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 $i = \sqrt{-1}$ ). Example 1: Simplify $\sqrt{x^{2}}$ This problem looks deceptively simple. Many students...

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## Absolute Value and Logarithms

by ms-academy | Math Learning

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...

## L’Hospital’s Rule

by ms-academy | Math Learning

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 $a=\infty$, $f(x)$ and $g(x)$ are differentiable for all...

## Algebraic Groups

by ms-academy | Math Learning

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...

## Algebraic Rings

by ms-academy | Math Learning

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...

## Find the Vertex by Using the Quadratic Formula

by ms-academy | Math Learning

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

by ms-academy | Math Learning

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

by ms-academy | Math Learning

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...