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What's So Special About Pi?

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The mathematical constant π, one of the most significant and well-known numbers in mathematics, is celebrated every year around the world on March 14th. π roughly equals 3.14 -- hence the choice of 3/14 for its commemorative holiday -- but it's best known as the ratio between a circle's circumference and its diameter. Pi Day is often observed by eating pie, so most people probably know it best as a day for delicious baked goods. But why celebrate π anyway? As it turns out, π is an incredibly special number with a lot of interesting properties, and it pops up in many, many formulas that dictate how our universe works and explain some of the deepest relations in mathematics. Here are some of our favorite unusual facts about π:

π is everywhere, but we don't know all its digits.

If you have a circular object handy, here's a quick experiment that can help you approximate π. Take your circle -- say, an empty pie tin -- and mark it on the rim with a little dot. Then, stand your pie tin on its edge so that the dot touches the floor, and then make another dot on the floor where the tin and the floor meet. Roll your pie tin in a straight line until the the dot on the rim meets the floor again -- that is, one full revolution -- and mark the floor again where it meets your circle. Congratulations: the distance between those two points on your floor, divided by the diameter of your pie, is π.

Measuring pi

Here's a visualization of your experiment. Image via Wikimedia.

 

Of course, you haven't measured π exactly: if you didn't roll your circle in a straight line, if your measuring stick isn't precise enough, or if your tin got a little squished, it can totally throw off your experiment. Even the width of the dot you make will introduce some error into your measurement. You've probably only got π to within one decimal place, or if you're very precise, maybe two.

Because of its many applications to physics, engineering, and architecture, mathematicians have long sought the value of π. However, since π is irrational, we cannot express it exactly as a fraction, and as a decimal, its expansion is infinitely long with no repeating pattern. Today, we've used complex algorithms to compute its value to over 20 trillion digits, but before the advent of high speed computers, we could only find about 1,000 digits of π. Luckily, for the purpose of scientific calculations, we only need about 30 digits of π or less -- but even still, we're unable to definitively write its exact value as a fraction or a decimal.

π isn't just irrational: it's transcendental.

Here's another headscratcher: let's say you use a compass to draw a circle. Is it possible, using that compass and a straightedge, to draw a square of the same area? This problem was known to the Greeks, and we call it "squaring the circle."

Squaring the circle

Can you "square the circle?" Image from Wikipedia.

 

If you're running to get your compass and straightedge, we can save you a little time and give you a sneak peek at the answer: it's impossible! As we said before, π is irrational, meaning it can't be expressed as a simple fraction. However, the reason we can't square the circle is that π is a very special type of irrational number called a transcendental number. The definition of transcendental numbers is hard to explain, but they have an interesting property: even though there are infinitely many transcendental numbers -- in fact, almost all numbers are transcendental -- it's very hard to prove that a number is transcendental, and therefore mathematicians have a lot of trouble finding transcendentals. π is one, as is e, the base of the natural logarithm, but there are many, many more, and the question of finding them is sure to puzzle mathematicians for years to come.

It pops up in some interesting sums:

In 1644, Italian mathematician Pietro Mengoli posed an interesting problem. What is the sum of the reciprocals of the perfect squares (1/1, 1/4, 1/9, 1/16, 1/25, etc.) all the way up to infinity? Turns out, this is equal to the square of π divided by 6:

Basel Problem
This and all formulas via Wikipedia.

What about the alternating reciprocals of all the odd numbers? That is, 1/1 - 1/3 + 1/5 - 1/7 + 1/9, and so on? Well, that's π divided by 4:

Leibniz formula for pi

And more places you might not expect.

In addition to being expressible as some beautiful infinite sums, π is also written as some interesting infinite products as well, such as this funky looking formula:

Vieta's formula

In the following product, note that the numerators are all the primes except 2, and the denominators are the closest multiples of 4 to each numerator:

 Euler product

And, like all irrationals, π can be expressed as an infinite continued fraction -- that's like a fraction on steroids. Check it out:

Continued fraction of pi.

It's a crucial part of one of the most beautiful theories in math.

Remember that other transcendental number e we talked about before? Turns out that the brilliant mathematician Leonhard Euler realized that e and π are related in one of the most deep identities in all of mathematics:

Euler's formula

Where i is an imaginary number, the square root of -1. How did Euler manage to relate ei, and π -- three numbers which, at first glance, seem to have little to do with each other -- to 0 and 1? Explaining Euler's famous identity is a bit beyond the scope of this post, which you can learn all about in a complex algebra class. But this mysterious and beautiful formula is one of the deepest mathematical expressions of all time, and is famous in the math community for its elegance and profundity.

It contains every finite number in its decimal expansion... maybe.

Some people have said that the digits of π contain all the secrets of the universe and, in a certain sense, they might be right. Some numbers, called normal numbers, are infinitely long and contain every digit -- and finite string of digits -- in equal measure, meaning that any string of numbers is just as likely to be found in their endless decimal tails as any other string of numbers. We don't know for sure if π is normal, but many mathematicians believe it is, and if this is proven to be so, then π will indeed be shown to contain every finite number. So could you find your birthday or your phone number in the digits of π? Sure -- but it would take you an awfully long time to look.

And maybe, we shouldn't even be using π at all

There's a community of mathematicians who feel that π isn't actually a sensible choice for the circle constant, and instead we should be using the ratio of a circle's circumference to its radius -- that is, 2π, or τ. Part of the justification for using τ is that 2π comes up in a lot of formulas in physics and math, so converting 2π to τ might make things a bit easier. However, τ has not caught on with most mathematicians, and π continues to be used and taught in math classes at all levels. But if your tastes tend towards τ for whatever reason, feel free to skip Pi Day and celebrate your circle constant on 6/28.

Got any other favorite π related facts? Shout them out in the comments!

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