The π FAQ
(also known as Why Pi is not special in the way you think it’s special)
By Andycyca, based on this GitHub repo, but with added links for E2.
This version: 2020-10-20
- What is this?
- 1 Is π random?
- 2 Is π infinite?
- 3 Do the digits of π digits go on forever?
- 4 How can we be sure that the digits of π go on forever?
- 5 How can we be sure that π is transcendental?
- 6 Is π irrational?
- 7 Is it true that π has no exact value?
- 8 Is it true that π is not defined?
- 9 Is it impossible to know the “real” value of π?
- 10 Is it true that I can calculate π with ruled paper and toothpicks?
- 11 Is it true that my name is somewhere in the digits of π?
- 12 Is it true that my phone number is somewhere in the digits of π?
- 13 Is it true that the complete works of (your favorite author) are somewhere in the digits of π?
- 14 Is it true that there are six nines in a row somewhere in π?
- 15 Is π a normal number?
- 16 What is a normal number?
- 17 It’s useless to calculate (whatever number) of digits of π
- 18 What about that law stating that π = 3.2?
- 19 Didn’t the State of Indiana ruled on the value of π?
- 20 Is it true that the Hebrew Bible establishes π equals three?
- 21 What is Pilish?
- 22 What is Piphilology?
- 23 There’s this poem that encodes the digits of π
- 24 There’s this novel that encodes the digits of π
- 25 There’s this movie…
- 26 In the movie π…
- 27 What is 748/238?
- 28 Common misconceptions surrounding π and other numbers
- 29 π is a special number because…
- 30 What is τ?
- 31 I think τ is better
- 32 A collection of stupid questions that should not be asked if the above FAQ has been read
What is this?
Now including Pi-adjacent questions!
The π FAQ (aka: “Why Pi is not special in the way you think it’s special”) is an archive of quick answers to some of the most infuriating questions asked by people who believe π is unique and special for the wrong reasons.
As is the case with FAQs, this will always be marked as being a “Work In Progress”. Andycyca welcomes any and all additions you may have to this list, but please bear in mind that this is not intended to be neither math-heavy nor citations-heavy.
- No it’s not. If you and I set to calculate π we should arrive at the exact same result, digit for digit.
- No it’s not. If you calculate π today and you calculate it tomorrow, you should arrive at the exact same result both days.
What you probably mean is that there’s no regular pattern to the digits of π in decimal notation.
What you probably mean is that the decimals of π do pass tests of statistical randomness
What you probably mean is that π is a normal number.
2 Is π infinite?
- No it’s not. There’s at least one number that is finite and larger than π, namely 4.
- No it’s not. There are may finite numbers larger than π.
What you probably mean is that you cannot write π exactly with a finite string of digits.
3 Do the digits of π digits go on forever?
4 How can we be sure that the digits of π go on forever?
- Smart people figured it out long ago. If you’re not prepared or willing to take my word, do try to read these proofs (they are scattered around the internet and calculus books) and try and disprove them. Most—if not all—mathematicians don’t bother themselves with trying to disprove this fact for anything other than amusement.
- In general, the proof(s) work by reductio ad absurdum:
Let’s assume π is rational;
Rational numbers should have this or that property;
π doesn’t have that property;
∴ Therefore, π is not rational.
5 How can we be sure that π is transcendental?
6 Is π irrational?
What this means is that π cannot be expressed as a fraction of two integers. Some numbers have digits that go on forever but are rational, like 0.333... = 1/3
7 Is it true that π has no exact value?
- No it’s not. The value of π is exact, but we cannot write it down exactly. We use approximations like 3.14 and 22/7, those are not exact.
- No it’s not. The value of π is exact, but the decimal representation of that value will always be inexact.
8 Is it true that π is not defined?
- No it’s not. The value of π is unique and well-defined. In other words, there’s no ambiguity as to what the value of π is.
9 Is it impossible to know the “real” value of π?
- No it’s not. There are ways to calculate π to any arbitrary accuracy. This means that π is a computable number.
- No it’s not. π can be calculated to any arbitrary precision through the use of infinite sums.
- No it’s not. Wikipedia has a page on Pi algorithms, take your pick.
- No it’s not and it matters little for practical uses. There are many approximations of π
10 Is it true that I can calculate π with ruled paper and toothpicks?
11 Is it true that my name is somewhere in the digits of π?
12 Is it true that my phone number is somewhere in the digits of π?
13 Is it true that the complete works of (your favorite author) are somewhere in the digits of π?
All of these questions hinge on whether π is normal (and on how exactly you “translate” from letters to numbers, but that’s relatively easy to do in a number of ways)
15 Is π a normal number?
- Naive definition: A simple, somewhat naive definition is a number that contains all finite sequence of digits in its decimal expansion.
- A less informal—but still incomplete—definition: In a normal number’s decimal expansion, all 1-digit strings are found with 10% probability each, and all 2-digit strings with 1% probability each, and 3-digit strings with 0.1% probability each… In other words, in a normal number all finite strings of length n are found with probability 10 − n (so in theory the numbers 000 to 999 happen with 1/1000 probability.
17 It’s useless to calculate (whatever number) of digits of π
- No it’s not. Calculating numbers like π to a certain precision is a way to benchmark computers and algorithms.
- No it’s not. Repeating the results of someone else can serve as a learning exercise.
- No it’s not. It’s fun.
18 What about that law stating that π = 3.2?
Despite what some Lawyers would like to believe, mathematical truth cannot be established by legislative attempts. Law doesn’t work that way. I know at least one Lawyer that is still bitter about being mathematically bullied in High School, a similar sentiment might be what prompts some people to think this way.
It was only a bill, written by a person sometimes cited as a “crank”. As mentioned in other discussions of this case:
(…) it seems clear to me that the author’s model of the world had more deviations from reality than the value of pi.
It wasn’t really about defining the value of π, rather a proposed way of squaring the circle, a problem stated by ancient geometers who didn’t have internet. We now know that solving that problem as stated is impossible.
In retrospect, the bill has been a net negative for the State of Indiana, given that it’s been a source for cheap laughs in the best cases, and a source for bad journalists to spin mediocre articles in the worst.
You can read the legalese at: Hallenberg, Arthur E. (1974). “House Bill No. 246 Revisited”. Proceedings of the Indiana Academy of Science. 84: 376–399.
20 Is it true that the Hebrew Bible establishes π equals three?
- Not really. It’s complicated.
- Could be an error in measuring inner and outer rims of a large basin known as the Molten Sea or Brazen Sea
- Determining π from physical measurements is always sure to be an approximation at best.
- Pilish is a style of constrained writing in which the lengths of consecutive words match the digits of the number π
23 There’s this poem that encodes the digits of π
24 There’s this novel that encodes the digits of π
25 There’s this movie…
26 In the movie π…
- …yes, the value at the beginning of the movie only has 8 digits “right” after the decimal point.
27 What is 748/238?
- 748/238 simplifies to 22/7 = 3.14285714286
- A funny reference at the end of Aronofsky’s movie titled “π”
28 Common misconceptions surrounding π and other numbers
29 π is a special number because…
(Hint: some of these also apply to that other number ϕ)
All of these are false (or misleading):
- A number and its representation (usually in decimal) are the same thing.
- There is only one way to write down the same number in decimal positional system.
- This is why lots of people get confused about 1 = 0.999999... Both are the same number, written in two possible ways.
- π is somehow special because its digits go on forever.
- π is somehow special because its digits never repeat.
- π is somehow special because its digits contain (your phone number/the name of your one true love/the secret recipe for KFC’s herb and spices mix)
- This is a feature of normal numbers, which greatly outnumber the non-normal numbers.
- π is somehow special because it can be calculated with a never-ending series of nested fractions.
- This is a feature of irrational numbers, which greatly outnumber the rational numbers. All irrationals can be written down as continued fractions.
- π is somehow special because it’s not the root of any “nice” polynomials.
- π is somehow special because…
- Ask yourself first: does that property also apply to π + 1? Does that property also apply to π + 2? Because it’s very likely there’s an infinite series of numbers that share that “special property”. π is not special because of these silly reasons.
30 What is τ?
31 I think τ is better
32 A collection of stupid questions that should not be asked if the above FAQ has been read
All of these I’ve found on the internet. Save for correcting the occasional typo, I present them verbatim, as someone else typed it.
- Could Pi Be Finite?
- Why is Pi infinite when you can just work it out exactly to 3 decimal places?
- Why is pi not infinite?
The Pi FAQ (also known as Why Pi is not special in the way you think it’s special) by Andycyca is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Permissions beyond the scope of this license may be available at https://github.com/andycyca.