Although the number Pi is made up of an endless, unpredictable sequence of digits, it’s not as random as we might think. Pi actually contains all sorts of surprising patterns.
An Enduring Mathematical Mystery
For thousands of years, mathematicians have tirelessly worked to uncover the mysteries of Pi (π). Known for its approximate value of 3.14, Pi extends for over 13 trillion digits after the decimal point – a testament to researchers’ efforts to calculate Pi with greater precision.
Since the 18th century, humans have known that they can never calculate all the digits of Pi. This is because Pi is an irrational number that continues infinitely without a
repeating pattern.
In 1888, logician John Venn – creator of Venn diagrams – visually attempted to demonstrate that the digits after Pi’s decimal point are entirely random. He plotted a graph representing the first 707 decimal places, using directed line segments to represent digits from 0 to 7, and then drew lines to show the path of each number. Venn did this work with pen and paper, but today, modern technology helps create more detailed and beautiful patterns.
Despite being formed from an endless, unpredictable string of numbers, Pi is not as random as we might assume. In fact, Pi contains all sorts of surprising patterns.
Order Within Apparent Randomness
Ordinary Yet Not Random
The reason we cannot call Pi “random” is that its digits are precisely and fixedly determined. For example, the second decimal place in Pi is always 4. Therefore, you cannot ask if other numbers could be placed in this position; it’s not a random placement.
But we can ask a related question: “Is Pi a normal number?” A decimal number is considered normal when every sequence of digits has an equal probability of appearing, making them seem random – even if they aren’t truly so.
By examining the digits of Pi and applying statistical tests, we can try to determine if Pi is a normal number. From tests conducted to date, this remains an open question without a definitive answer.
For example, in 2003, Yasumasa Kanada published a statistical table of the occurrences of different digits in the first trillion digits of Pi:
Digit
Occurrences (approx.)
0
99,999,485,134
1
99,999,945,664
2
100,000,480,057
3
99,999,787,805
4
100,000,357,857
5
99,999,671,008
6
99,999,807,503
7
99,999,818,723
8
100,000,791,469
9
99,999,854,780
Total
1,000,000,000,000
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His research results imply that these numbers appear to be fairly evenly distributed. However, this is not enough to prove that all of Pi’s digits are normal.
Every Sequence of Digits
A surprising fact is that if Pi is a normal number, then any sequence of digits you can name, you will find within Pi. For example, at position 768 in the digits of Pi, there are six 9s appearing consecutively. The probability of this happening (given that Pi is normal) and each string of n digits having an equal chance of occurring, is 0.08%.
After Richard Feynman’s Nobel Prize, this sequence of six nines became known as the “Feynman Point.” Feynman once joked that if he had to recite the digits of Pi, he would name them and then just say “and so on.”
Other interesting sequences have also been found. At position 17,387,594,880, you find the sequence 0123456789, and surprisingly, at position 60, you find these ten digits arranged in order.
Those who search and study Pi often ask: “Where can I find specific digits in Pi?” If you want to check if certain digits are in Pi, you can use a free online tool called Pi birthdays.
Pi (symbol: π) is a mathematical constant whose value is equal to the ratio of a circle’s circumference to its diameter. This constant has an approximate value of 3.14159265358979. It has been represented by the Greek letter π since the mid-18th century.
