(-1)a=-a Proof

Suzhoumeite

2 min read

·

Jul 03, 2024

14

Share

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.