Conjugates & division
The conjugate of z = a + bi is z̄ = a − bi, and the key fact is that z·z̄ = a² + b² is always real and non-negative. That is what makes division work: multiply numerator and denominator by the conjugate of the denominator and the denominator becomes real, exactly as rationalising a surd removes a root.
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Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Write the quotient as a fractionPut the division in the form (a + bi)/(c + di).
- Multiply top and bottom by the conjugate of the DENOMINATORMultiply by (c − di)/(c − di), which is 1 and so changes nothing.
- Simplify the denominatorIt becomes c² + d², a real number.
- Expand the numerator and splitWrite the result as a single a + bi with both parts over the same real denominator.
Worked example
Simplify (2 + 3i)/(1 − i).
- Multiply top and bottom by the conjugate 1 + i.
- Denominator: (1 − i)(1 + i) = 1 − i² = 1 + 1 = 2.
- Numerator: (2 + 3i)(1 + i) = 2 + 2i + 3i + 3i² = 2 + 5i − 3 = −1 + 5i.
- So the quotient is (−1 + 5i)/2.
Answer. −1/2 + (5/2)i.
Where marks get dropped
These are the specific errors that cost credit on conjugates & division questions — QED's rubric penalises each of them separately.
- Multiplying by the conjugate of the NUMERATOR, which leaves the denominator complex and achieves nothing.
- Getting the sign wrong in (c − di)(c + di) = c² + d². The cross terms cancel and −d²i² becomes +d².
- Leaving the answer as a fraction with a complex denominator. The required form is a + bi with real a and b.
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Conjugates & division — frequently asked questions
What are the key conjugate properties?
Conjugation distributes over everything: (z+w)‾ = z̄ + w̄, (zw)‾ = z̄w̄, and z is real exactly when z = z̄.
How does the conjugate relate to the modulus?
z·z̄ = |z|², so the conjugate is how you compute the modulus algebraically and how you invert: z⁻¹ = z̄/|z|².
Why do complex roots come in conjugate pairs?
Because for a polynomial with REAL coefficients, conjugating the equation p(z) = 0 gives p(z̄) = 0. Real coefficients are essential — the claim fails otherwise.
The rest of Complex Numbers
Arithmetic, the Argand plane, polar form and De Moivre. Each subtopic below has its own method, worked example and mark-losing traps.
- 1The imaginary unit i & complex arithmetic
- 2Conjugates & division
- 3Modulus, argument & the Argand diagram
- 4Polar & exponential form re^{iθ}
- 5Multiplying & dividing in polar form
- 6De Moivre’s theorem & powers
- 7nth roots & roots of unity
- 8Solving polynomial equations over ℂ
- 9The fundamental theorem of algebra & conjugate roots
- 10Euler’s formula & trigonometric identities
- 11Loci & regions in the complex plane
- 12Complex multiplication as rotation & scaling
- 13Applications: phasors & AC circuits
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