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Pursuit problems are some of the most elegant challenges in classical mechanics. Two objects moving, one chasing, the other evading. The math gets surprisingly deep. This problem comes from David Morin’s “Introduction to Classical Mechanics,” and it’s a personal favorite. A fox chases a rabbit. Both run at the same speed. But the rabbit doesn’t…
Weierstrass introduced a shocking idea in 1872. There exist functions that are continuous everywhere but differentiable nowhere. Not just at a few points. At no points whatsoever. This was a direct attack on mathematical intuition. Before Weierstrass, most mathematicians assumed that continuous functions were “smooth” except at isolated trouble spots. Weierstrass proved them wrong with…
Memory is the human ability to store, retain, and recall information when needed. Instead of getting lost in technical definitions, let’s explore ten practical methods to enhance your memory: 1. Simple Repetition Method This classical method is widely used for memorizing poems or any text by reading it repeatedly. The more you read, the more…
I’ve spent years collecting free math resources, including a variety of calculus text books. Not the sketchy PDFs from torrent sites. Not the scanned copies with missing pages. Real Calculus textbooks that professors and universities have made freely available. In this comprehensive guide, you will find a selection of calculus textbooks that cover everything from…
Let $ \mathbf{Q}$ be the set of rational numbers. It is well known that $ \mathbf{Q}$ is an ordered field and also the set $ \mathbf{Q}$ is equipped with a relation called “less than” which is an order relation. Between two rational numbers there exists an infinite number of elements of $ \mathbf{Q}$. Thus, the system…
Looking for Free Algebra and Topology PDF E-books? I’ve compiled a comprehensive list of high-quality, free e-books covering Algebra, Topology, and other related mathematics topics. These resources are ideal for both students and teachers seeking valuable content for their studies or teaching. Download these now and enhance your learning experience! Let’s start. Abstract Algebra Online by…
The D’Alembert Ratio Test is one of the most useful tools for determining whether an infinite series converges or diverges. If you’re studying real analysis or calculus, you’ll use this constantly. Statement of D’Alembert’s Ratio Test A series \( \sum u_n \) of positive terms is convergent if, from and after some fixed term, \(…
When a molecule absorbs light, something has to happen to that energy. It doesn’t just disappear. The molecule can use it for a chemical reaction. It can release it as heat. Or it can spit the light back out, sometimes in beautiful ways we call fluorescence and phosphorescence. The Jablonski diagram maps all of this.…
Problem solving is more than just finding answers. Learning how to solve problems in mathematics is simply to know what to look for. Mathematics problems often require established procedures. To become a problem solver, one must know What, When and How to apply them. To identify procedures, you have to be familiar with the different…
I was reading a book on ancient mathematics problems from Indian mathematicians. Here I wish to share one problem from Bhaskaracharya‘s famous creation Lilavati. Who was Bhaskaracharya? Bhaskara II, who is popularly known as Bhaskaracharya, was an Indian mathematician and astronomer from the 12th century. He’s especially known for the discovery of the fundamentals of…
Henry Poincaré was trying to save the Old classical theory of Physics by Suitable Adjustments & Modifications in it. When the experiments, like Michelson Morley Experiment, in search of the ether drift failed, it began to be increasingly realized that there was no such thing as an absolute or privileged frame of reference and that…
Pick any positive integer. If it’s even, halve it. If it’s odd, triple it and add one. Repeat. You’ll always end up at 1. That’s the Collatz Conjecture. It sounds like something you could prove in an afternoon. Mathematicians have been trying for nearly 90 years. Nobody’s succeeded. You might also hear it called the…
