“And it was Euler1 who discovered the most beautiful formula in all mathematics:
eiπ + 1 = 0
a mysterious and ineffable expression connecting the five most important numbers in the universe.”2
This formula is a special case of Euler’s famous equation:
eix = cos x + i sin x
To derive the special case, from the above equation, let x = π (pi) . . .
eiπ = cos π + i sin π
Given that cos π = -1 and sin π = 0, then . . .
eiπ = -1
Adding “1” to each side of the equation:
eiπ + 1 = 0
Paul Nahin has written an entire book devoted to Euler’s formula wherein he states that the formula sets “the gold standard for mathematical beauty.” He continues:
“The special case is a compact expression of exquisite beauty. I think eiπ + 1 = 0 is beautiful because it is true even in the face of enormous potential constraint. The equality is precise; the left-hand side is not ‘almost’ or ‘pretty near’ or ‘just about’ zero, but exactly zero. That five numbers, each with vastly different origins, and each with roles in mathematics that cannot be exaggerated, should be connected by such a simple relationship, is just stunning. It is beautiful. And unlike the physics or chemistry or engineering of today, which will almost surely appear archaic to technicians of the far future, Euler’s formula will still appear, to the arbitrarily advanced mathematicians ten thousand years hence, to be beautiful and stunning and untarnished by time.”3
He also adds, in lighthearted levity, this limerick:
I used to think math was no fun
’Cause I couldn’t see how it was done
Now Euler’s my hero
For I now see why zero
Equals ei pi + 1.
Take another look at this form of the formula:
eiπ = -1
Since e and pi are both irrational numbers,4 and i is an imaginary number, let us stop and wonder that an irrational number—raised to a power that is the product of an imaginary and an irrational number—turns out to be a natural whole number.
One final tidbit:
A quarterly journal called the Mathematical Intelligencer once invited its readers to vote on what they believed was the most beautiful theorem in mathematics. The readers of this scholarly journal were, for the most part, academic and industrial mathematicians and, when the votes were in, the winner was . . .5
You guessed it:
eiπ + 1 = 0
Notes
1. Leonhard Euler (1707-83) was a Swiss mathematician who is esteemed as one of the greatest mathematicians of all time.
2. David Berlinski, A Tour of the Calculus, p. 91. The five numbers:
“e”: the base of natural logarithms (approximate value of 2.71828 . . .)
“i”: an imaginary number, the square root of -1
“π”: the ratio of the circumference of a circle to its diameter (3.14159 . . .)
“1”
“0”
3. Paul Nahin, Dr. Euler’s Fabulous Formula, p. xxxii. The limerick is on page 12. This is a substantial book, running 380 pages.
4. An irrational number is a number not expressible as an integer or as a quotient of two integers.
5. The journal presented a total of twenty-four theorems for consideration. Euler nudged out Euclid (and his proof that the number of primes is infinite) to take top honors.
HyC