Education News & Blogs – NASA Jet Propulsion Laboratory

While world record holders may have memorized more than 70,000 digits of pi, a JPL engineer explains why you really only need a tiny fraction of that for most calculations – even at NASA.

Update: October 24, 2022 – This article, originally written in 2016, has been updated to reflect the latest values for NASA’s Voyager 1 spacecraft, which continues to venture farther into interstellar space. The author, Marc Rayman, has ventured on too, from the chief engineer for NASA’s Dawn mission, which concluded successfully in 2018, to the chief engineer for mission operations and science at NASA’s Jet Propulsion Laboratory.

The decimals of pi are listed out on an orange background with a large pi symbol in the background.

This graphic shows more than 500 of the infinite number of decimals in pi. Image credit: NASA/JPL-Caltech | + Expand image

We received this question from a fan on Facebook who wondered how many decimals of the never-ending 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 [360 or even more]?”

Here’s JPL’s Chief Engineer for Mission Operations and Science, Marc Rayman, with the answer:

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 asked about. Consider these examples:

The most distant spacecraft from Earth is Voyager 1. As of this writing, it’s about 14.7 billion miles (23.6 billion kilometers) away. Let’s be generous and call that 15 billion miles (24 billion kilometers). Now say we have a circle with a radius of exactly that size, 30 billion miles (48 billion kilometers) 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 94 billion miles (more than 150 billion kilometers). 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 30-billion-mile (48-billion-kilometer) diameter circle would be wrong by less than half an inch (about one centimeter). Think about that. We have a circle more than 94 billion miles (more than 150 billion kilometers) around, and our calculation of that distance would be off by no more than the width of your little finger.

Put your pi math skills to the test with this problem from NASA's Pi Day Challenge. Can you use pi to determine what fraction of a signal from Voyager 1 reaches Earth? Image credit: NASA/JPL-Caltech | + Expand image | › View lesson page
We can bring this closer to home by looking at our planet, Earth. It is more than 7,900 miles (12,700 kilometers) in diameter at the equator. The circumference is roughly 24,900 miles (40,100 kilometers). That's how far you would travel if you circumnavigated the globe – and didn't worry about hills, valleys, and obstacles like buildings, ocean waves, etc. How far off would your odometer be if you used the limited version of pi above? The discrepancy would be 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 more than 30,000 times thinner than a hair!

Image credit: NASA | + Expand image
Let's go to the largest size there is: the known universe. The radius of the universe is about 46 billion light years. Now let me ask (and answer!) 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? It turns out that 37 decimal places (38 digits, including the number 3 to the left of the decimal point) would be quite sufficient. Think about how fantastically vast the universe is. It’s certainly far beyond what you can see with your eyes even on the darkest, most beautiful night of sparkling stars. It’s yet farther beyond the extraordinary vision of the James Webb Space Telescope. And the vastness of the universe is truly far, far, far beyond what we can even conceive. Now think about how incredibly tiny a single atom is. Isn’t it amazing that we wouldn’t need to use many digits of pi at all to cover that entire unbelievable range?

If you were to hold a single grain of sand at arm's length, you could cover the entire area of space taken up by this image, which was captured by the James Webb Space Telescope and contains thousands of galaxies. The oldest-known galaxy identified in the image is 13.1 billion years old. Image credit: NASA, ESA, CSA, STScI | + Expand image | › More about the image | Text description (PDF)
Pi is an intriguing number with interesting mathematical properties. It’s fun to think about its truly endless sequence of digits, and it may be surprising how often it appears in the equations scientists and engineers use. But there are no questions – prosaic or esoteric – in humankind’s noble efforts to explore or comprehend the marvels of the cosmos, from the unimaginably smallest scales to the inconceivably largest, that could require very many of those digits.

Hear more from Marc in his inspiring TEDx talk, “If It Isn’t Impossible, It Isn’t Worth Trying” and in his Dawn Journal, where he wrote frequent updates about the Dawn mission’s extraordinary extraterrestrial expedition to the protoplanet Vesta and dwarf planet Ceres.

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Infographic: Planet Pi

This poster shows some of the ways NASA scientists and engineers use the mathematical constant pi (3.14) and includes common pi formulas.

Recursos en español

18 Maneras en Que la NASA Usa Pi

Pi nos lleva lejos en la NASA. Estas son solo algunas de las formas en que pi nos ayuda a explorar el espacio.

TAGS: Pi, Pi Day, Dawn, Voyager, Engineering, Science, Mathematics

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NASA/JPL Edu supports science, technology, engineering and mathematics (STEM) education by providing NASA and JPL activities, resources and materials to educators and students in grades K-12 as well as STEM internships and research opportunities to higher education students and faculty.

A NASA space telescope mission is giving astronomers a whole new way to peer into the universe, allowing us to uncover long-standing mysteries surrounding objects such as black holes. Find out how it works and how to engage students in the science behind the mission.

Some of the wildest, most exciting features of our universe – from black holes to neutron stars – remain mysteries to us. What we do know is that because of their extreme environments, some of these emit highly energetic X-ray light, which we can detect despite the vast distances between us and the source.

Now, a NASA space telescope mission is using new techniques to not only scout out these distant phenomena, but also provide new information about their origins. Read on to learn how scientists are getting exciting new perspectives on our universe and what the future of X-ray astronomy holds.

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How They Did It
How It Works
Why It's Important
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How They Did It

In 2021, NASA launched the Imaging X-Ray Polarimeter Explorer, or IXPE, through a collaboration with Ball Aerospace and the Italian Space Agency. The space telescope is designed to operate for two years, detecting X-rays emitted from highly energetic objects in space, such as black holes, different types of neutron stars (e.g., pulsars and magnetars) and active galactic nuclei. In its first year, the telescope is focusing on roughly a dozen previously studied X-ray sources, spending hours or even days observing each target to reveal new data made possible by spacecraft's scientific instruments.

IXPE isn't the first telescope to observe the universe in X-ray light. NASA's Chandra X-ray Observatory, launched in 1999, has famously spent more than 20 years photographing our universe at a wavelength of light exclusively found in high-energy environments, such as where cosmic materials are heated to millions of degrees as a result of intense magnetic fields or extreme gravity.

Using Chandra, scientists can assign colors to the different energy levels, or wavelengths, produced by these environments. This allows us to get a picture of the highly energetic light ejected by black holes and tiny neutron stars – small, but extremely dense stars with masses 10-25 times that of our Sun. These beautiful images, such as from Chandra’s first target, Cassiopeia A (Cas A for short), show the violent beauty of stars exploding.

A blue halo of squiggly lines surrounds an explosion of colors extending out from the center of the supernova. Closest to the center is a circular splatter of orange surrounded by green and yellow and finally a hazy purple.

This image of the supernova Cassiopeia A from NASA’s Chandra X-ray Observatory shows the location of different elements in the remains of the explosion: silicon (red), sulfur (yellow), calcium (green) and iron (purple). Each of these elements produces X-rays within narrow energy ranges, allowing maps of their location to be created. Image credit: NASA/CXC/SAO | › Full image and caption

While Chandra has earned its name as one of “The Great Observatories,” astronomers have long desired to peer further into highly energetic environments in space by capturing them in even more detail.

IXPE expands upon Chandra’s work with the introduction of a tool called a polarimeter, an instrument used to understand the shape and direction of the light that reaches the space telescope's detectors. The polarimeter on IXPE allows scientists to gain insight into the finer details of black holes, supernovas, and magnetars, like which direction they are spinning and their three-dimensional shape.

A blue halo of squiggly lines surrounds a fuzzy donut-shaped haze of magenta with splatters of blue and white throughout.

This image of Cassiopeia A was created using some of the first X-ray data collected by IXPE, shown in magenta, combined with high-energy X-ray data from Chandra, in blue. Image credit: NASA/CXC/SAO/IXPE | › Full image and caption

While scientists have just begun putting IXPE's capabilities to use, they're already starting to reveal new details about the inner workings of these objects – such as the magnetic field environment around Cas A, shown in a newly released image.

The supernova remnant is shown as a blob of blue with swirls of brighter blues and large splatters of white. Dashed lines on top of the image flow from the center outward. Dividing the supernova and lines into quarter sections of a circle, the top right section has lines that flow directly northeast. The section at the bottom right has lines that flow nearly southeast but curve northwards slightly The section at the bottom left has lines that flow straight up from the bottom edge of the supernova, curve around the center and then flow back down. And the section at the top left has lines that flow from the center directly west, others that curve around the center and flow diagonally northwest and others that flow from the center to the north. Small sections of the lines are highlighted in green at the 1 o'clock, 2 o'clock, 4 o'clock, 7 o'clock and 11 o'clock portions of the supernova.

The lines in this newly released image come from IXPE measurements that show the direction of the magnetic field across regions of Cassiopeia A. Green lines indicate regions where the measurements are most highly significant. These results indicate that the magnetic field lines near the outskirts of the supernova remnant are largely oriented radially, i.e., in a direction from the center of the remnant outwards. The IXPE observations also reveal that the magnetic field over small regions is highly tangled, without a dominant preferred direction. Observations such as this one can help scientists learn how particles shooting out from supernovae interact with the magnetic field created by the explosion. Image credits: X-ray: Chandra: NASA/CXC/SAO; IXPE: NASA/MSFC/J. Vink et al. | + Expand image | › Full image and caption

“For the first time, we will use every collected photon of light to tell us about the nature and shapes of objects in the sky that would be dots of light otherwise,” says Roger Romani, a Stanford professor and the co-investigator on IXPE.

How It Works

Generally, when light is produced, it is what we call unpolarized, meaning that it oscillates in every direction. For example, our Sun produces unpolarized light. But sometimes, light is produced in a highly organized fashion, oscillating only in one direction. In astronomy, this arises when magnetic fields force particles to incredibly high speeds, creating highly organized, or polarized, light.

This is what makes objects like the supernova Cas A such enticing targets for IXPE. Exploded stars like Cas A generate massive energetic waves when they go supernova, giving scientists a view of how particles shooting out at immense speeds interact with the magnetic fields from such an event. In the case of Cas A, IXPE was able to determine that the x-rays are not very polarized, meaning the explosion created very turbulent regions with multiple field directions.

While the idea of polarized or organized light may sound abstract, you may have noticed it the last time you were outside on a sunny day. If you’ve tried on a pair of polarized sunglasses, you may have noticed that the glare was greatly reduced. That’s because as light scatters, it bounces off of reflective surfaces in all directions. However, polarized lenses have tiny filters that only allow light coming from a narrow band of directions to pass through.

The polarimeter on IXPE works in a similar way. Astronomers can determine the strength of an object's magnetic field by using the polarimeter to measure how much of the light detected by the telescope is polarized. Typically, the more polarized the light the stronger the magnetic field at the source.

Astronomers can even go a step further to measure the direction this light is oscillating by measuring the angle of the light that reaches the telescope. Because the polarized light leaves the source in a predictable fashion – namely perpendicular from its magnetic field – knowing the angle of the oscillating light provides information about the axis of rotation and potentially even the surface structure of objects such as neutron stars and nebulae.

Side by side animations showing a rope moving from side to side through an open window and a rope moving up and down through an open window. As the window closes, fewer of the waves in the rope moving up and down make it through the window whereas the rope moving from side to side is undisturbed.

In this demonstration, the rope represents light waves and the open window represents a polarimeter. Depending on the angle of the light waves (rope), more or less information makes it through the polarimeter (window) the narrower it is. By measuring the amount of light received through the polarimeter, IXPE can determine the angle and the polarization of the light. Image credit: NASA/JPL-Caltech | + Expand image

Imagine, for example, that you were holding one end of a piece of rope secured to an object at the other end. If you swung the rope side to side to make horizontal waves, those waves would be able to make it through a narrow target like a window. If you started to shut the window from the top, narrowing the opening, the waves could conceivably still make it through the opening. However, if you made veritcal waves by waving the rope up and down, as the window closed, fewer and fewer waves would make it through the opening. Likewise, by measuring the light that makes it through the polarimeter to the detector on the other side, IXPE can determine the angle of the light received.

To collect this light, IXPE uses three identical mirrors at the end of a four meter (13 foot) boom. The light received by IXPE is carefully focused on the spacecraft’s polarimeter at the other end of the boom, allowing scientists to collect those crucial measurements.

During the IXPE launch broadcast, commentators discuss the components of the spacecraft and how it measures polarization. | Watch on YouTube

Why It's Important

Building on Chandra's observations from the past two decades, IXPE's novel approach to X-ray science is pulling the curtain back even farther on some of the most fascinating objects in the universe, providing first looks at how and where radiation is being produced in some of the most extreme environments in the universe. IXPE's measurements of Cas A are just the beginning, with even more mysterious targets ready to be explored.

Take it from Martin Weisskopf, the principal scientist on IXPE and project scientist for Chandra, who has spent his 50-year career working in X-ray astronomy, who says, “IXPE will open up the field in ways we’ve been stuck only theorizing about."

Teach It

Explore more on how NASA uses light to map our universe, and dig deeper into some of the celestial features it allows to study, such as blackholes and neutron stars.

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