Pi: The Most Important Number in the Universe?

Transcript from a lecture by Professor Edward B. Burger, Ph.D.

One of the most important numbers in our universe is the number Pi or π. While the origins of π are not known for certain, we know that the Babylonians approximated π in base 60 around 1800 B.C.E.

illustration featuring the symbol for the number pi

A Definition

The definition of π centers around circles. In fact, it’s the ratio of the circumference of a circle to its diameter—a number just a little bit bigger than 3. We’ll explore humankind’s odyssey to compute, approximate, and understand this enigmatic number, π. These attempts throughout the ages truly transcend cultures.

drawing of Johann Lambert, the Swiss mathematician was the first to prove that pi was an irrational number.
Johann Lambert, the Swiss mathematician was the first to prove that pi was an irrational number.

The constant π helps us understand our universe with greater clarity. In fact, the definition of π inspired a new notion of measurement of angles, a new unit of measurement. This important angle measure is known as “radian measure” and gave rise to many important insights in our physical world. As for π itself, Johann Lambert showed in 1761 that π is an irrational number, and later, in 1882, Ferdinand von Lindemann proved that π is not a solution to any polynomial equation with integers. However, many questions about π remain unanswered.

Learn More: Geometry—Polygons and Circles

Experimenting with Pi

diagram illustrating the relationship between circumference and diameterAny discussion of the origins of pi must begin with an interesting experiment involving circles that we can all try. Take any circle at all and take the length of the circumference—which is the length around—and measure it in terms of the diameter, which is the length across. You will end up with three diameters and just a little bit more, and if you look really closely, it’s actually a little bit more than 1/10 of the way extra. So, this little experiment shows us that that ratio of the circumference to the diameter is going to be a number that’s around, or a little bit bigger than, 3.1. No matter what the size of the circle is, the circumference is slightly greater than three times its diameter.

We’ve given this fixed, constant value a name, and we call it π. So, let’s say this more precisely. The number π is defined to equal the ratio of the circumference of any circle to its diameter across. This ratio is constant. No matter what size of circle we try this with, that number will be always the same. It begins 3.141592653589, and it keeps going.

painting portrait of William Jones, the Welsh mathematician who first used the symbol for pi
William Jones, the Welsh mathematician

We’ll first take a look at the early history of π and the ancient struggle to pin down its exact value—first, a word about the symbol π. We use the Greek letter π for this number, because the Greek word for “periphery” begins with the Greek letter π. Now, the periphery of a circle was the precursor to the perimeter of a circle, which today we call circumference. The symbol π first appears in William Jones’s 1709 text A New Introduction to Mathematics. The symbol was later made popular by the great 18th-century Swiss mathematician Leonhard Euler around 1737.

Learn More: Number Theory—Prime Numbers and Divisors

From Babylon to the Bible

Moving from its name to its value, we have evidence that the Babylonians approximated π in base 60 around 1800 B.C.E. In fact, they believed that π = 25/8, or 3.125—an amazing approximation for so early in human history. The ancient Egyptian scribe Ahmes, who is associated with the famous Rhind Papyrus, offered the approximation 256/81, which works out to be 3.16049. Again, we see very impressive approximation to this constant. There’s even an implicit value of π given in the Bible. In 1 Kings 7:23, a round basin is said to have 30-cubit circumference and 10-cubit diameter. Thus, in the Bible, implicitly it states that π equals 3 (30/10).

The Indian mathematician and astronomer Aryabhata approximated π, in c. 500 CE, with the fraction 62,832/20,000, which is 3.1416—a truly amazing estimate.

Not surprisingly, as humankind’s understanding of number evolved, so did its ability to better understand and thus estimate π itself. In the year 263, the Chinese mathematician Liu Hui believed that π = 3.141014. Approximately 200 years later, the Indian mathematician and astronomer Aryabhata approximated π with the fraction 62,832/20,000, which is 3.1416—a truly amazing estimate. Around 1400, the Persian astronomer Kashani computed π correctly to 16 digits.

How to Measure Angles with Pi

Let’s break away from this historical hunt for the digits of π for a moment, and consider π as an important number in our universe. Given π’s connection with measuring circumferences of circles, scholars were inspired to use it as a measure of angle distance. Now, let’s consider a circle having radius 1. Radius is just the measure from the center out to the side. It’s half the diameter.

The traditional units for measures of angles are, of course, degrees. With degrees, one complete rotation around the circle has a measure of 360 degrees, which, by the way, happens to approximately equal the number of days in one complete year and which might explain why we think of once around as 360.

This is a transcript from the video series Zero to Infinity: A History of Numbers. It’s available for audio and video download here.

Instead of the arbitrary measure of 360 to mean once around the circle, let’s figure out the actual length of traveling around this particular circle, a circle of radius 1, once around. So what’s the length? What’s the circumference of that? Well, let’s see. If we have a radius of 1, then our diameter is twice that, 2, and so we know that the once-around will be 2 times π, because the circumference is π times the diameter.

Once around will be 2π. One full rotation around, which is an angle of 360 degrees, would be swept out with circumference length of 2π in this particular circle. In fact, what would be halfway around? Well, that would be 180 degrees, and we would sweep out half of the circumference, which, in this case, would be π. Ninety degrees would sweep out a quarter of the circle, and for this particular circle, that would have length π/2, or one-half π.

We’re beginning to see that every angle corresponds to a distance measured partway or all the way around this particular circle of radius 1. In other words, for any angle, we can measure the length of the arc of this circle swept out by that angle.

This arc length provides a new way of representing the measure of an angle, and we call this measure of angles “radian measure.” So, for example, 360 degrees = 2π radians, those are the units; 180 degrees equals π radians, and 90 degrees would equal π/2 radians. Remember, all these measures are always based on a special circle that has radius 1.

Learn More: Geometry and the Transformation Tactic

Radian Measures and the Power of Pi

It turns out that this radian measure is much more useful in measuring angles for mathematics and physics than the more familiar degree measure. This fact is not too surprising, since radian measure is naturally connected through the circumference length with the angle, rather than the more arbitrary degree measure that has no mathematical underpinnings, but just represents an approximation through a complete year.

The term radian first appeared in print in the 1870s, but by that time, great mathematicians, including the great mathematician Leonhard Euler, had been using angles measured in radians for over a hundred years. Well, beyond angle measures, π is central in our understanding of our universe.

In fact, the number π appears in countless important formulas and theories, including the Heisenberg uncertainty principle and Einstein’s field equation from general relativity. So it’s a very, very important formula, a very important number indeed.

From the lecture series Zero to Infinity: A History of Numbers
Taught by Professor Edward B. Burger, Southwestern University