Mathematics Archives - Projeda

Vectors

Vectors (or Vector Quantities) are the combination of two scalars. (Two units, each only with a magnitude only.) Direction and Distance are two common scalars, that combine to form Direction, a vector. In order to describe any type of motion […]

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 […]

Commutative Property

The Commutative Property is defined as a fundamental axiom of a Field, meaning that in mathematics where this property is applied, we can expect an identical result regardless of the sequence in which elements are combined. The “order-independence” of a […]

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, […]