Algebra Archives - Projeda

Inverse Property

The Inverse Property is the axiom that ensures we can “undo” any operation. Within a Field, every element must have a corresponding “opposite” that, when combined with it, results in the Identity Element (0 for addition, 1 for multiplication). The […]

Associative Property

In University Algebra and Trig Course, the Associative Property is the axiom that governs the “grouping” of elements within a single operation. While the Commutative Property tells us that order doesn’t matter, the Associative Property tells us that the placement […]

Distributive Property

The Distributive Property is the “bridge” axiom that links the two fundamental operations of a Field: Addition and Multiplication. It is the only axiom that describes how these two distinct operations interact with one another. Formal Definition The Distributive Property […]

Types of Real Numbers

There are a number of Types of Real Numbers, subsets within the set of real numbers, that are useful classifications that highlight distinct differences within the Real Numbers. These different classifications of real numbers are: (In the following description of […]

Real Numbers

Real Numbers (denoted by the symbol $\mathbb{R}$) are presented not just as a collection of points on a line, but as a specific algebraic structure: a Complete Ordered Field. While the Rational Numbers ($\mathbb{Q}$) allow us to perform basic arithmetic, […]

Field Axioms

The Field Axioms provide the definitive structural foundation for the number systems we use in Algebra and Trigonometry. While lower-level math focuses on how to perform calculations, higher-level algebra focuses on why those calculations are valid by defining the properties […]

Vertical Line Test

The Vertical Line Test is the geometric realization of the fundamental definition of a function. As you progress through your STEM track, you will find that the shift from seeing a “drawing” to seeing a “mapping” is critical for higher-level […]

Order of Operations

In mathematical systems, the Order of Operations is the set of rules that ensures a single, unambiguous result for any given expression. Without these rules, a problem like $x = 2 + 5 \times 3$ could be 21 or 17, […]

Calculus — Mathematical Functions

A function is a unique relationship that can be made between numbers that enables us to relate one group of numbers to another group of numbers, or to create a group of numbers from another group using a specific relation. […]

Functions & Graphs — Introduction

Calculus is the mathematics that describes and investigates rates of change which, fundamentally, is going to rely on a deep understanding of functions. More generally, it requires an understanding of data which allows a mathematical analysis of said data. In […]