## **The 20 Most Important Mathematical Equations**

A journey through the most elegant and influential formulas in mathematics

**1. Euler's Identity**  $\text{Contextsts five fundamental constraints (e., i., 7, 0), reveziling} $ the profound relationship between exponential functions and trigonometry. **2. Pythagorean Theorem** In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture. *e* + *iπ* 1 = 0 *a* +  $c^2 + b^2 = c^2 $ 

## **3. The Fundamental Theorem of Calculus**

## **4. Maxwell's Equations**

$$\int\_{a}^{b} f(\mathbf{x}) \, d\mathbf{x} = F(b) - F(a)$$

Establishes that differentiation and integration are inverse  $\text{Operations. If } \mathbb{F} \text{ is an antiderivative of } \mathfrak{f} \text{, the definite integral.} $  $\mathsf{esquals} \text{ } \mathsf{F}\langle \mathsf{b} \rangle \text{ - } \mathsf{F}\langle \mathsf{a} \rangle$ .  $\mathsf{Revoluctionized}$   $\mathsf{matthematical problem} $ solving.

∇ ⋅ **E** = *ε*0 *ρ* ∇ ⋅ **B** = 0 ∇ × **E** = − ∂*t* ∂**B** ∇ × **B** = *μ*0**J** + *μ*0*ε*<sup>0</sup> ∂*t* ∂**E**

Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones.