Knowledge organisers

Complex Numbers and Quadratic Equations: CBSE Class 11 Maths knowledge organiser

Everything to know about complex numbers and quadratic equations on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Complex number: z = a + ib with i² = −1; a is the real part and b the imaginary part.
Conjugate: The conjugate of a + ib is a − ib.
Modulus: |a + ib| = √(a² + b²), the distance from the origin on an Argand plane.

Key formulas

\(i^2=-1\); \(i^{4k}=1\)	\((a+ib)(c+id)=(ac-bd)+i(ad+bc)\)
Modulus; conjugate	\(\|z\|=\sqrt{a^2+b^2},\) \(z\bar z=\|z\|^2,\) \(z^{-1}=\frac{\bar z}{\|z\|^2}\)
\(\|z_1z_2\|=\|z_1\|\|z_2\|\); \(\overline{z_1z_2}=\bar z_1\bar z_2\); Argand: \(a+ib\) ↦ \((a,b)\)

Worked example

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

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

Common mistakes

\(i^2 = 1\), 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 \(a, b\).

You should be able to…

Simplify powers of i, add, multiply and take conjugates and moduli.
Divide complex numbers, compare real and imaginary parts and plot numbers on the Argand plane.
Combine the properties of modulus and conjugate in longer calculations and in circuit or rotation contexts.
Solve equations in z = x + iy and interpret them on the Argand plane.

The printable sheet

Complex Numbers and Quadratic Equations knowledge organiser for CBSE Class 11 Maths: one A4 page of key definitions, formulas, a worked example and common mistakes — Complex Numbers and Quadratic Equations knowledge organiser (CBSE Class 11 Maths), A4. Download the PDF.

Revise it next

Other CBSE Class 11 Maths topics: Sets · Relations and Functions · Trigonometric Functions · Linear Inequalities · Permutations and Combinations · Binomial Theorem · Sequences and Series · Straight Lines · Conic Sections · Introduction to Three Dimensional Geometry · Limits and Derivatives · Statistics · Probability · All CBSE Class 11 Maths organisers