Complex arithmetic using polar form

Specialist Mathematics · Unit 2 — Complex numbers, further proof, trigonometry, functions and transformations · Complex arithmetic and algebra

Learning objectives (6)

LO-1Convert between Cartesian form and polar form.LO-2Express a complex number in Cartesian form 𝑧 = 𝑎 + 𝑏𝑖 and polar form. 𝑧 = 𝑟 (cos(𝜃) + 𝑖 sin(𝜃)) or 𝑧 = 𝑟 cis(𝜃)LO-3Sketch and use complex numbers in polar form as polar coordinates. Specialist Mathematics 2025 v1.4 LO-4Understand and use multiplication, division of complex numbers in polar form and the geometric interpretation of these. 𝑧1 𝑧2 = 𝑟1 𝑟2 cis(𝜃1 + 𝜃2) 𝑧1 𝑧2 = 𝑟 1 𝑟 2 cis(𝜃1 − 𝜃2)LO-5Understand the difference between the argument, arg(𝑧), and the principal argument, Arg(𝑧) of a non-zero complex number 𝑧. arg(𝑧) = Arg(𝑧) + 2𝜋𝑛, 𝑛 ∈ ℤ LO-6Use the modulus |𝑧| of a complex number 𝑧 and the principal argument Arg(𝑧) of a non-zero complex number 𝑧. |𝑧| = √𝑎2 + 𝑏2 Arg(𝑧) = 𝜃, tan(𝜃) = 𝑏 𝑎, −𝜋 < 𝜃 ≤ 𝜋, 𝑎 ≠ 0

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