Completing the Square HT Goodwill Overview of This Article Introduction A brief discussion about what completing the square is and what we use it for. Focuses on using the technique for other reasons than solving equations. Background Math Development of the patterns we use for completing the square. This part is important since it introduces […]

10 ways tutoring can help you Finding a good math tutor can be essential to your success in school. Tutoring can do much more than just improving math grades. Here are 10 ways in which you can benefit from tutoring. 1. Improve your confidence When working one-on-one outside of the classroom setting, you might find […]

Modular Arithmetic and Fermat’s Little Theorem Modular arithmetic is a way of counting in which the numbers wrap around after reaching a certain value. The clock is often used as an analogy. While time always progresses forward, the 12-hour clock “resets” to 1 after passing 12 (13 o’clock is equivalent to 1 o’clock). If we replace […]

Introduction to Algebraic Rings An algebraic ring is one of the most fundamental algebraic structures. It builds off of the idea of algebraic groups by adding a second operation  (For more information please review our article on groups). For rings we often use the notation of addition and multiplication because the integers are a good […]

Introduction to Algebraic Groups One of the most fundamental algebraic structures in mathematics is the group. A group is a set of elements paired with an operation that satisfies the following four conditions: I. It is closed under an operation (represented here by “+”, although it does not necessarily mean addition): For all elements a […]

Countably Infinite Sets Cardinality is a term used to describe the size of sets. Set A has the same cardinality as set B if a bijection exists between the two sets. We write this as |A| = |B|. One important type of cardinality is called “countably infinite.” A set A is considered to be countably infinite […]

L'Hospital's Rule L’Hospital’s Rule is a useful way to evaluate tricky limits. It is most often used for limits of indeterminate form. The rule is as follows: If $f(x)$ and $g(x)$ are differentiable on some interval around the number $a$ (or if $a=∞$, $f(x)$ and $g(x)$ are differentiable for all $x>ε$ for some $ε$), and […]

Absolute Value and Logarithms Absolute values often turn up unexpectedly in problems involving logarithms. That’s because you can’t take the log of a negative number. Let’s first review the definition of the logarithm function: Logb x = y ⇔ by = x (The double arrow is a bi-conditional, which means that one side is true if […]

Absolute Value and Square Roots   Absolute values often show up in problems involving square roots. That’s because you can’t take the square root of a negative number without introducing imaginary numbers (those involving i = √-1 ).   Example 1: Simplify √x². This problem looks deceptively simple. Many students would say the answer is […]

How to Hire the Right Math Tutor   The Three Biggest Pitfalls in Hiring a Tutor, and How to Handle Them   When searching for a tutor there are three major things you need to have in place to ensure a good experience. Having a good tutor can be a great benefit– far more bang for your […]