Complex Numbers and Quadratic Equations Class 11: notes and important questions

Complex Numbers: revision notes

1. The number \(i\)

\(x^2 + 1 = 0\) has no real solution, so we introduce \(i\) with \(i^2 = -1\). Powers repeat in a cycle of four: \(i, -1, -i, 1\); so \(i^{4k} = 1\), \(i^{4k+1} = i\), \(i^{4k+2} = -1\), \(i^{4k+3} = -i\). For \(a \gt 0\), \(\sqrt{-a} = i\sqrt a\).

2. Complex numbers

\(z = a + ib\) with real \(a = \operatorname{Re} z\) and \(b = \operatorname{Im} z\). Two complex numbers are equal exactly when their real parts are equal and their imaginary parts are equal.

3. Algebra of complex numbers

• Add and subtract parts: \((a + ib) \pm (c + id) = (a \pm c) + i(b \pm d)\).
• Multiply like binomials and use \(i^2 = -1\): \((a + ib)(c + id) = (ac - bd) + i(ad + bc)\).
• Divide by multiplying top and bottom by the conjugate of the denominator. The inverse of \(z \neq 0\) is \(\dfrac{\bar z}{|z|^2}\).
• Caution: \(\sqrt a\sqrt b = \sqrt{ab}\) is not true when \(a\) and \(b\) are both negative: \(\sqrt{-4}\sqrt{-9} = (2i)(3i) = -6\).

4. Modulus and conjugate

\(\bar z = a - ib\), \(|z| = \sqrt{a^2 + b^2}\). Properties: \(z\bar z = |z|^2\), \(|z_1z_2| = |z_1||z_2|\), \(\left|\dfrac{z_1}{z_2}\right| = \dfrac{|z_1|}{|z_2|}\), \(\overline{z_1 \pm z_2} = \bar z_1 \pm \bar z_2\), \(\overline{z_1z_2} = \bar z_1\bar z_2\).

5. The Argand plane

\(z = a + ib\) is the point \((a, b)\): real axis horizontal, imaginary axis vertical. \(|z|\) is the distance from the origin; \(|z_1 - z_2|\) is the distance between the points; \(\bar z\) is the reflection in the real axis. Multiplying by \(i\) turns a point through \(90^\circ\) anticlockwise about the origin.

Worked example 1

Find \(i^{-39}\).

\(i^{-39} = i^{-40}\,i = i\).

Worked example 2

Write \(\dfrac{5}{2 + i}\) in the form \(a + ib\).

\(\dfrac{5(2 - i)}{4 + 1} = 2 - i\).

Worked example 3

Find real \(x, y\) with \((x - y) + i(x + y) = 1 + 5i\).

\(x - y = 1\), \(x + y = 5\): \(x = 3\), \(y = 2\).

Common errors

• \(i^2 = 1\), or forgetting to replace \(i^2\) by \(-1\).
• Rationalising with \(a + ib\) instead of the conjugate \(a - ib\).
• \(|a + ib| = a + b\), or including \(i\) in the modulus: \(|3 - 4i| = \sqrt{9 + 16}\).
• Using \(\sqrt a\sqrt b = \sqrt{ab}\) with negative \(a, b\).

Exam tips

• Give every final answer in the form \(a + ib\).
• To show a complex number satisfies an equation, substitute and collect real and imaginary parts separately.