How would the world look like if the most popular irrational number, pi, was tweaked to be just 3.14? The question is, in some sense, similar to the speculation of how the world would look like if the gravitational constant were tweaked, but in essence is meaningless. Is the fact the value of pi isn’t some knob we can tweak independently, as it lies at the core of the basic mathematical operations such as addition and multiplication. It is also related to prime numbers as well as to geometry. Changing it would mean mathematics becomes inconsistent, with an example being that circles would turn into polygons.
As Alon Amit, Ph.D. in Mathematics states on Quora,
“Counterfactuals are generally dubious, but this one is particularly egregious. It’s like asking what the world would be like if good was bad, and 2 was 4,294,967,296.
We sometimes get questions about physical constants changing, and those questions make sense because there’s no real reason for the constants to be what they are. But ? is mathematically derivable; it kinda needs to be what it is. You can’t, through the power of reason alone, figure out what the gravitational constant or the speed of light are, but you can figure out what Pi is.”
If rounding off pi means ending the concept of irrational, never-ending sequences forever, then we’ll have to say goodbye to almost every physical and theoretical concept, from electromagnetism to complex numbers, quantum mechanics, and Fourier analysis!
Aside from the mess of mathematical contradictions and paradoxes, changing pi would ensue changes in a host of fundamental forces and constants of the universe. But pi affects too many things at a time to make a meaningful statement on the exact impact of the change in the universe.
The scenario can’t be compared with changing the gravitational coupling constant “constant” as they are very different entities. Physicists can imagine universes with different parameter values but doing the same with mathematics means you are asking for a universe where 1 is 7, pink is black and good is bad.
Here’s a video elaborating on the history of a certain “mathematician” trying to approximate pi and “square the circle.”