Algebra 1

Algebra is a fundamental branch of mathematics that explores the relationships, structures,

and operations involving numbers and abstract symbols. It provides essential tools for

reasoning, problem-solving, and formalizing mathematical ideas. Algebra 1 lays the

groundwork for further studies in mathematics and its applications in science, engineering,

economics, and computer science.

The course begins with an introduction to the principles of logic and set theory, which form

the basis of mathematical reasoning. Students learn how to use truth tables, quantifiers, and

different forms of reasoning, as well as how to describe and manipulate collections of objects

using sets.

It then explores functions and relations, including injective, surjective, and bijective

mappings, direct and inverse images, and restrictions and extensions of functions. Binary

relations such as reflexive, symmetric, antisymmetric, and transitive ones are studied, along

with order and equivalence relations and the concept of equivalence classes.

A major part of the course is devoted to algebraic structures, covering internal laws of

composition, groups, rings, and fields. Students encounter examples such as the finite groups

ℤ/nℤ and the permutation group S₃, as well as the fields ℤ/pℤ, ℝ, and ℂ. These topics

illustrate how algebraic structures can model a wide range of mathematical phenomena.

The final part of the course deals with polynomial rings, including their construction,

arithmetic operations, divisibility, Euclidean division, greatest common divisor (GCD), and

least common multiple (LCM). The study of irreducible polynomials, factorization, and the

roots of polynomials—along with their multiplicities—completes the introduction to modern

algebraic methods.

Algebra 1 encourages students to think abstractly, reason precisely, and recognize connections

between mathematical concepts and their applications in various disciplines.