That unnameable confusing thing about Field extensions

A first incursion into the concept of field extensions proves to be rather confusing. And I’ll tell you why. Things which seem difficult, and sometimes impossible to prove, are proved very easily using concepts of vector spaces. For example, we know that is a finite extension of . Prove that is also algebraic over .Continue reading “That unnameable confusing thing about Field extensions”

Generalizing the product of sets to a universal property

The concept of product and co-product can be generalized to a universal property. However, “how” to do so is given in a rather unclear manner in most books and internet links. An example would be I have somehow patched together a coherent explanation. I hope this helps anyone starting out in Category Theory. LetContinue reading “Generalizing the product of sets to a universal property”

Finally, a valid generalisation of the Third Isomorphism Theorem

Let be a commutative group. Let and be subgroups of . If , then . However, what if ? We will consider the general scenario, where and are any subgroups of , provided is not a proper subgroup of . Then In the case that , we have . I have arrived upon this resultContinue reading “Finally, a valid generalisation of the Third Isomorphism Theorem”

A generalization of the quotient group: an exhaustive analysis of all possible cases

We know that if is a group and is a normal subgroup of , then is a quotient group. Today, we shall try and explore some fundamental questions that have plagued my understanding of Algebra for a long long time. What if is just a subset of , and not necessarily a subgroup? Does meanContinue reading “A generalization of the quotient group: an exhaustive analysis of all possible cases”

Aristotle and I I so remember trying to re-interpret nature based on my own empirical observations and intuition, philosophizing and hatching explanations. I can’t even remember when I started this. Maybe in class 2? When I used to take a sheet of paper and pencil and write and write and write about how clouds form, how insects move,Continue reading “Aristotle and I”

My original proof of “continuous mappings on compact metric spaces are uniformly continuous”

The proof of “continuous mappings on compact metric spaces are uniformly continuous” is rather convoluted and opaque, as given in most real analysis textbooks (Rudin included). There is a lot of unnecessary bookkeeping, and the treatment is not really motivated. Given below is my own proof of the theorem, which is inspired by the existingContinue reading “My original proof of “continuous mappings on compact metric spaces are uniformly continuous””

Explaining the beginner’s problems with Topology

When one suddenly starts studying compactness and connectedness and other topological concepts in college, one is likely to get confused. Where did all these concepts come from? Then, seemingly intuitive properties of start being proven using these alien notions. Forming a big picture which includes these concepts seems difficult. One still thinks about the realContinue reading “Explaining the beginner’s problems with Topology”

Why ax+by+cz=d is a plane

Why is a plane in three dimensions? Because it just is? Not good enough. There are proofs for this. They involve normals and dot products and other fancy things that you’re not entirely sure are legit (if you’re in high school). I’m going to try a very different approach to convince you that this isContinue reading “Why ax+by+cz=d is a plane”

Vector addition: my mortal enemy

I have always, A.L.W.A.Y.S. found vectors confusing. WHY do they add up in that funny manner. Why do we learn about them at all? They just seem to be a piece of complicated machinery that just makes life difficult for high school and college students. It was only on learning abstract mathematics that I somehowContinue reading “Vector addition: my mortal enemy”

Choosing a new coordinate system

I often used to wonder why one coordinate system is more appropriate than another in particular situations. For example, when dealing with circular motion, we are often advised to use polar coordinates. I used to wonder why we can’t use only coordinates. My teacher used to say using polar coordinates makes things easier. But IContinue reading “Choosing a new coordinate system”