This lecture covers the priority of operations in relation to integers, emphasizing the importance of parentheses to indicate the order of operations and distinguish between different notations. It explains how to simplify notations using rules, such as the use of parentheses to indicate the order of operations and the importance of not combining multiple operations. The concept of opposites is also discussed, highlighting the visual representation of absolute values. Through examples, the lecture demonstrates calculations involving multiple levels of parentheses and how to simplify them using properties like distributivity and commutativity.