If z = 5 + 3i, which of the following is the complete conjugate of z?

Rationale

5 - 3i is the complete conjugate of z.

The complex conjugate of a complex number is obtained by changing the sign of its imaginary part. In this case, for the complex number z = 5 + 3i, the imaginary part is +3i, and its conjugate is 5 - 3i.

A (-5 - 3i) YOUR ANSWER

This option represents a complex number with both real and imaginary parts negated. It is not the conjugate of z since the conjugate only changes the sign of the imaginary part, not the real part.

B (-5 + 3i) YOUR ANSWER

Here, the real part is negative, while the imaginary part remains positive. This is not the conjugate of z either, as it incorrectly alters the real part of the original number.

C (-3 + 5i) YOUR ANSWER

This choice rearranges the numbers and changes both the real and imaginary parts. Therefore, it does not represent the conjugate of z, which requires only a sign change of the imaginary component.

D (3 + 5i) YOUR ANSWER

In this option, both parts are altered, with the real part being modified to 3. This does not correspond to the definition of a conjugate, which should retain the original real part of z.

E (5 - 3i) CORRECT

This is the correct choice as it retains the real part of z (5) and changes the sign of the imaginary part from +3i to -3i, perfectly fitting the definition of the complex conjugate.

Conclusion

The complete conjugate of a complex number involves flipping the sign of its imaginary part while keeping the real part unchanged. For the given complex number z = 5 + 3i, the correct conjugate is 5 - 3i, as it reflects this property accurately. All other choices incorrectly modify either the real part, the imaginary part, or both, failing to meet the criteria for a conjugate.

If z = 5 + 3i, which of the following is the complete conjugate of z?

5 - 3i is the complete conjugate of z.

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