AS 91577 · 5 credits · External
Complex numbers
Apply the algebra of complex numbers in solving problems
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This standard asks you to work with complex numbers — numbers that have both real and imaginary parts — and use algebra to solve problems with them. You'll need to convert between different forms, find arguments and moduli, apply important theorems, and check which solutions are actually valid. It's about using mathematical techniques confidently and showing your working clearly so examiners can see your reasoning.
What to do, grades and common mistakes
- ·Convert complex numbers between rectangular form (a + bi) and polar form (r cis θ), and recognise when each form is useful
- ·Multiply and divide complex numbers accurately in both forms, including using conjugates to simplify division
- ·Find the modulus and argument of complex numbers, including those not in the first quadrant, and plot them on Argand diagrams
- ·Apply de Moivre's theorem to simplify powers of complex numbers and solve equations like z^n = r cis θ
- ·Check the validity of your solutions — especially with surds and quadratic equations — to ensure they actually work in the original problem
You can manipulate complex numbers in both forms, apply the quadratic formula or complete the square, find moduli and arguments correctly, apply de Moivre's theorem, and simplify expressions by grouping real and imaginary parts.
