Most Elegant Numerical Relationship (W07)

The Most Elegant Relationship in the Universe

We deal with numbers every day since humans began life on Earth knowingly or unknowingly. This article talks about four powerful numbers forming the most elegant relationship in the universe. The numbers are 1, π, e, and i. The last three numbers could be seen as higher mathematics. This article takes us to the land of higher mathematics on a plain English cruise. It is not hard to appreciate the elegance in 3 minutes.

π (pronounced pi) suggests that all circles in the universe share one property in common. The circumference of a circle is longer than the diameter by a ratio called π. This is geometry. Obviously, circle is the most common shape in the universe although not everything is circular but close to being circular. The Earth, the Sun, the Moon, the birthday cake, orange, and the list goes on. The value of π is about 3.1416 (decimal place is irrationally long).

e is more often known as the natural exponential function rather than as a number although it has a value of 2.71828 (decimal place is irrationally long).

The number was presumably discovered by a top mathematician Leonhard Euler (1707-1783) for prescribing some phenomena in nature such as the reduction of sound level over distance, the cooling of a heated rod over time, and the decay of radioactive substances over time. All these examples can be prescribed by e-rt where r is the rate and t is time or distance. No other numbers than e can provide this facilitation with elegance. Instead of decay, we can apply e to growth such as the growth of our bank balance of a fixed deposit over time in the form of Pert where P is the principal, r is the interest rate, and t is the number of time periods. This is analytical mathematics.

Surprisingly, i cannot be expressed as a number similar to π and e because it is imaginary and not real. Numerically it is equal to the square root of -1 but nobody has been able to compute its value as of this moment 2023-05-06. Call it a symbol and this takes us to algebra. The symbol appears often in astronomy when the radiation from the heavens arrives on Earth as waves. We express waves in the polar coordinate of en = sin(n) + i cos(n) where n is the radian of the wave at any one moment. This Imaginary number has been officially accepted for a couple of hundred years as part of a complex number. Complex numbers are essential in quantum physics which deals with things in the universe of the smallest size. We do not see things in the quantum world with our naked eyes, but we do when the quantum behavior manifests in daily life. If i multiplies itself, we see -1. This imaginary number empowers the development of quantum physics and astronomy.

Many well-produced movies got us leaving the cinema seat with the deepest emotion ever by closing the movie with the most touching screen. Here it is: e iπ = -1. Can we believe our eyes? This amazing equation puts geometry, analysis, and algebra together. It was derived by Leonhard Euler from the polar coordinate expression shown above (by substituting n with iπ). Euler is one of the top names in the history of mathematics. Nevertheless, not many people know of these powerful numbers and their elegant relationship. Those who do are the celebrated few. Congratulations, you are one. We are one step closer to understanding the Universe.

Acknowledgement: The inspiration to write this article originated from a Princeton University Press book entitled “e: the story of a number” by Eli Maor published in 1994.