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