What is the importance of $\pi$ in mathematics?

Question

In what specific fields is $\pi$ relevant in mathematics and how is its accuracy important?

Is there any field in which its precision leads to some results despite others?

Answer 1

The Quest for Pi. David H. Bailey, Jonathan M. Borwein, Peter B. Borwein and Simon Plouffe, Mathematical Intelligencer, vol. 19, no. 1 (Jan. 1997), pg. 50–57

In one of the earliest accounts (about 2000 BC) of π, the Babylonians used the approximation $3 \ 1/8 = 3.125$. At this same time or earlier, according to an account in an ancient Egyptian document, Egyptians were assuming that a circle with diameter nine has the same area as a square of side eight, which implies $π = 256/81 = 3.1604\ldots$ In the 1700s the mathematician Euler, arguably the most prolific mathematician in history, discovered a number of new formulas for π. Among these are $$ \dfrac{\pi^2}{6} = \sum_{n=1}^{\infty}\dfrac{1}{n^2}, \quad \dfrac{\pi^4}{90} = \sum_{n=1}^{\infty}\dfrac{1}{n^4} $$ These formulas aren’t very efficient for computing π, but they have important theoretical implications and have been the springboard for notable research questions, such as the Riemann zeta function hypothesis, that continue to be investigated to this day. One motivation for computations of π during this time was to see if the decimal expansion of π repeats, thus disclosing that π is the ratio of two integers (although hardly anyone in modern times seriously believed that it was rational). This question was conclusively settled in the late 1700s, when Lambert and Legendre proved that π is irrational. Some still wondered whether π might be the root of some algebraic equation with integer coefficients (although as before few really believed that it was). This question was finally settled in 1882 when Lindemann proved that π is transcendental...

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Answer 2

$\pi$ itself appears in many equations, from circles to spheres, and even the Euler identity ($e^{i\pi}+1=0$), which has been crowned as the most beautiful equation in math. So, yes, $\pi$ is very important.

The digits? Not as much. Some people are searching for a pattern or a regularity to the numbers, but even from centuries of pondering on $\pi$, we have yet to conclude any patterns from it. Learning more digits is both an exercise in seeing if we can find a pattern, and just a competition to see who can find the most. It isn't as useful as other fields of mathematics, but it still has unsolved mysteries for future Mathematicians to solve.

If you want more examples of formulas using $\pi$, check out this link: Formulas Using $\pi$