or 25 billion miles in diameter) and we want to calculate the circumference, which is pi times the radius times 2. Using pi rounded to the 15th decimal, as I gave above, that comes out to a little more than 78 billion miles. We don't need to be concerned here with exactly what the value is (you can multiply it out if you like) but rather what the error in the value is by not using more digits of pi. In other words, by cutting pi off at the 15th decimal point, we would calculate a circumference for that circle that is very slightly off. It turns out that our calculated circumference of the 25 billion mile diameter circle would be wrong by 1.5 inches. Think about that. We have a circle more than 78 billion miles around, and our calculation of that distance would be off by perhaps less than the length of your little finger.
- We can bring this down to home with our planet Earth. It is 7,926 miles in diameter at the equator. The circumference then is 24,900 miles. That's how far you would travel if you circumnavigated the globe (and didn't worry about hills, valleys, obstacles like buildings, rest stops, waves on the ocean, etc.). How far off would your odometer be if you used the limited version of pi above? It would be off by the size of a molecule. There are many different kinds of molecules, of course, so they span a wide range of sizes, but I hope this gives you an idea. Another way to view this is that your error by not using more digits of pi would be 10,000 times thinner than a hair!
- Let's go to the largest size there is: the visible universe. The radius of the universe is about 46 billion light years. Now let me ask a different question: How many digits of pi would we need to calculate the circumference of a circle with a radius of 46 billion light years to an accuracy equal to the diameter of a hydrogen atom (the simplest atom)? The answer is that you would need 39 or 40 decimal places. If you think about how fantastically vast the universe is — truly far beyond what we can conceive, and certainly far, far, far beyond what you can see with your eyes even on the darkest, most beautiful, star-filled night — and think about how incredibly tiny a single atom is, you can see that we would not need to use many digits of pi to cover the entire range.
Read more from Marc Rayman on the Dawn Journal, where he writes monthly updates about the Dawn spacecraft currently exploring the dwarf planet Ceres to provide scientists with a window into the dawn of the solar system.
Can you use pi like a NASA scientist?› Take the Pi in the Sky Challenge!
TAGS: Pi, Pi Day, Dawn, Voyager, Engineering, Science, Mathematics
Teachable Moments | March 16, 2016
Pi Day the NASA Way
UPDATE: March 16, 2016 – The answers the the NASA Pi Day Challenge are now available as an illustrated answer key. Download a poster version of the answer key on the "Pi in the Sky 3" activity page.
This post was originally published on March 9, 2016
In the News
Pi Day, the informal holiday beloved by math enthusiasts – and even by the math averse – is almost here! March 14 marks the yearly celebration of the mathematical constant (pi), which represents the ratio of a circle’s circumference to its diameter. More than just a number for mathematicians, pi has all sorts of applications in the real world, including on missions developed by NASA’s Jet Propulsion Laboratory. And as a holiday that encourages more than a little creativity – whether it’s making pi-themed pies or reciting from memory as many of the never-ending decimals of pi as possible (the record is 70,030 digits) – it’s a great way to have fun and celebrate the M in STEM.
Why March 14?
Pi is what’s known as an irrational number, meaning its decimal representation never ends and it never repeats. It has been calculated to more than one trillion digits, but 3.14 is often a precise enough approximation, hence the celebration occurring on March 14, or 3/14 (when written in US month/day format). The first known celebration occurred in 1988, and in 2009, the US House of Representatives passed a resolution designating March 14 as Pi Day and encouraging teachers and students to celebrate the day with activities that teach students about pi.
Why It’s Important
Pi Day is lots of fun, but its importance lies in the role that pi plays in the everyday work of scientists and engineers at JPL.
Fred Calef, a geospatial information scientist at JPL, uses pi to make measurements – like perimeter, area and volume – of features on Mars. “I use pi to measure the circularity of features, or how round or compact they are," said Calef. "Craters become more elliptical if the projectile hits the surface at a lower angle, so I use pi to measure how round a crater is to see if it impacted at a low angle.”
"We use pi every day commanding rovers on Mars," said Hallie Gengl, a rover planner for the Mars Exploration Rover Opportunity, "Everything from taking images, turning the wheels, driving around, operating the robotic arm, and even talking to Earth.”
Bryana Henderson, who specializes in planetary ices, uses lasers to explode ice samples and study their composition. "I use pi to calculate the width of my laser beam, which in turn can be used to calculate the amount of energy, or fluence, that hits my ice sample," said Henderson. "A larger fluence equals a bigger explosion in the ice, so this is a very important parameter for us."
The Pi Day Challenge
JPL has released the third installment of its popular Pi Day challenge, which gives students and the public a chance to put their pi skills to the test to solve some of the same problems NASA scientists and engineers do. The set of four illustrated math problems are compiled into a graphic (as well as classroom handouts) designed for students in grade 4 through high school – but fun for all!
› Check out this year's Pi Day challenge!
This year’s problem set shows how pi can be used to map the surface of Saturn’s hazy moon Titan, track the Mars Reconnaissance Orbiter as it explores the Red Planet, keep Earth’s satellites powered as Mercury transits the sun, and put the Juno spacecraft into orbit around Jupiter.
“For Pi Day, we like to give students and the public a glimpse into how math is used at JPL through questions that feature current events involving our space missions,” said Ota Lutz, an education specialist at JPL who helped create the problem set. “For instance, to put the Juno spacecraft into orbit around Jupiter on July 4, engineers will have to slow the spacecraft just the right amount. In the Pi Day challenge, students use pi to calculate that change in velocity.”
In the challenge, students will also use pi to calculate how much sunlight is blocked by our solar system’s innermost planet as it passes between Earth and the sun. This year, Pi Day comes just a few months before the May 9 transit of Mercury, making this a timely problem.
On March 16, the answers to all four problems and the steps needed to find those answers will be released in a companion infographic on the Pi Day challenge activity page.
In addition to the Pi Day challenge, JPL is inviting the public to share their Pi Day pictures and stories online. On March 14, JPL will join in on the fun with Pi Day photos and stories from the lab.
› Share Your Pi Day photos and storiesTo see a compilation of all 12 Pi Day challenge questions optimized for mobile devices and screen readers, visit: http://www.jpl.nasa.gov/edu/nasapidaychallenge
Explore More
Pi Day Challenges
Multimedia
Facts and Figures
Missions
Websites
TAGS: pi, pi day, math, mathematics