Best of Math — June 2025

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   Article

   The Palindrome·48w

   The 10 Most Important Lessons 20 Years of Mathematics Taught Me

   A mathematician with 20 years of experience shares ten key lessons about learning and mastery. The core insights include the importance of understanding fundamentals before breaking rules, learning through hands-on problem solving rather than passive consumption, and recognizing that there are no shortcuts to expertise. The author emphasizes taking things slow to build deep understanding, tackling complexity one step at a time, and finding the right perspective to solve problems. Other key points include the power of asking questions without shame, the primacy of hard work over talent, and the importance of forging your own path rather than blindly following others' advice.

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   Article

   Hacker News·49w

   Art of Problem Solving

   A comprehensive collection of visual mathematical proofs demonstrating various theorems and identities through geometric diagrams and illustrations. Covers summations (including odd numbers, triangular numbers, and Fibonacci sequences), geometric series, fundamental geometry theorems like the Pythagorean theorem, and advanced concepts like Fermat's Little Theorem. Each proof uses visual arrangements and geometric transformations to make abstract mathematical concepts more intuitive and accessible.

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   Article

   The Palindrome·50w

   The Obstacle is the Path

   A mathematician's journey from being labeled as academically inadequate to earning a PhD reveals how external validation can both hinder and motivate personal growth. The author describes pivotal moments including a conversation with Fields medallist Cédric Villani that led to the realization that true success comes from defining your own metrics rather than chasing others' approval. The narrative emphasizes that traditional markers like publication counts, citations, and awards don't define success - instead, personal fulfillment comes from pursuing what genuinely matters to you, whether that's becoming an inspiring teacher or mastering challenging skills through persistence.

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   Article

   freeCodeCamp·48w

   How to Write a PHP Script to Calculate the Area of a Triangle

   A comprehensive guide demonstrating three different methods to calculate triangle area using PHP: the base-height formula, Heron's formula for when all three sides are known, and the trigonometric formula using two sides and an included angle. Each approach includes complete PHP code examples, mathematical explanations, time/space complexity analysis, and guidance on when to use each method.

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   Article

   Towards Data Science·48w

   Part 2: Matrix-Matrix Multiplication

   Matrix-matrix multiplication is explained through visual X-diagrams that show how input values flow through transformations. The article derives the multiplication formula by demonstrating that multiplying matrices A*B creates a combined transformation equivalent to applying B first, then A. This visualization clearly explains why matrix multiplication is non-commutative (A*B ≠ B*A) and shows how special matrices like scale, shift, permutation, and triangular matrices behave when multiplied together, with their properties preserved in the resulting products.

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