I am not convinced. You take for granted the existence of multiplication of integers. But how is it defined? What is meant by a*b if a and be are negative integers? Since you haven't said what it means, you can't come to any conclusion about it.

This explanation is insightful, and I appreciate how you’re trying to break it down! Let me simplify and clarify the reasoning even further for better understanding: 1. Why does multiplying two negatives give a positive? Mathematically, it ensures the rules of arithmetic are consistent. The explanation revolves around how addition, subtraction, and multiplication work together. 2. Breaking it down with your example: • You start with the equation x + (-1) \cdot x = 0 , which means adding x and (-1) \cdot x gives zero. • Substituting x = -1 : -1 + (-1) \cdot (-1) = 0 . To balance the equation, the term (-1) \cdot (-1) must equal 1 . This keeps the math consistent and logical. 3. Intuitive explanation: • Think of multiplying negatives as flipping directions: A negative flips the sign of a number, and flipping twice (negative × negative) returns it to the original positive direction. 4. Conclusion: Your explanation using -1 + (-1) \cdot (-1) = 0 is a good way to show how the rule holds! Multiplying negatives gives a positive because it aligns with how numbers behave in arithmetic.