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Category: Math

The Monstrous Function That Broke Calculus

2025-01-24
The Monstrous Function That Broke Calculus

In the 19th century, Karl Weierstrass unveiled a function that sent shockwaves through the mathematical community. This function, continuous everywhere but differentiable nowhere, resembled an infinitely jagged sawtooth, defying intuition and challenging the very foundations of calculus. Its seemingly paradoxical properties forced mathematicians to rigorously redefine continuity and differentiability, ultimately leading to the development of modern analysis. This 'mathematical monster' not only holds theoretical significance but also finds practical applications in fields like Brownian motion, showcasing the boundless possibilities within mathematics.

(www.quantamagazine.org)
Math Weierstrass function mathematical analysis

Century-Old Math Problem Solved: Proving the Irrationality of ζ(3)

2025-01-09
Century-Old Math Problem Solved: Proving the Irrationality of ζ(3)

This article recounts the legendary story of mathematician Roger Apéry's 1978 proof that ζ(3) (the Riemann zeta function at 3) is irrational. His proof was met with skepticism and even caused chaos at the conference where it was presented. However, Apéry was ultimately proven correct. For years, mathematicians struggled to expand Apéry's method with little progress. Recently, Calegari, Dimitrov, and Tang developed a more powerful method, proving the irrationality of a series of zeta-like values, including ζ(3), solving a decades-old problem. This breakthrough lies not only in its result but also in the generality of its approach, providing new tools for future irrationality proofs.

(www.quantamagazine.org)
Math irrational numbers

Infinity's Size: Mathematicians Get Closer to Answering How Many Real Numbers Exist

2025-01-09
Infinity's Size: Mathematicians Get Closer to Answering How Many Real Numbers Exist

For decades, mathematicians believed determining the total number of real numbers was an unsolvable problem. A new proof suggests otherwise. The article details how mathematicians Asperó and Schindler proved that two axioms previously considered competing foundations for infinite mathematics actually imply each other. This finding strengthens the case against the continuum hypothesis and indicates an extra size of infinity exists between the two that, 143 years ago, were hypothesized to be the first and second infinitely large numbers. While this result has generated excitement and debate within the mathematical community, the arguments surrounding the sizes of infinite sets are far from settled.

(www.quantamagazine.org)
Math continuum hypothesis infinity