Math

Weierstrass introduced the idea that there exist functions that are continuous for every value of $x$ but do not possess a derivative at any value of $x$. We now consider the celebrated Weierstrass function, which exhibits this property. In this note, I will demonstrate that such a function exists using Weierstrass’s construction. The Weierstrass Function…

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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…

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…

In this article, we will formulate the D’ Alembert’s Ratio Test on convergence of a series. Let’s start. Statement of D’Alembert Ratio Test A series $ \sum {u_n}$ of positive terms is convergent if from and after some fixed term $ \dfrac {u_{n+1}} {u_n} < r < {1} $ , where r is a fixed…

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…

The Collatz Conjecture is one of the Unsolved problems in mathematics, especially in Number Theory. The Collatz Conjecture is also termed as 3n+1 conjecture, Ulam Conjecture, Kakutani’s Problem, Thwaites Conjecture, Hasse’s Algorithm, and Syracuse Problem. Collatz Conjecture Statement If you keep repeating this procedure, you shall reach the number 1 at last. Illustrations » Starting…

Topic: Beta & Gamma functions The Gamma function and Beta functions belong to the category of special transcendental functions and are defined in terms of improper definite integrals. Definitions of Beta and Gamma functions are given below. But before that, let’s quote the statement of Dirichlet’s theorem so that we can work around Liouville’s extension…

In a busy marketplace, two women sell marbles differently: one sells three marbles for a Rupee, and the other sells two marbles for a Rupee. When they combine their unsold marbles and try to sell them together, they expect to make the same amount of money. But to their surprise, they end up one Rupee…

In 1904, the french Mathematician Henri Poincaré (en-US: Henri Poincare) posed an epoch-making question, which later came to be termed as Poincare Conjecture, in one of his papers, which asked: If a three-dimensional shape is simply connected, is it homeomorphic to the three-dimensional sphere? Henri Poincare – 1904 So what does it really mean? How…

Derivative of x squared As we know, the derivative of x squared, i.e., differentiation of $ x^2$ , with respect to $ x$, is $ 2x$. i.e., $ \dfrac{d}{dx} x^2 = 2x$ A Curious Case Suppose we write $ x^2$ as the sum of $ x$ ‘s written up $ x$ times. i.e., $ x^2…