What do you know about pi?
Maybe you would say it’s a handy constant, helpful when trying to figure out a circumference or area. Maybe you’d say it’s irrational and transcendental. Maybe you’d say you know the first five digits, 3.1415 (or maybe you even know the first 10 or 15 or more). But whether you know the first 5 or the first 5000 digits, there are a lot of questions we can ask about the digits that make up pi.
We could wonder about whether a specific string of digits (such as your birthday) exists somewhere in pi or even if all possible finite strings of digits appear somewhere in pi.
The first 100 digits of pi have plenty of birthdays!
Happy birthday to everyone celebrating their birthday on pi day! Your birthday is underlined. We also found Leonhard Euler’s birthday, April 15, and highlighted it in green.
Mathematician Emmy Noether’s birthday, March 23, is also in pi! We highlighted it in purple.
Looking deeper into pi’s digits, we noticed mathematician Alberto Calderón’s birthday (September 14, 1920) in the 295-7th digits of pi, highlighted in yellow:
Even later in the sequence of pi, we even found mathematician Benjamin Banneker’s birthday (November 9, 1731) in the 3254-7th digits of pi, highlighted in blue:
The first 1000 digits of pi are here if you want to look for your birthday!
You could also ask what seems like a simple question: does the digit 7 appear infinitely often in pi? Probably yes, right? Would you expect 5's to show up more or less often than 7's, or equally often? Do you expect each digit to show up 1/10th of the time?
Amazingly, we don't know the answer to those questions, not for sure, anyway.
All these questions have to do with the sequence of digits that makes up this famous constant, and it turns out that, despite its fame, there’s still a lot to learn about the digit distribution of pi. The answers are wrapped up in mathematical concepts called normality and simple normality. In mathematics, simply normal means every digit appears equally often as we go out to infinity, and normal means that every possible finite string of digits appears equally often as we go out to infinity. To make this more formal a number is simply normal in an integer base b if its infinite sequence of digits is distributed uniformly in the sense that each of the b digit values has the same natural density 1/b. A number is said to be normal in base b if, for every positive integer n, all possible strings n digits long have density 1/bn.
All these questions have to do with the sequence of digits that makes up this famous constant, and it turns out that, despite its fame, there’s still a lot to learn about the digit distribution of pi. The answers are wrapped up in mathematical concepts called normality and simple normality. In mathematics, simply normal means every digit appears equally often as we go out to infinity, and normal means that every possible finite string of digits appears equally often as we go out to infinity. To make this more formal, a number is simply normal in an integer base b if its infinite sequence of digits is distributed uniformly in the sense that each of the b digit values has the same natural density 1/b. A number is said to be normal in base b if, for every positive integer n, all possible strings n digits long have density 1/b^n.
As an example, the number 0.12345678901234567890123… (where we just repeat all ten digits in consecutive order) is simply normal, but not normal. Each digit appears equally often as we go out to infinity, but the string 11 never appears!
To get a better picture of how the digit distribution looks for pi in particular, we have color coded the digits and plotted the digit distribution for the first 10, then 100, 1000, and finally 10,000 digits of 𝛑. We see as the number of digits we are looking at increases, the slices start to look more similar in size. If this holds true as we head out to infinity then that would mean that pi is simply normal in base 10.
When thinking about math, it is often helpful to think about statements in terms of stronger or weaker. Stronger statements give us more information, but they are also harder to prove, while weaker statements are easier to prove but don’t tell us as much about the subject or topic we are interested in. You will notice that saying a number is normal gives us a lot more information about its digit distribution than simply normal does. Normality tells us something about how often all possible finite strings appear, instead of just how often each digit occurs. That’s a lot more information!
If every possible finite string occurs at least once, well then we can certainly find our birthday. We can find everyone’s birthday. We could find the whole works of Shakespeare encoded in pi — or any book ever written for that matter. Normality even implies that all finite information is encoded in a normal number. What’s more, if a number is normal, then it will automatically be simply normal, and if a number is simply normal then it has to have infinitely many of each digit.
However this is where our wild wondering about what can be found encoded in pi comes to an abrupt disruption. It turns out that it has yet to be shown that there’s even infinitely many 7’s in pi, much less whether it is simply normal or, beyond that, normal.
So why do we think it’s normal anyway?
When we zoom in on a specific number like pi, it can be hard to show anything about its normality. However, zooming out and looking at the whole set of normal numbers and the whole set of non-normal numbers, mathematicians CAN prove things about these sets and we can even try to measure these sets. The mathematical machinery that we use to try to measure sets is called (maybe you guessed it) measure theory. While we don’t have time to go into how we measure a set like normal or non-normal numbers, it turns out that the set of numbers that are NOT-normal has “measure 0.” (Itching to see an actual proof of this fact? Don’t worry, just jump to the further reading section at the end of this post.) Intuitively this means that almost every real number is normal, because when we measure the set of numbers that are not normal it’s vanishingly small (measure 0). Since almost all numbers are normal, and we have no special reason to think pi would not be normal, we can guess that it probably is.
All that’s to say, as special and unique as pi is, it’s likely that it’s just a normal number like (almost all of) the rest of them.
While it would be nice to settle the question more definitively, the unknown is part of the joy and frustration that comes with thinking about open problems, problems without any known solution. Sometimes the lines between the known and unknown appear in unexpected places. The frontier of mathematics is not a smooth, well defined arc, but rather a wild, craggy expanse. Even when it comes to pi, there’s still much more to explore.
Further Reading:
If you still have room for more musings about numbers here is a great Numberphile video that goes over different types of numbers and includes a nice discussion on normal numbers (starts right about the 8:15 mark). It’s also helpful to see where normal fits in with other properties we may be better acquainted with such as rational/irrational, or transcendental:
And finally for those wanting to check out an actual proof of the fact that non-normal numbers have measure 0 here are a couple options:
If you have a working knowledge of measure theory under your belt here is a proof for the fact that non-normal numbers have measure.
Or if you are a math undergrad student and haven’t seen measure theory yet, but have an understanding of calculus, here is a neat approach that tackles proving the same fact without measure theory; instead you’ll just need some knowledge of sequences and series, as well as how to integrate step functions on an interval.
Looking for the solution to the Pi Day Math Problem?
Check out this blog post!