MathematicsChapter 1: Complex Numbers

Argand Plane and Polar Form

Plot z = x + i y as the point (x, y).

Learning objectives

State the main ideas and notation used in argand plane and polar form.
Apply the rules of argand plane and polar form in a clear, step-by-step solution.
Check signs, conditions, and the final answer for mathematical accuracy.

Definition

Plot z = x + i y as the point (x, y). The modulus r = |z| is the distance from the origin. The argument theta is the angle from the positive real axis to the ray OP, usually in radians. Polar form: z = r (cos theta + i sin theta). Euler: e^{i theta} = cos theta + i sin theta, so z = r e^{i theta}. Meaning: same point, described by length and turn instead of x and y.

Mathematical form

r = sqrt(1+1) = sqrt(2)

theta = pi/4 (first quadrant)

z = x + i y as the point (x

z = r (cos theta + i sin theta). Euler

Where / symbols

ARGAND PLANE AND POLAR FORM: Plot z = x + i y as the point (x, y). The modulus r = |z| is the distance from the origin. The argument theta is the angle from the positive real axis to the ray OP, usually in radians. Polar form: z = r (cos theta + i sin theta). Euler: e^{i theta} = cos theta + i sin theta, so z = r e^{i theta}. Meaning: same point, described by length and turn instead of x and y.
1. FIND r AND theta: r = sqrt(x^2+y^2). tan theta = y/x, then pick the quadrant from the signs of x and y. Do not blindly take tan^{-1}(y/x) if x < 0.
2. PRINCIPAL ARG: Arg(z) is the argument in (-pi, pi] (or [0, 2pi) if the book uses that — copy the book's interval). arg is any angle differing by 2 pi k. Zero has no argument.

Relationship

r = |z|
z = r (cos theta + i sin theta)

Overview

From (x,y) to r and theta:

FIND r AND theta

r = sqrt(x²+y²). tan theta = yx, then pick the quadrant from the signs of x and y. Do not blindly take tan^-1(yx) if x < 0.

Examples 1+i: r=sqrt(2), theta=pi/4 · -1: r=1, theta=pi · i: r=1, theta=pi/2

Principal Arg

Arg(z) is the argument in (-pi, pi] (or [0, 2pi) if the book uses that — copy the book's interval). arg is any angle differing by 2 pi k. Zero has no argument.

Examples Arg(i) = pi/2 · Arg(-1) = pi · Arg(1-i) = -pi/4

WHAT — ARGAND PLANE AND POLAR FORM

RULE — 1. FIND r AND theta

r = sqrt(x²+y²). tan theta = yx, then pick the quadrant from the signs of x and y. Do not blindly take tan^-1(yx) if x < 0.

METHOD — 2. PRINCIPAL ARG

Arg(z) is the argument in (-pi, pi] (or [0, 2pi) if the book uses that — copy the book's interval). arg is any angle differing by 2 pi k. Zero has no argument.

WORKED EXAMPLE

Worked: polar form of 1 + i. r = sqrt(1+1) = sqrt(2) theta = pi/4 (first quadrant) 1+i = sqrt(2) (cos pi/4 + i sin pi/4)

How to answer a argand plane and polar form question

Identify the given quantities, notation, and required result.
Choose the matching rule and write it before substitution.
Work one algebraic step at a time, keeping brackets and signs clear.
Check restrictions, simplify the result, and verify it where possible.

Worked example

polar form of 1 + i.
r = sqrt(1+1) = sqrt(2)
theta = pi/4 (first quadrant)
1+i = sqrt(2) (cos pi/4 + i sin pi/4)

In simple words

theta must match the quadrant. (-1)+i is not pi/4.

Key points

r = |z|
z = r (cos theta + i sin theta)
quadrant first
one main angle

Exam preparation

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