How Many Decimal Digits of Pi Do We Really Need?

The decimal digits of Pi have no known ending: 3.14159265358979323846…

Pi is a mathematical constant that is defined as the ratio of the circumference of a circle to its diameter (see image below for a graphical illustration). In mathematics, this is known as an irrational number, which is a fancy way of saying that it cannot be represented accurately using fractions of integers. Pi’s origins trace back to the Greek mathematical genius Aristotle when he tried to use polygons to determine the circumference of a circle. He eventually narrowed down the value of Pi between 3.1408 and 3.4129. All this happened in 250 B.C. Fast forward a few years to 2021, and throw in whole generations of mathematical geniuses in between 250 B.C and 2021 A.D, we have managed to precisely calculate Pi up to well over 50 trillion digits after the decimal point. And the end is still not in sight yet.

All this muscle-flexing around the topic of calculating Pi digits brings us to the following questions: How many digits do we really need in real-life applications? Is it really necessary to calculate Pi precisely up to 50 trillion digits or more? In this article, I’ll be trying to answer these questions by working out practical examples as well as by touching upon the industrial standards around the usage of Pi.

This essay is supported by Generatebg

What Does My Calculator Use?

Once upon a time, I used to work as an engineer. Engineers are creatures who are equipped with things known as ‘scientific calculators’. These are calculators that are more capable than the normal ones, including some minor programming capabilities. I am the proud owner of the one you see in the image below: a CASIO fx-991ES PLUS. It is a handy little thing. Such a scientific calculator comes with Pi stored in as a default mathematical constant. And as you can see from the image, my scientific calculator uses only the first 9 digits of Pi after the decimal point. To be precise, the ninth digit is rounded up. Does this mean that even engineers don’t need more than 9 digits of Pi after the decimal point? Why don’t we find out by trying stuff out on our own?

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Using Decimal Digits of Pi Practically

Let us say that we are interested in calculating the circumference of a circle that is 1 Kilometre (Km) in diameter (D). To see how many digits of Pi we need to accurately calculate the circumference, let us consider two versions of Pi:

Pi_Short = 3.141593 (6 digits after the decimal place)

Pi_Long = 3.141592653589793 (15 digits after the decimal place)

The formula for the circumference is Pi*D (where D is the diameter of the circle). When we calculate the circumference of the circle using the two versions of the Pi we considered, we get the following:

Based on this analysis, we can see that for a circle that is 1Km in diameter, the difference between the two versions of the Pi is less than 0.04 cm. What if we consider a super-long version of Pi? Let’s do just that:

Pi_Super_Long = 3.1415926535897932384626433 (25 digits after the decimal place)

When we use this version of Pi to calculate the circumference, we get the following:

We originally noticed that we improved accuracy by less than 0.04 cm when we switched from Pi_Short to Pi_Long. In order to determine how much more accuracy we gained by switching from Pi_Long to Pi_Super_Long, we need to subtract (C1-C2) from (C1-C3). If we do that, we get the following:

As you can see, we have gained an extra accuracy of 0.000001021 cm by going through all this trouble.

In real-life applications, when we consider a scale of 1km diameter for a circle, errors of the order of 1cm or lesser are more than acceptable in most cases. So, I would say that 6 decimal digits of Pi are more than sufficient for most real-life applications. But who am I to say that? Let’s look at what industry standards have to say next.

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Industry Standards on Decimal Digits of Pi

When it comes to industries that are sensitive to calculation accuracy, the space industry is right at the top of the list. So, I ask the question: how many digits of Pi does NASA use? The answer: NASA uses the first 15 digits of Pi (our Pi_Long from before). NASA’s interest seems to be up to a circular diameter of 125 billion kilometres. And if they use the first 15 decimal places of Pi, the error comes to around 3.81 cm (in the context of 12500000000000000 cm). So, NASA is okay with designing systems that consider this level of tolerance rather than loading their computers with higher digits of Pi.

Another industry that is very sensitive to calculation accuracy is the motorsport industry (where I used to work). The Federation Internationale de l’Automobile (FIA), the governing body of Formula 1, uses the first four digits of Pi. That’s right, back here on earth, F1 cars run fine with the first four digits of Pi.

Why Calculate Over 50 trillion Decimal Digits of Pi?

If NASA needs just the first 15 decimal digits of Pi and Formula 1 needs just the first 4, why do we hear people (or computers) calculating over 50 trillion digits of Pi? The answer lies in the human obsession with Pi as a mathematical constant. Computing higher digits of Pi requires complex mathematical procedures and algorithms. Academic and professional teams use Pi as an excuse to test and prove their latest clever algorithms and supercomputer architectures. Other than that, for practical applications, we currently do not need any more than the first 15 digits of Pi.

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Further reading that might interest you: Why Is 3 A Special Denominator In Division? and What Exactly Is Zero Raised To The Power Zero?