What is the complex conjugate of z = a + bi?

What is the complex conjugate of z = a + bi?

• A. z* = a + bi
• B. z* = a - bi
• C. z* = -a + bi
• D. z* = -a - bi

Answer: (B)

The main idea is that complex conjugation flips the sign of the imaginary part while keeping the real part the same. For a + bi, with a and b real, the complex conjugate is a − bi. This is why the conjugate is a − bi: it preserves the real component a and changes the imaginary component from +bi to −bi. As a quick check, multiplying the number by its conjugate gives (a + bi)(a − bi) = a^2 + b^2, a real value. For a concrete example, if z = 3 + 4i, its conjugate is 3 − 4i. The forms that keep the imaginary part unchanged or flip the real part or flip both parts don’t match the standard conjugation rule in general.

The main idea is that complex conjugation flips the sign of the imaginary part while keeping the real part the same. For a + bi, with a and b real, the complex conjugate is a − bi. This is why the conjugate is a − bi: it preserves the real component a and changes the imaginary component from +bi to −bi. As a quick check, multiplying the number by its conjugate gives (a + bi)(a − bi) = a^2 + b^2, a real value.

For a concrete example, if z = 3 + 4i, its conjugate is 3 − 4i. The forms that keep the imaginary part unchanged or flip the real part or flip both parts don’t match the standard conjugation rule in general.