Additive inverse of a complex number

The additive inverse of a complex number a + bi is the complex number x + yi such that a + bi + x + yi = 0

Solve for x and y using a + bi + x + yi = 0

a + bi + x + yi = 0

Subtract x from each side of the equation

a + bi + x - x + yi = 0 - x

a + bi + yi = 0 - x

Subtract yi from each side of the equation

a + bi + yi - yi = 0 - x - yi

a + bi = - x - yi

Now, you can compare the real parts and the imaginary parts of a + bi = - x - yi

We get a  = -x and b = -y

When a = -x, x = -a

When b = -y, y = -b

The additive inverse is x + yi = -a + -bi

Other examples showing how to find the additive inverse of a complex number

Example #1:

Find the additive inverse of 4 + -5i

The additive inverse of 4 + 5i is the complex number x + yi such that

4 + -5i + x + yi = 0

4 + -5i = - x - yi

Compare the real parts and the imaginary parts of 4 + -5i = - x - yi

We get -x = 4 and -y = -5

When -x = 4, x = -4

When -y = -5, y = 5

The additive inverse of 4 + -5i is the complex number -4 + 5i 

Example #2:

Find the additive inverse of -2 + -6i

The additive inverse of -2 + -6i is the complex number x + yi such that

-2 + -6i + x + yi = 0

-2 + -6i = - x - yi

Compare the real parts and the imaginary parts of -2 + -6i = - x - yi

We get -x = -2 and -y = -6

When -x = -2, x = 2

When -y = -6, y = 6

The additive inverse of -2 + -6i is the complex number 2 + 6i