The Number Line
Imagine a straight line stretching infinitely in both directions. In the middle is Zero. Everything to the right is positive (+), and everything to the left is negative (-). Positive numbers represent "having," while negative numbers represent "owing" or "direction."
Rules for Addition
- Same Sign: Add the numbers and keep the sign. (\(5 + 3 = 8\) and \(-5 + (-3) = -8\)).
- Different Signs: Subtract the smaller number from the larger one and keep the sign of the larger one. (\(5 + (-3) = 2\) and \(-5 + 3 = -2\)).
The Secret of Subtraction
Subtraction is simply adding the opposite. \[a - b = a + (-b)\] \[a - (-b) = a + b\]
Worked Examples
Example 1: Adding Negatives
Evaluate: \(-12 + (-8)\)
- Both are negative. Add \(12 + 8 = 20\).
- Keep the negative sign.
- Result: -20
Example 2: Subtracting a Positive from a Negative
Evaluate: \(-10 - 5\)
- Rewrite as addition of the opposite: \(-10 + (-5)\).
- Both are negative. Add \(10 + 5 = 15\).
- Result: -15
Example 3: Subtracting a Negative
Evaluate: \(7 - (-3)\)
- Two negatives in a row make a positive: \(7 + 3\).
- Result: 10
Example 4: Mixed Signs
Evaluate: \(-4 + 9\)
- The signs are different. Subtract smaller from larger: \(9 - 4 = 5\).
- The larger number (9) was positive, so the result is positive.
- Result: 5
The Bridge to Quantum Mechanics
In Quantum Mechanics, we talk about Potential Energy Barriers. If a particle is at an energy of 5, and it hits a barrier of energy 10, we calculate the difference: \(5 - 10 = -5\). This negative result tells us that the particle is "below the barrier"—it doesn't have enough energy to go over. This simple subtraction of signed numbers is the first step in discovering Quantum Tunneling, where particles can sometimes pass through barriers they shouldn't be able to.