# Prove: If a + b = 0 then b = -a. (This states that the additive inverse of a real number is unique.)

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We have to prove that if a + b = 0, then b = -a

a + b = 0

add -a to both the sides, this can be done for any values of a and b

=> a + b - a = 0 - a

=> b = -a

**This proves that if a + b = 0, b = -a or that the additive inverse of a number is unique.**

To prove that the additive inverse is unique, we'll have to deny first this assumption.

We'll assume that there are at least two additive inverse of the same real number, b and c, such as:

a + b = 0 (1)

a + c = 0 (2)

We'll equate (1) and (2):

a + b = a + c

We'll subtract a both sides:

b = c

**We notice that the numbers assumed to be the additive inverse of the real number a, are equal, therefore, the additive inverse of a real number is unique. Also, we'll apply the symmetric property of equality, such as if a = -b => b = -a => a + (-a) = 0.**