Proof: (-1)a = -a
In this article, we will provide a proof for the algebraic property (-1)a = -a.
What is the problem?
The problem states that when we multiply -1 with any number a, the result is equal to -a. This seems intuitive, but we need to prove it mathematically.
Proof
Let's start with the definition of -a, which is the additive inverse of a. By definition, -a is a number that when added to a, results in 0:
a + (-a) = 0 ... (1)
Now, let's multiply both sides of equation (1) by -1:
(-1)(a + (-a)) = (-1)(0)
Using the distributive property of multiplication over addition, we get:
(-1)a + (-1)(-a) = 0
Simplifying the right-hand side of the equation, we get:
(-1)a + a = 0 ... (2)
Now, we can rewrite equation (2) as:
(-1)a = -a
Thus, we have proved that (-1)a = -a.
Conclusion
In conclusion, we have successfully proved that (-1)a = -a using the definition of the additive inverse and the distributive property of multiplication over addition. This property is a fundamental concept in algebra and is used extensively in various mathematical operations.