The negative times negative equals positive principle is a fundamental rule in mathematics. When we multiply two negative numbers together, the result is always positive. For example, negative five times negative three equals positive fifteen. This rule ensures mathematical consistency and is essential for algebraic operations.
Let's examine all the sign rules for multiplication. When multiplying two positive numbers, the result is positive. When multiplying a positive and a negative number, the result is negative. When multiplying a negative and a positive number, the result is also negative. Finally, when multiplying two negative numbers, the result is positive. This last rule is highlighted because it's the focus of our discussion.
We can visualize negative multiplication using a number line. Starting at zero, moving left represents negative direction. If we take two steps of negative three, we reach negative six. However, when we multiply negative two by negative three, we reverse this process. The negative sign means we reverse the direction, so instead of moving left, we move right, ending up at positive six. This demonstrates why negative times negative equals positive.
The negative times negative equals positive rule is essential for mathematical consistency. It preserves the distributive property, which states that a times the sum of b and c equals a times b plus a times c. Let's see why this matters. If we calculate negative two times the sum of three and negative three, we get negative two times zero, which equals zero. Using the distributive property, this should also equal negative two times three plus negative two times negative three. We know negative two times three is negative six. For the equation to equal zero, negative two times negative three must equal positive six. This proves that negative times negative must equal positive to maintain mathematical consistency.
The negative times negative equals positive principle has many real-world applications. In physics, it helps us understand forces and motion. In economics, it's used for calculating compound interest on debts. For example, consider temperature: if the temperature drops 3 degrees Celsius per hour, then 2 hours ago it was 6 degrees warmer than now. This is calculated as negative 3 times negative 2 equals positive 6. The key takeaways are: two negatives make a positive, this rule is essential for mathematical consistency, it appears in real-world problems, and it forms the foundation for advanced mathematics. Understanding this principle is crucial for success in algebra and beyond.