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Worked solution

Write conjugates and reflect across the real axis

Write down the conjugate of (i) z1=1+3iz_1=1+3i and (ii) z2=−1−4iz_2=-1-4i.

CJC · Example 5

(i) Conjugate of z₁

Answer

z1∗=1−3iz_1^*=1-3i.

Keep the real part 1 and reverse the sign of the imaginary coefficient 3. On the Argand plane, (1,3)(1,3) reflects in the real axis to (1,−3)(1,-3).

Expand diagram

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Equal-unit Argand axes show vectors from O to z1=(1,3) and its conjugate z1*=(1,-3), with dashed grid and equal distances above and below the real axis. The real coordinate remains 1; the imaginary coordinate changes from 3 to -3.
Equal-unit Argand axes show vectors from O to z1=(1,3) and its conjugate z1*=(1,-3), with dashed grid and equal distances above and below the real axis. The real coordinate remains 1; the imaginary coordinate changes from 3 to -3.
Equal-unit Argand axes show vectors from O to z1=(1,3) and its conjugate z1*=(1,-3), with dashed grid and equal distances above and below the real axis. The real coordinate remains 1; the imaginary coordinate changes from 3 to -3.
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(ii) Conjugate of z₂

Answer

z2∗=−1+4iz_2^*=-1+4i.

Keep the real part −1-1 and reverse the imaginary coefficient from −4-4 to 44. Conjugate pairs have the same real part and imaginary parts of equal magnitude and opposite signs.

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Solution notes

Calculator procedure from pages 13–14: MODE → a+bi; MATH → CMPLX → 1:conj(. Enter 1+3i and close the parenthesis to obtain 1−3i. Repeat with −1−4i to obtain −1+4i. These are the two source screen outputs.