Todos los furrys deben ser tratados como le peor. #antifurry

The Faces of e Leonhard Euler introduced the symbol e in the 18th century, but the number itself appeared earlier in the work of Jacob Bernoulli, while studying compound interest. Euler later showed how naturally this constant appears across mathematics.

Richard Feynman called Euler's formula "the most remarkable formula in mathematics." And there's a fascinating engineering reason why this equation is so important. e^(iθ) = cos(θ) + i·sin(θ) It connects exponential functions, imaginary numbers, trigonometry, and rotation through a single mathematical relationship. But its usefulness goes much further. Consider a sinusoidal signal: x(t) = A cos(ωt + φ) Using Euler's formula, we can represent it as the real part of a complex exponential: A e^(i(ωt + φ)) Why would engineers use this representation? Because differentiating e^(iωt) simply multiplies it by iω. Phase shifts can be represented through complex multiplication. For linear, time-invariant electrical circuits in sinusoidal steady state, this allows differential equations to be reduced to algebraic equations. Engineers use this mathematics in phasor analysis, AC circuit calculations, impedance, and frequency-response analysis. The same complex-exponential representation is fundamental to Fourier analysis, signal processing, communication systems, and the study of mechanical vibrations. Feynman also described the complex-plane representation of numbers as the unification of algebra and geometry. And that's what makes Euler's formula so remarkable. An equation connecting imaginary numbers and circular motion provides a practical way to analyze real electrical signals, oscillations, and physical systems.

The prize for the most remarkable formula in all of mathematics involves 0, 1, π, e, and i = √−1. When we think of the famous numbers of mathematics, we think of 0, 1, π, e, and the imaginary number i = √−1. This is a result attributed to Euler.

Euler's formula, e^(iθ) = cos θ + i sin θ, puts e^(iθ) on the unit circle at angle θ: its real part is cos θ and its imaginary part sin θ, two waves of period 2π. The reason is one derivative, d/dθ e^(iθ) = i·e^(iθ). Multiplying by i turns a vector through 90°, so the velocity stays perpendicular to the radius with length 1, and the point runs round the circle at unit speed, covering arc θ by time θ.

One of the strangest constants in number theory is ζ(3) = 1 + 1/2³ + 1/3³ + 1/4³ + ⋯ ≈ 1.202056903159594… Euler found elegant formulas for ζ(2), ζ(4), ζ(6), … in terms of powers of π. But the odd values proved far more mysterious. Then in 1978, Roger Apéry unexpectedly proved that ζ(3) is irrational. His argument was initially surprising because it relied on sequences that looked unlikely to have the required integer properties, yet the proof held. Even today, ζ(3) stands as a reminder that the odd zeta values behave very differently from the even ones.

For decades, mathematicians struggled with a deceptively simple question: what is the exact value of 1 + 1/4 + 1/9 + 1/16 + ⋯ This became known as the Basel Problem. The series was known to converge, but finding its exact sum proved remarkably difficult. In 1735, Leonhard Euler, then only 28, found the solution= π²/6. What makes Euler’s result so striking is the unexpected connection it reveals. The left side is built entirely from the squares of positive integers, while the answer involves π, one of the fundamental constants of geometry. Euler’s solution showed that an infinite series arising from arithmetic could encode a quantity associated with the circle. And this is one of the beautiful lessons of mathematics: apparently unrelated structures can be connected by a single exact identity. The Basel Problem did not merely produce a numerical answer; it opened a deeper view of the relationship between number theory, analysis, and geometry.