Some Beautiful Mathematical Formulas
Mathematics is not just about numbers and calculations. It is also about discovering beautiful relationships between seemingly different ideas.
Euler's Identity
eiπ + 1 = 0
Euler's Formula
eiθ = cosθ + i sinθ
Pythagorean Theorem
a² + b² = c²
Quadratic Formula
x = (−b ± √(b² − 4ac)) / (2a), a ≠ 0
Binomial Theorem
(a + b)n = Σk=0n C(n,k)an−kbk
Geometric Series
1 + r + r² + r³ + ··· = 1/(1−r), |r| < 1
Signed Area Under a Curve
∫ab f(x) dx
Fundamental Theorem of Calculus
∫ab f(x) dx = F(b) − F(a), where F′(x) = f(x)
De Moivre's Formula
(cosθ + i sinθ)n = cos(nθ) + i sin(nθ)
Fibonacci Sequence
Fn = Fn−1 + Fn−2, n ≥ 2; F₀ = 0, F₁ = 1
Golden Ratio
φ = (1 + √5)/2
These formulas show how mathematics connects algebra, geometry, trigonometry, calculus, complex numbers, and even patterns found in nature.
Mathematics is where simplicity and complexity meet. ∞
Conditions are part of the formula.
- In a² + b² = c², the triangle is right-angled and c is its hypotenuse.
- In Euler’s formula and De Moivre’s formula, θ is measured in radians. The elementary De Moivre identity shown here is used for integer n.
- The finite binomial expansion shown here uses a non-negative integer n.
- A definite integral gives signed area. For a continuous function and a ≤ b, geometric area between the graph and the x-axis is ∫ₐᵇ |f(x)| dx.
- For the usual elementary form of the Fundamental Theorem of Calculus, take f continuous on [a,b] and F an antiderivative of f.