The Argand diagram — A/L Combined Mathematics

What must I remember about The Argand diagram?

• Argand Diagram: Geometric representation of complex numbers in a plane where the horizontal axis is the Real Axis (Re) and vertical axis is the Imaginary Axis (Im).
• Vector representation: Complex number z = x + iy corresponds to position vector OP = (x, y) = x*i + y*j.
• Geometric addition and subtraction: Addition z1 + z2 corresponds to vector addition via the Parallelogram Law; subtraction z1 - z2 corresponds to directed displacement vector from z2 to z1.
• Polar (Modulus-Argument) Form: z = r(cos θ + i sin θ) = r * cis(θ), where r = |z| ≥ 0 and θ = Arg(z) is the Principal Argument: -π < θ ≤ π.
• Multiplication and Division in Polar form: z1 * z2 = r1*r2 * [cos(θ1 + θ2) + i sin(θ1 + θ2)]; modulos multiply (|z1 z2| = r1 r2) and arguments add (Arg(z1 z2) = θ1 + θ2 ± 2kπ); z1 / z2 = (r1/r2) * [cos(θ1 - θ2) + i sin(θ1 - θ2)].

Test yourself (1 minute)

1. Principal Argument Arg(z) Quadrant Rules

   If x > 0: θ = arctan(y/x); If x < 0, y ≥ 0 (Quadrant II): θ = arctan(y/x) + π; If x < 0, y < 0 (Quadrant III): θ = arctan(y/x) - π; If x = 0, y > 0: θ = π/2; If x = 0, y < 0: θ = -π/2.

2. Multiplication by i as a 90° rotation

   Multiplying z by i rotates the vector OP counterclockwise by 90° (π/2 radians) without altering its modulus (|i*z| = |z|).

3. Internal/external division in Argand plane

   Point P dividing segment z1z2 in ratio m : n has complex number z = (m*z2 + n*z1) / (m + n).

Practise The Argand diagram questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 21

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Where does The Argand diagram fit in the syllabus?

Year 2 · Term 1