The Rules of Multiplication
Multiplying signed numbers follows a very rigid logic:
- Positive \(\times\) Positive = Positive (\(2 \times 3 = 6\))
- Negative \(\times\) Negative = Positive (\(-2 \times -3 = 6\))
- Positive \(\times\) Negative = Negative (\(2 \times -3 = -6\))
Memory Trick: "The friend of my friend is my friend. The enemy of my enemy is my friend. The friend of my enemy is my enemy."
Division Rules
Division follows the exact same rules as multiplication because division is just multiplying by a fraction.
Worked Examples
Example 1: Multiplying Mixed Signs
Evaluate: \(4 \times (-5)\)
- Calculate \(4 \times 5 = 20\).
- One is positive, one is negative. The result is negative.
- Result: -20
Example 2: Multiple Negatives
Evaluate: \((-2) \times (-3) \times (-4)\)
- Step 1: \((-2) \times (-3) = 6\) (Negative \(\times\) negative = positive).
- Step 2: \(6 \times (-4) = -24\) (Positive \(\times\) negative = negative).
- Result: -24 (Note: An odd number of negative signs always leads to a negative result).
Example 3: Division
Evaluate: \(-100 \div (-25)\)
- Calculate \(100 \div 25 = 4\).
- Both are negative. The result is positive.
- Result: 4
Example 4: Division by Zero
Evaluate: \(-5 \div 0\)
- You cannot divide by zero. It is Undefined. In physics, if your equation divides by zero, it often means your theory has reached a "Singularity."
- Result: Undefined
The Bridge to Quantum Mechanics
In Chapter 4, we will learn about the imaginary number \(i\). The definition of \(i\) is based on the rule we learned today: \(i \times i = -1\). This is a shock because we just learned that a negative times a negative should be positive! By breaking this rule, we create a new type of number that allows us to describe Waves. Without the foundational understanding of why negative times negative equals positive, the invention of \(i\) would have no meaning.