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Understanding the properties of real numbers is crucial for success in advanced algebra, calculus, and foundational computer science. By the end of this course, you will be able to confidently apply core concepts like the Euclidean Division Lemma, prove the irrationality of numbers, and accurately determine the nature of decimal expansions. What you'll learn: * Understand the fundamental differences between rational and irrational numbers and their unique properties. * Apply the Euclidean Division Algorithm to efficiently solve problems involving divisibility and highest common factors (HCF). * Master the Fundamental Theorem of Arithmetic for prime factorization and finding the least common multiple (LCM). * Practice techniques for rigorously proving that specific numbers, such as square roots, are irrational. * Analyze and predict whether a rational number's decimal expansion is terminating or non-terminating repeating. * Learn how these number theory concepts form the basis for cryptographic and computational algorithms. The course begins by defining number systems and key terminology. We then move into practical applications of division algorithms and factorizations, concluding with rigorous proofs and the analysis of decimal expansions. This course is designed for absolute beginners, including high school students preparing for exams and adult learners seeking to refresh their foundational mathematical knowledge. No prior advanced math knowledge is required. Start building your strong mathematical foundation today.