The ratio between a circle’s circumference and its diameter is always constant, no matter how big or small the circle gets. This ratio has been known in some form for almost 4000 years. This number has appeared in the work of ancient Greek, Babylonian, Indian, and Chinese mathematicians. It was even used to build the pyramids of ancient Egypt.
THE LAST DIGIT OF PI
Since decimals were nonexistent, mathematicians like Archimedes in 250 BC first estimated the number by inscribing larger and larger polygons into a circle to the point where the shape became nearly indistinguishable from it. By the year 1400 it had been estimated to as far as 10 decimal places. Some thinkers, like mathematician Ludolph Van Ceulen, were so obsessed with Pi that they made calculating Pi their life’s work. Van Ceulen had managed to work out 35 digits of Pi by the end of his life and had them inscribed on his tombstone. By the 20th century the ENIAC computer was able to calculate pi to 2000 digits. So when did we figure out the actual value instead of just estimating it?
The answer, dear reader, is never.
It just goes on and on, an irrational number, one that can never be expressed as the ratio between 2 integers. You can come close, but no matter how precise the fraction is… you will always be a small percentage off, adding more and more digits after the decimal. Instead of writing out all these digits every time, we’ve just come to refer to it using the greek letter “π”.
Pi has long tormented the greatest minds of the ancient world and drove mathematicians to madness. Over 2000 years ago, the greek mathematician Pythagoras and his followers saw numerical relationships as the key to understanding the secrets of reality. But in their investigation of geometric shapes, one of Pythagoras’s students discovered that some very important ratios, such as π and the square root of 2, could not be expressed in simple numbers. They had no repeating patterns and were seemingly infinite, a concept which terrified the academics of his time. Legend has it that the discovery of these numbers was so disturbing that one member of the Pythagorean cult, Hepasus, was killed for divulging their existence. The discoveries of Hepasus so enraged Pythagoras, who held that all things were made up of ratios (numbers being the basis of his ontology as well), that he murdered the student and swore the rest of the Pythagoreans to secrecy.
Long ago infinity was too wild to be tamed by mathematics, to wild to be used in equations. Aristotle believed that time was eternal and could go on forever but he refused to accept that the universe was infinite sized. He believed the earth was at the center of the universe, however, without an end, there can be no middle. So he banned infinity from his mathematics. To Aristotle, infinity evoked the formless chaos from which the universe was thought to have emerged, a primordial state with no natural laws or limits, devoid of all form or content. Aristotle resererved the status of “infinity” only for the divine, but now, we associate it with one of the most mysterious numbers in the universe.
So is the number Pi infinite?
Not necessarily. Pi is only categorized as a “possible infinity”, meaning we won’t actually know until we reach the end. It may just be a really big finite number, but we’ll never know until we actually calculate it in its entirety. But will that eventually be possible with our current technology?
SUPERCOMPUTERS AND CURRENT PROGRESS
We have not yet calculated Pi, but modern supercomputers are currently able to work out a mind-boggling 10 Trillion digits. Our quantum computers are even able to calculate it up to 2 quadrillion digits! It may not be complete, but what we have so far is good enough for physicists. It is more than enough of the digits they need to measure the circumference of the observable universe. The number we’ve calculated is overkill. As a matter of fact, we could be as accurate as the length of 1 hydrogen atom if we just used the 39 digits of Pi. Even engineers working on the orbits of satellites you only need a small handful of digits.
If that’s the case, why do we even bother calculating Pi to 2 quadrillion digits?! It doesn’t seem to have any particular use or day-to-day application for scientists and engineers. For most numerical calculations involving π, a handful of digits provide sufficient precision. Despite this, people have worked strenuously to compute π to quadrillions of digits. This effort may be partly ascribed to the human compulsion to break records, and such achievements with π often make headlines around the world.
In a way, the true pursuit of calculating Pi is simply just meant to test the power of our newest supercomputers and numerical analysis algorithms. We do it for the glory. It is a way to demonstrate how capable they are and how far we’ve come as a species. In futurology, it could be taken as a way to measure how advanced a civilization is in terms of computing power, perhaps an informatic analog to the Kardashev scale.
THE COMING SINGULARITY: WHAT CAN PI TELL US ABOUT THE FUTURE?
Revisiting the great mathematician, Archimedes, we realize just how fast the progress of our civilization is accelerating.
“And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about.” – 1 Kings 7:23
Since the dawn of biblical times, mankind possessed very little more than the first digit. Many civilizations 4000 years ago just used the number “3”, a single digit. 2000 years later, Archimedes, using more advanced techniques, was able to estimate up to 5 digits and 1000 years after that those digits doubled. A few hundred years later, the first early vacuum tube computers of the 1940s were able to give us thousands of digits. In mere decades after, millions of digits.
Now every few years we get the next paradigm shift in information technology, an exponential trend accelerating even beyond the patterns predicted by Moore’s law. Groundbreaking computational methods are discovered each month, it is becoming harder and harder for the average person to keep up with our runaway technological growth. Billions of digits of Pi, trillions, quadrillions… a Googolplex?
The power to compute pi, like many other technical aspects of our society, is converging toward a familiar horizon where our ability to anticipate the future breaks down. A point in human history called “the technological singularity”. I suppose Pi, even if it’s not as mystical as the ancients believed it to be, can be a way to track our progress towards increasingly better technologies. I can’t reveal to you the nature of infinity or what the future will bring, but perhaps one day, with the right technology, maybe we will be able to reach the last digit…
I leave you with this masterpiece: https://www.youtube.com/watch?v=E36qMxXGo3A