Are there other real solutions than x = π ?

## Beyond 65 digits, π serves no practical purpose

For spatial engineers’ highest accuracy calculations, used in interplanetary navigation, 3.141592653589793 is more than sufficient. Let’s understand why more decimals aren’t needed.

Consider these examples:

• Voyager 1, the farthest spacecraft from Earth, is about 14.7 billion miles away. Using π rounded to the 15th decimal, the circumference of a circle with a radius of 30 billion miles would be off by less than half an inch.

• Earth’s circumference is roughly 24,900 miles. The discrepancy using limited π would be smaller than the size of a molecule, over 30,000 times thinner than a hair.

• The radius of the universe is about 46 billion light years. To calculate the circumference of a circle with a radius of 46 billion light years to an accuracy equal to the diameter of a hydrogen atom, only 37 decimal places are necessary.

• With just 65 decimal places, we could determine the size of the observable universe within a Planck length, the shortest measurable distance.

While π’s digits are endless, for microscopic, macroscopic or cosmic endeavors, very few are necessary.

## Feynman π Point

The Feynman point occurs at the 762nd decimal of π, displaying **six consecutive nines** (999999). Named after physicist Richard Feynman, he humorously shared, “*I once memorized 380 digits of π as a high-school kid. My ambitious goal was the 762nd decimal, where it goes ‘999999.’ I’d recite it, reach those six 9’s, and cheekily say, ‘and so on!’ implying π is rational (which it is not).*“

## Pervasive Pi: Unveiling the Quantum Connection

The formula developed by British mathematician John Wallis defines π as an infinite product of integer ratios. This formula was recently discovered in a quantum mechanics equation for hydrogen atom energy states, emphasizing its pervasiveness in math and science.

More number facts.

## Zeroes in 𝜋

The decimal representation of 𝜋 is thought to have random digits that are uniformly distributed. Since there is a 0.9 probability for any digit in a random decimal sequence to be non-zero, it’s fascinating to investigate when a 0 or a sequence of 0s will appear in this sequence.

Remarkably, a string of eight zeroes (00000000 !) emerges at position 172,330,850 in the decimal representation of 𝜋, counting from the digit immediately following the decimal point.

Below is the string, along with the digits surrounding it:

## A Magic Right Triangle

A triangle with sides ** π** (3.14),

**(2.71), and the**

*e***(1.61) is almost a right triangle!**

*Golden Ratio*

## Four Constants in Four 4’s

The infamous problem of representing numbers with four 4’s appeared for the first time in 1881 in a London science journal. In 2001, a team of mathematicians from Harvey Mudd College found that we can even get four 4’s to approximate four notable constants: the ** number e**,

*,*

**π***, and*

**acceleration of gravity***.*

**Avogadro’s number**## The Kepler Triangle, Phi and Pi

A “Kepler triangle” is a right triangle having edge lengths in a geometric progression, in which the common ratio is √ϕ, where ϕ represents the golden ratio.

Well, let’s construct a square with side length √ϕ that inscribes a Kepler triangle, that is, a right triangle with edges 1 : √ϕ : ϕ (or approximately 1 : 1.272 : 1.618), as shown in the picture. Draw then the circumcircle of the Kepler triangle (highlighted in orange in the picture) whose diameter is the hypotenuse of the triangle.

Then, the perimeters of the square (4√ϕ≈5.0884) and the circle (πϕ≈5.083) coincide up to an error less than 0.1%. From this, we can get the approximation coincidence π≈4/√ϕ