The first one is from an identity called Euler's Identity. It is an equation that involves π and a few other prominent numbers. Maybe even the most prominent and important numbers. Here is that equation:
Here is a clip from Wikipedia that does a pretty good explanation of this amazing equation:
Whether you are a mathematician or not, this is amazing. the five most important numbers all in one equation with all of the basic mathematical operations... Weird.
Euler's identity is considered by many to be remarkable for its mathematical beauty. Three basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. The identity also links five fundamental mathematical constants:
- The number 0.
- The number 1.
- The number π, which is ubiquitous in trigonometry, geometry of Euclidean space, and mathematical analysis.
- The number e, the base of natural logarithms, which occurs widely in mathematical analysis (e ≈ 2.71828).
- The number i, imaginary unit of the complex numbers, which contain the roots of all nonconstant polynomials and lead to deeper insight into many operators, such as integration.
Furthermore, in mathematical analysis, equations are commonly written with zero on one side.