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Does π contain all possible number combinations?

  • Last Updated : 03 Aug, 2021

Pi is believed by some people to be a number that in its infinite length contains all the possible number combinations that could ever be. Which includes but is not limited to your phone number, you Mastercard’s PIN and any numerical values that have not even been discovered yet. But is that belief entirely true? Let’s find out.

Introduction :
Pi is a mathematical constant, defined as the ratio of a circle’s circumference to its diameter. Pi is an irrational number, which means that it is non-terminating and non-repeating. The value of Pi goes like this 3.14159265359…. this number is supposed to have an infinitely long decimal expansion which never repeats. 

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Previous Proofs :
To test the validity of the assertion we must first gather the information that we already have that is proven and then find whether that information can be used to prove the assertion further.
It has been proved that Pi is irrational, and so it stands that it never terminates and never repeats. To support this, the longest sequence calculated till date is 50 trillion digits by Timothy Mullican,  which took 303 days to do the calculations using a computer.
So we can safely say that “Pi will generate infinite amount of numbers (non-terminating), which will be distinct (non-repeating).” But is this phrase enough to prove the assertion?
To cut to the chase, no. To give an example, have a look at this number 2.718281828459045… this number is the mathematical constant e, it is non-terminating and non-repeating, but it has not been proved yet that it contains all the combinations of numbers in base 10.
So we come to a revelation that there is one more condition that should be satisfied for the assertion to be true. 



What is Missing?
Pi need to be non-terminating and non-repeating but the likelihood of each digit occurring throughout the decimal expansion should also be equal. Simply put, out of all the digits 0, 1, 2,…., 9, the probability of each of these 10 digits to be the next number in the sequence should be 10%. 

In mathematics we have some terms defined for these type of numbers. Lets discuss some terminology that we are going to use throughout the next section of this article. 

  1. Disjunctive Numbers – 
    A disjunctive number or disjunctive sequence is an infinite sequence (over a finite alphabet of characters) in which every finite string appears as a substring (chunk of the original string.) 
    Example – If we take a finite alphabet/set of characters (in this case 0s and 1s) then the Champernowne binary number 0 1 00 01 10 11 000 001….  is called a disjunctive number because any combination of a finite chunk of 0s and 1s (a substring) that you can think of, will be present as a part of this infinitely long number
    If we add one more property to a disjunctive number we get a Normal number.
  2. Normal Numbers –
    A normal number for a radix/base b, contains every possible combination of numbers, but each combination occurs with equal likelihood to other combinations of that length.
    Example – Champernowne constant for base 10: 1234567891011121314151617181920212223….

Conclusions –
Normal number is the last condition that we wanted Pi to fulfil. So, is Pi a normal number? No. Let’s learn why. 

Pi is an infinitely long number and to prove that all the digits occur with equal likelihood in its decimal expansion, would be impossible because of its infinite length, at least for now. 
You might interject with the statement that “We have calculated Pi to a very large length and we have observed the occurrence of each digit at least once, which implies that every digit has a probability of occurring, no matter how small. So in the infinitely long sequence of Pi, there is a chance that every possible combination will occur at some point because of the presence of a probability.”  
To answer that, statistical proofs are there that show Pi having the property of being normal for a large length, that is 22 trillion. But the thing is that we are again restricted by Pi being infinitely long, yes every digit has occurred with equal likelihood for a very very long sequence but we cannot say for sure that for an even longer sequence all of the 10 digits will keep popping out with equal probability. What if at some point all that we get are 0s and 1s. 

Below is the table of frequency of digits in Pi for the first 10,000,000 digits –

DigitFrequency
0999440
1999333
21000306
3999965
41001093
51000466
6999337
71000206
8999814
91000040

Pi being a normal number has not yet been proven. Mathematicians believe and assume Pi to be normal, but no one has proven it yet and therefore, we cannot assume it to be true. To clarify one thing, neither has it been proven to be a normal number nor has it been proven to not be a normal number. So as it stands for now, the assertion is false unless its proven to be otherwise in the future.

So the property of Pi to contain all possible combinations depends on whether Pi can be proven to be a normal number or not. Similar to Pi is Euler’s number and root 2, that hold the same mystery with them.

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