Complex Numbers

A-Level Further Mathematics Year 1: Complex Numbers

Throughout the unit, students understand what an imaginary and complex number is and how to add, subtract, multiply and divide with them. Later, as learning progresses they solve cubic and quartic equations with complex roots.

Lessons on Complex Numbers

4 Part Lesson
Imaginary Numbers
4 Part Lesson
Complex Conjugate
4 Part Lesson
Roots of Quadratic Equations
4 Part Lesson
Solving Cubic and Quartic Equations

Prerequisite Knowledge


  • Calculate exactly with surds
  • Simplify surd expressions involving squares and rationalise denominators
  • Identify and interpret roots, intercepts, and turning points of quadratic functions graphically
  • Deduce roots algebraically and turning points by completing the square
  • Recognise, sketch and interpret graphs of quadratic functions
  • Solve quadratic equations (including those that require rearrangement) algebraically by factorising, completing the square and using the quadratic formula; find approximate solutions using a graph

Success Criteria

  • Solve any quadratic equation with real coefficients;
  • Solve cubic or quartic equations with real coefficients (given sufficient information to deduce at least one root for cubics or at least one complex root or quadratic factor for quartics)
  • Add, subtract, multiply and divide complex numbers in the form ixy+ with x and y real;
  • Understand and use the terms ‘real part’ and ‘imaginary part’
  • Understand and use the complex conjugate;
  • Know that non-real roots of polynomial equations with real coefficients occur in conjugate pairs.

Teaching Points

  • i = -1 and i2 = -1
  • A complex number is of the form z=x + iy, where x and y are real and i = (-1)
  • The complex conjugate of z = x + yi is z* = x – yi
  • To add and subtract with complex numbers you collect the real and imaginary parts separately.
  • To multiply complex numbers, multiply out the brackets and simplify using i2 = -1.
  • Complex roots of polynomials with real coefficients always occur in conjugate pairs.


  • When multiplying a complex number by its conjugate, students can make mistakes multiplying out the two imaginary parts; for instance –3i × –3i = –3 rather than +9i2 = -9
  • When performing arithmetic with complex numbers, some students write down the final correct answer without working. Therefore, indicating they had used the complex mode on a calculator, which gains no marks.
  • When finding the roots of a cubic or quartic equation is given one of its complex roots, students incorrectly use the sum of the roots as b rather than -b.
  • When given a complex number of the form z = a + bi and asked to work out z2 the most common loss of marks is not separating the real and imaginary parts.

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