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...
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Modular Arithmetic & 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 12 with 0,...
Find the Vertex by Using the Quadratic Formula
The Quadratic Formula is primarily used to identify the roots (\(x\)-intercepts) of a quadratic function. What many people don't know is that you can also easily find the vertex of the function by simply looking at the Quadratic Formula! Graphing Quadratic Functions When graphing quadratic...
Completing the Square
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 the notation...
Cardinality and 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 if a bijection...
The Quadratic Equation
The quadratic equation is always the answer. Time and time again, while tutoring students, I encounter a resistance to using an unfailing tool called the quadratic equation. This simple algebraic mathematical statement allows a student to find the roots of any clumsy second order polynomial with...
The Natural Logarithm Rules
The natural logarithm, whose symbol is ln, is a useful tool in algebra and calculus to simplify complicated problems. In order to use the natural log, you will need to understand what ln is, what the rules for using ln are, and the useful properties of ln that you need to remember. What is the...