Lectures related to Integers and Rings

Rational Numbers: Construction and Properties

Covers the construction and properties of rational numbers, including fractions and equivalence, with proofs on commutativity.

Integers: Sets, Maps, and Principles

Introduces sets, maps, divisors, prime numbers, and arithmetic principles related to integers.

Fractions of Rings

Explores the concept of fractions of rings and their uniqueness in ideals and quotients.

Multiplication in Rational Numbers

Covers the extension of addition and multiplication operations in rational numbers.

Division and Multiplication in Rational Numbers

Explores division and multiplication in rational numbers, emphasizing the concept of multiplying by the inverse.

Functions and Integers

Covers functions, integers, GCD, and reasoning by recurrence, including the Euclidean algorithm and induction principle.

Elementary Algebra: Numeric Sets

Explores elementary algebra concepts related to numeric sets and prime numbers, including unique factorization and properties.

Properties of Division

Covers the properties of division in integers and the relationship between divisibility and unique quotients.

Cartesian Product and Induction

Introduces Cartesian product and induction for proofs using integers and sets.

Function Composition and Integers

Covers function composition and integers, including properties and examples.

Shor's algorithm: factoring integers

Covers the basics of Shor's algorithm for factoring integers and the steps involved in the quantum algorithm.

Number Theory: GCD and LCM

Covers GCD, LCM, and the Euclidean algorithm for efficient computation of GCD.

Number Theory: Greatest Common Divisor and Prime Factorization

Introduces greatest common divisor, prime factorization, and the Euclidean Algorithm.

Logic and Proof Techniques

Covers integers, rationals, logic, proof techniques, functions, and relations using examples and truth tables.

Arithmetic Fundamentals: Equivalence and Irreducibility

Covers the fundamental theorem of arithmetic, focusing on equivalence and irreducibility of integers.

Basic Properties

Covers basic properties of natural numbers, including order relations and inverses.

Linear Algebra: Reciprocity and Equivalence

Explores reciprocity, equivalence, and proof techniques in linear algebra, emphasizing logical reasoning and mathematical rigor.

Mathematical Induction: Basics and Applications

Introduces mathematical induction principles and applications, including inequalities, divisibility, subsets, and strong induction.

Complexity & Induction: Algorithms & Proofs

Covers worst-case complexity, algorithms, and proofs including mathematical induction and recursion.

Number Theory: Operations and Equivalence

Covers operations and equivalence relations in number theory, including addition, subtraction, multiplication, division, and properties of neutral and inverse elements.