Category Archives for "Algebra"

The Definitive Higher Math Guide on Long Division and Its Variants — for Integers

In mathematics, long division is a powerful procedure through which division can be carried out with pencil and paper. As powerful as it is, it is generally taught at the pre-college level without much consideration on its underlying theory or its alternatives… In this guide though, we’ll take a look at both of those, and […]

Desmos: A Definitive Guide (How to Perform Cool Computations and Creating Great Graphs Using Online Graphing Calculator)

Think you’re fond of of graphing and computing stuffs? Great! Because you might remember this thing called the Texas Instrument TI-83 from the old days. Sure, while programmable calculators in general are still pretty much popular these days, the graphing calculators from the 21st-century are also coming in waves as we speak — potentially disrupting the market of scientific computing and educational […]

Infinite Limits and the Behaviors of Polynomials at the Infinities — A Theoretical Musing

Regardless of your early surrounding or schooling background, we know for one that there are two kinds of mathematical objects that are kind of hard to miss in life. The names? Polynomial and Infinity! While the former might have sounded a bit like the name of a snake, polynomials is a one-of-its-kind mathematical entity whose perfection defies our mathematical imagination. For […]

The Ultimate Guide to Logarithm — Theory & Applications

For the very vast majority of humans on earth, there is a topic found in the good old math textbooks that many of us still even dread contemplating about, as it seems to mess with our brain in a rather particular way.  The name? $\displaystyle \text{Logarithmus}$ — or Logarithm in English to be sure! As terrible-sounding as it […]

Quadratic Factorisation: A General Method that Conquers Each and Every Quadratic Trinomial

When it comes to quadratic polynomials — which seem to come in all shapes and forms — most of us have spent at least a semester just learning about how to maneuver around them. Indeed, when we look at a standard math curriculum on quadratic expressions, we would find topics such as square of a […]

Modular Arithmetic: Some Advanced Tricks in Finding the Least Positive Residue

We got a least-residue question from David a few weeks ago, but never really took the time to address it properly. Here’s the bombshell David dropped us a bit earlier: Find the least positive residue of $\displaystyle 12^{345}$ in mod $35$. Wow. That seems like a huge number. How on earth do you reduce it […]

How to Solve a System of Linear Equations with an Additional Parameter?

Today, Connor asked the following question involving a system of linear equation with an additional parameter: Given that $x_1 + hx_2 = 4$  and $3x_1 + 6x_2 =8$, find all pairs of the form $(x_1,x_2)$ satisfying both equations. OK…What do we do with that additional $h$? Well, read on!