A beautiful equation

I recently picked up a book about Paul Erdős, who was a Hungarian mathematician with a very interesting and peculiar life. Amongst many references to great mathematicians and their contribution in the book, Euler’s work stood out to me. Leonhard Euler was a prolific Swiss mathematician who made many profound contribution to mathematics. Despite losing his vision in 1771, he continued his work; much like Beethoven who wrote music he never heard, Euler wrote math theories he never saw. I don’t claim to know or understand Euler’s work. But I am fascinated by his identity formula, which has been called the “most beautiful equation”, a “jewel” and the “the most remarkable formula in mathematics”. His formula shows the relationship between five important mathematical constants (or identities), and written as:

$e^{i\pi}+1=0$

$e$: The first identity that's extensively used in calculus
$\pi$: The ratio of the circles circumference to its diameter. An extensively used constant in trignometery
$i$: The complex imaginary unit that's extensively used in electrical and computer engineering for circuit analysis and other applications
$0$and $1$: The addictive identity and the first natural number, both widely used in all domains of mathematics

In one stroke of genius, the man connected five mathematical identities across fields of trigonometry, calculus, and complex numbers. Euler’s equation is constantly used in many areas of mathematics and physics and it helps formulating the basic laws of fluid mechanics, number theory, complex analysis, and analytical mechanics. In application, Euler’s identity has had a great influence in the success of computer science, particularly computer graphics. In electronic, it is constantly used in circuit analysis and robotics. And in aerospace, it is used in navigation, flight dynamics, and orbital mechanics. It has helped scientists understand planetary retrograde motion. It is deservingly referred to as the “greatest equation ever”, and several books are written about this seemingly simple, yet profound mathematical formula.