By Edward B. Burger, Ph.D, Southwestern University
One of the most important numbers in our universe is the number Pi or π. Explore humankind’s odyssey—attempts throughout the ages that truly transcend cultures—to compute, approximate, and understand this enigmatic number.
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. The definition of π centers around circles. It’s the ratio of the circumference of a circle to its diameter—a number just a little bit bigger than three.
The constant π helps us understand our universe with greater clarity. The definition of π inspired a new notion of the 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.
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Experimenting with Pi
Any discussion of the origins of pi must begin with an 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 closely, it’s a little bit more than 1/10 of the way extra. This experiment shows us 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.
This is a transcript from the video series Zero to Infinity. Watch it now, on The Great Courses.
This fixed, constant value was given a name, and we call it π. How do we say it 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 the circle we try this with, that number will be always the same. It begins 3.141592653589, and it keeps going.
The symbol π comes from the Greek letter π, because the Greek word for “periphery” begins with the Greek letter π. 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, and the symbol was later made popular by the great 18th-century Swiss mathematician Leonhard Euler around 1737.
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From Babylon to the Bible
Moving from its name to its value, evidence exists 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 an 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, it implicitly states that π equals 3 (30/10).
Not surprisingly, as humankind’s understanding of numbers 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 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. 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 happens to approximately equal the number of days in one complete year and which might explain why we think of once around as 360.
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. What’s the length and the circumference of that? 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. Halfway around 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.
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We’re beginning to see that every angle corresponds to a distance measured part- 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.” 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.
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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 unsurprising. Radian measure is naturally connected through the circumference length with the angle, rather than the more arbitrary degree measure that has no mathematical underpinnings. It 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.
The number π appears in countless important formulas and theories, including the Heisenberg uncertainty principle and Einstein’s field equation from general relativity. It’s an important formula and number across the world.