what is the theorem that proves 0 - (-b) = b

The theorem that proves \( 0 - (-b) = b \) is based on the **Additive Inverse Property** and the **Identity Property of Addition**. Here's the relevant theorem:

### **Theorem:**
For any real number \( b \), the expression \( 0 - (-b) = b \).

### **Proof:**

1. **Start with the Left-Hand Side:**
   \[
   0 - (-b)
   \]
   **Justification:** This is the expression we want to simplify and show that it equals \( b \).

2. **Definition of Subtraction:**
   \[
   0 - (-b) = 0 + (-(-b))
   \]
   **Justification:** Subtracting a number is the same as adding its additive inverse. Here, subtracting \(-b\) is the same as adding the opposite of \(-b\), which is \(-(-b)\).

3. **Double Negative Property:**
   \[
   0 + (-(-b)) = 0 + b
   \]
   **Justification:** The negative of a negative number is the original positive number, so \(-(-b) = b\).

4. **Identity Property of Addition:**
   \[
   0 + b = b
   \]
   **Justification:** The Identity Property of Addition states that adding zero to any number does not change its value.

5. **Conclusion:**
   \[
   0 - (-b) = b
   \]
   **Justification:** We have shown that the expression simplifies to \( b \), which completes the proof.

This theorem leverages basic properties of real numbers to demonstrate that subtracting a negative number from zero results in the positive version of that number.

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How does the Identity Property of Addition apply to other arithmetic operations?
2. Can you prove that \( b - 0 = b \) using similar properties?
3. How would the proof change if \( b \) were a negative number?
4. What is the significance of the Additive Inverse Property in solving equations?
5. How would you extend this theorem to prove \( a - (-b) = a + b \) for any real number \( a \)?

**Tip:** Understanding the fundamental properties of numbers, like the additive inverse and identity properties, is key to proving many arithmetic and algebraic theorems.

Explore the theorem that proves 0 - (-b) = b with a detailed mathematical proof. This problem covers the Additive Inverse Property and the Identity Property of Addition, essential concepts in real number arithmetic. Suitable for students in Grades 7-9.

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