Prove and fill in the blanks: For all real numbers a, -(-a) = a. (Cancellation property of additive inverses. 1. a is a real number  2. -a and -(-a) are real numbers  3. -(-a) = - (-a) + 0 4. _____________ = -(-a) + [(-a) + a]  5. _____________ = [-(-a) + (-a)] + a   6. ___________ = 0 + a.  7. _____________ = a  8. -(-a) = a

Let's go through the proof step by step, identifying the axioms or theorems that justify each step:

1. **a is a real number.**  
   **Justification:** This is given by the problem statement.

2. **-a and -(-a) are real numbers.**  
   **Justification:** **Closure property of real numbers under addition.**  
   (Since a is a real number, its additive inverse -a is also a real number. Similarly, -(-a) is also a real number.)

3. **-(-a) = -(-a) + 0.**  
   **Justification:** **Identity property of addition.**  
   (Adding 0 to any number gives the number itself.)

4. **-(-a) + 0 = -(-a) + [(-a) + a].**  
   **Justification:** **Additive inverse property.**  
   (The sum of a number and its additive inverse is 0, i.e., \(0 = (-a) + a\).)

5. **-(-a) + [(-a) + a] = [-(-a) + (-a)] + a.**  
   **Justification:** **Associative property of addition.**  
   (The grouping of the numbers in addition can be changed.)

6. **[-(-a) + (-a)] + a = 0 + a.**  
   **Justification:** **Additive inverse property.**  
   (The sum of a number and its additive inverse is 0, i.e., \(-(-a) + (-a) = 0\).)

7. **0 + a = a.**  
   **Justification:** **Identity property of addition.**  
   (Adding 0 to any number gives the number itself.)

8. **-(-a) = a.**  
   **Justification:** **Cancellation property of additive inverses.**  
   (This final step is the conclusion of the proof.)

Would you like more details on any specific step or concept?

Here are 5 related questions to expand your understanding:

1. How does the identity property of addition simplify expressions in algebra?
2. Can you prove that \(a + (-a) = 0\) using the properties of real numbers?
3. Why is the associative property of addition important in algebraic proofs?
4. How does the closure property apply to other operations like multiplication?
5. What is the significance of the cancellation property of additive inverses in solving equations?

**Tip:** Understanding the properties of real numbers, such as identity and inverse properties, is crucial when proving algebraic identities. These foundational concepts are used frequently in mathematical proofs.

Explore the proof and fill in the blanks for the cancellation property of additive inverses in real numbers. Understand the properties of real numbers and the steps involved in proving -(-a) = a. Suitable for Grades 7-9.

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