Question

the additive inverse of a complex number ( z ) is a complex number ( z_a ) such that ( z + z_a = 0 ). find the additive inverse of each complex number.

a. ( z = 1 + i )

the additive inverse is ( z_a = square ).

b. ( z = 3 - i )

the additive inverse is ( z_a = square ).

c. ( z = -2 + 8i )

the additive inverse is ( z_a = square ).

Explanation:

Step1: Recall additive inverse definition

For a complex number \( z = a + bi \), its additive inverse \( z_a \) satisfies \( z + z_a = 0 \). So, \( z_a=-z=-a - bi \).

Step2: Solve for part a (\( z = 1 + i \))

Here, \( a = 1 \), \( b = 1 \). Then \( z_a=- (1 + i)=-1 - i \).

Step3: Solve for part b (\( z = 3 - i \))

Here, \( a = 3 \), \( b=-1 \). Then \( z_a=- (3 - i)=-3 + i \).

Step4: Solve for part c (\( z = -2 + 8i \))

Here, \( a=-2 \), \( b = 8 \). Then \( z_a=- (-2 + 8i)=2 - 8i \).

Answer:

a. \(-1 - i\)
b. \(-3 + i\)
c. \(2 - 8i\)