Everything You Need to Know About Pi

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Podcast Transcript

Every year on March 14, the world celebrates one of the most important mathematical constants: pi.

It is a number which appears all over nature, even in places you wouldn’t expect it. It is also a number that has been known, or at least had been approximated, by civilizations for thousands of years.

We might as well start the discussion of pi with its definition. Pi is nothing more than the ratio of the circumference of a circle to its diameter.

On one hand, it is extremely simple, but at the same time, it is devilishly complicated.

The problem is that the circumference of a circle isn’t divided evenly by its diameter.

Finding out exactly what this ratio was has been the subject of inquiry for centuries.

Almost every early civilization knew about the importance of this ratio. How they differed is in how they approximated the number, and the methods they used to figure it out.

A Babylonian tablet that dates back to about 1700 BC has an approximation of pi of 3.125. Earlier Babylonian approximations just used three.

An Egyptian text, known as the Rhind Papyrus, had a value that works out to 3.1605. Some people who have analyzed the Great Pyramid have determined that the Egyptians used the ratio of 22/7 as an approximation for pi.

Ancient Indian mathematicians wrote in the Shatapatha Brahmana that pi was approximately 339/108 which works out to 3.139.

The techniques used to find these approximations in the ancient world were primary geometric and physical. They would use either a compass and a straightedge or physically create a circle and measure.

Believe it or not, these ancient measurements were not that bad. Given the type of engineering they were doing and the level of precision involved, they were able to get pi to within one percent.

However, what is good enough for hand-work construction doesn’t cut it for pure mathematics.

Much of the story of pi from here on out is all about finding better ways to calculate the number, and ever greater precision in the number of digits it can be calculated.

The first big step in calculating pi was independently discovered by both Chinese and Greek mathematicians.

The Chinese mathematician Liu Hui (lou way) and the Greek mathematician Archimedes both realized that you could approximate the circumference of a circle by creating ever-larger polygons inside of it.

For example, a hexagon inside a circle is clearly smaller, but an octagon is bigger. A decagon would be even closer to a circle, and so on, and so on.

In the year 245, Liu Hui eventually calculated a polygon with 3,072 sides and came up with a value of ? of 3.1416.  Both of these techniques were early forms of integral calculus.

In 480, another Chinese mathematician named Zu Chongzhi used the same technique as Liu Hui and calculated a 12,288 sided polygon. His value of pi was correct down to seven decimals.

This was a huge leap in calculating pi and one which would stand for 800 years.

Liu Hui’s algorithm worked, but there was a practical limit to how many sides of a polygon you could measure….but there were still a few more decimal places to be had using this method.

In 1424, the Persian astronomer Jamsh?d al-K?sh? calculated pi to 16 digits by calculating the equivalent of a polygon with 30 octillion sides.

It is amazing that something so large can only get you to 16 digits, but that is the reality of pi.

Here I should mention just how good 16 digits is. The head engineer at NASA has publicly stated that they only need to use 15 decimal points of pi when they are doing calculations. With that level of precision, you could calculate a circle with a circumference of 78 billion miles, and have a margin of error with the length of your little finger.

In 1596, Dutch mathematician Ludolph van Ceulen managed to calculate pi to 20 digits, and later 35 digits, using the polygon technique.

This was pretty much the limit for using polygons as it was simply too hard to calculate.

The next big innovation in calculating pi was the use of infinite series. Infinite series are, as the name suggests, adding up an infinite number of fractions. The more you add up, the close it converges to the number you are trying to approximate.

This technique is much easier to calculate than trying to determine the area of ever-larger polygons.

For example, the co-inventor of calculus Gottfried Leibniz came up with a series that converges to pi.

Basically for times one over every odd number, with alternating adding and subtracting each term.

There are actually many different infinite series that converges to some multiple of pi. The way they differ is in how quickly they converge. The development of calculus led to an exploding in these infinite series.

In 1699, English mathematician Abraham Sharp calculated pi using a modification of the Leibniz series out to 71 digits.

Here I should note that at this point, no one was calling this number, pi.

The first use of the Greek letter pi to represent the ratio of a circle’s circumference to its diameter was by the Welsh mathematician William Jones in 1706.

71 digits of pi are far more than anyone could ever possibly use. To put it into perspective, if you use 40 digits of pi, you could calculate a circle the size of the observable universe with a margin of error the size of a hydrogen atom.

As pi was getting calculated to ever more precise values, mathematics advanced and mathematicians started asking questions about it.

For example, did the digits of pi ever repeat? Could pi be represented in some sort of polynomial equation?

Several attributes of pi have been proven.

One of which is the fact that pi is a transcendental number. This means it is an irrational number that is not algebraic. For example, the square root of two is irrational but isn’t transcendental.

The fact that pi is transcendental was proven in 1882 by German mathematician, Ferdinand von Lindemann.

This actually resolved what was perhaps the longest unsolved problem in the history of mathematics: squaring the circle.

The very earliest mathematics was geometry done with a straightedge and a compass. There was a surprisingly large amount of mathematical proofs that could be done with such simple tools.

One problem that confounded everyone from Archimedes to Leonardo da Vinci was trying to create a circle with the exact same area as a square using a compass and a straightedge.

No one ever found out a way to do it, and it turned out that it was impossible, because pi is transcendental.

Likewise, it was proven that the numbers of pi never repeat, although there may be short segments of numbers that do repeat themselves.

The numbers also meet the criteria of being random.

Calculations of pi kept getting better. It passed 100 digits in 1706, 200 digits in 1844, and over 400 by the end of the 19th century.

It was the great Indian mathematician Srinivasa Ramanujan who created several rapidly converging infinite series which could increase the number of decimal points by eight at a time with every addition to the series.

This radically changed the ability to calculate pi.

In 1946, pi was calculated to 460 digits by hand, which was the end of hand calculation records.

After that, the computers started to take over. In fact, calculating pi was a way to test the performance of new computers.

In 1949, Levi Smith and John Wrench calculated pi to 1,100 digits using a desktop calculating device.

Just months later, one of the first electrical computers, ENIAC, calculated pi to 2,037 digits in 70 hours.

In 1961, an IBM 7090 calculated pi to 100,000 digits under 9 hours.

With ever more powerful computers and improved algorithms, the ability to calculate pi just exploded.

The 1 million digit threshold was crossed in 1973, 10 million in 1983, 100 million in 1987, and 1 billion in 1989.

As of the time of recording, the record for calculating the digits of pi was set in 2021 by a team at The University of Applied Sciences of Eastern Switzerland. They have calculated pi out to 66.8 trillion digits.

With so many random digits, memorizing pi has become a competitive activity. Memorizing pi even has its own name: Piphilology.

Most people can easily memorize pi out to 10 or 20 digits as it isn’t that harder than a phone or credit card number.

The current Guinness World Record for pi memorization is 70,000 digits. The feat was accomplished by Indian Rajveer Meena in 9 hours and 27 minutes in 2015.

Pi is a universal constant. If we should ever encounter an alien intelligence, they should be just as aware of pi as we are. However, it is entirely possible that if they contacted us, they wouldn’t let us know they were there by sharing pi with us.

Some mathematicians claim that it is because pi isn’t the number we should be using.

The reason is that you almost never encounter the diameter of a circle in mathematics.

What defines a circle is its radius. If the radius is the important measurement, why do we use the diameter?

The real number that we should care about, some suggest, would be the ratio of the circumference and the radius, which would be equal to two times pi.

This number, 6.28318…, has been dubbed Tau.

If you’ve ever studied a sine or cosine function, you’ll know it makes a complete cycle every 2pi. Likewise, if you’ve ever worked in radians, one complete circle is likewise 2pi.

If aliens send us a number that shows some universal mathematical constant, it might be tau instead of pi.

Likewise, I should really be doing an episode on Tau day, which takes place on June 28.

Even if the tau advocates are right, pi is still an important component of it, and pi will still be used as it has for over two thousand years, as one of the significant mathematical constants.