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Achievement objective M8-9

In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems and model situations that require them to:

  • manipulate complex numbers and present them graphically.


  • Uses and manipulates surds and other irrational numbers.
  • Interprets and uses graphical representations of complex numbers, using polar and rectangular form on an Argand diagram.
  • Uses and manipulates complex numbers in rectangular form and polar form, and makes links with their graphical representation.
  • Finds roots over the complex number system for polynomial equations with real coefficients, making links with their graphs.
  • See key mathematical ideas on NZmaths.


M8-9 links from M7-6, M7-7, M8-7.

Possible context elaborations

  • Explore the need for, and relevance of, different number systems leading to the development of i and complex numbers.
  • Carry out mathematical operations on expressions involving i, for example, powers of i.
  • Convert between rectangular and polar form.
  • Carry out operations on rcisө.
  • Find the locus of the point z if (z+i)/(z-i) is purely imaginary.
  • Use de Moivre’s theorem to solve equations of the form zn = a and display these solutions.
  • Solve equations of the form zn=rcis, or zn=a+bi where a, b are real and n is a positive integer.
  • Find the 5th roots of -32 and show them on an Argand diagram.
  • Find the real and complex roots of a cubic equation with real coefficients.
  • Represents a locus using an equation with complex variables.
  • Activity: Big Bang Theory - Fractals and complex numbers
  • Complex numbers investigation
  • Complex numbers matching activity: Review activity

Assessment for qualifications

NCEA achievement standards at levels 1, 2 and 3 have been aligned to the New Zealand Curriculum. Please ensure that you are using the correct version of the standards by going to the NZQA website.

The NZQA subject-specific resources pages are very helpful. From there, you can find all the achievement standards and links to assessment resources, both internal and external.

The following achievement standard(s) could assess learning outcomes from this AO:

Last updated September 14, 2018