Try each one on paper first, then check a single step. That is worth far more than reading a finished solution.
Example 1 — dividing complex numbers
Express 1−i3+2i in the form x+yi.
- 1Multiply top and bottom by the conjugate of the denominator, 1+i.This clears i from the bottom, like rationalising a surd.
- 2Numerator: (3+2i)(1+i)=3+3i+2i+2i2
- 3=3+5i−2=1+5i2i2=−2.
- 4Denominator: (1−i)(1+i)=1−i2=2Always real and positive.
- 521+5i=21+25iSplit into real and imaginary parts as asked.
Example 2 — De Moivre's theorem
Use De Moivre's theorem to evaluate (1+i)8.
- 1First convert to modulus–argument form.
- 2∣1+i∣=1+1=2 and arg(1+i)=4πThe point (1,1) lies on the diagonal of the first quadrant.
- 31+i=2(cos4π+isin4π)
- 4De Moivre: raise the modulus, multiply the argument.
- 5(1+i)8=(2)8(cos2π+isin2π)=16(1+0)8×4π=2π, a full turn.
- 6=16Check: (1+i)2=2i, so (1+i)4=−4 and (1+i)8=16 ✓
Example 3 — a quadratic with complex roots
Solve x2−4x+13=0.
- 1Discriminant: 16−52=−36, which is negative.So the roots are complex.
- 2x=24±−36
- 3−36=6i−1=i, so −36=36−1.
- 4x=24±6i=2±3iDivide both parts by 2.
- 5The roots are conjugates, as real coefficients require.Check: (2+3i)+(2−3i)=4 ✓ and (2+3i)(2−3i)=4+9=13 ✓