Complex Numbers

This achievement standard involves manipulating real and complex numbers, and solving equations.complex

  • Manipulate real and complex numbers, and solve equations.
  • Manipulation will be based on a selection from:

    • conversion between polar and rectangular forms of real and complex numbers

    • simplification of sums, differences, products, and quotients of surds or complex numbers expressed in rectangular form

    • simplification of products or quotients of complex numbers expressed in polar form

    • use of De Moivre’s theorem in the simplification of expressions such as .
  • Equations will be based on a selection from:

    • quadratic

    • cubic (limited to rational roots only)

    • exponential, such as

    • logarithmic, such as log(x + 5) = 1.34 (any base).
  • Candidates will be expected to have a knowledge of:

    • the remainder and factor theorem

    • the process of completing the square.
  • Solve more complicated equations.
  • Assessment will be based on a selection of those equations identified for achievement, plus:

    • irrational equations, such as x + 2 = 2

    • cubic equations with one integer root and two complex roots

    • equations of the form zn = a, zn = r cis ,

zn = a + b i where a, b are real and n is a positive integer.

  • Solve problem(s) involving real or complex numbers.
  • Problems will require a chain of reasoning.
  • Problems may include:

    • algebraic proofs

    • loci – geometric representation of complex numbers

    • multi-step equations

    • binomial expansions for small positive integer exponents

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