Argand Plane and Polar Form

Argand Plane and Polar Form

What is it?

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.

In simple words

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

Important terms

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 = yx, then pick the quadrant from the signs of x and y. Do not blindly take tan^-1(yx) 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.

What you should learn

• 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.

Examples

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

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.

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.

Full school-level explanation

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.

From (x,y) to r and theta:

WHAT — 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.

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)

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.

Remember

• quadrant first

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.

Remember

• one main angle

Examples

• Arg(i) = pi/2
• Arg(-1) = pi
• Arg(1-i) = -pi/4

How to answer a argand plane and polar form question

How it works

• 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.

Remember

• r = |z|
• z = r (cos theta + i sin theta)
• sqrt(2) cis pi/4
• quadrant
• quadrant first
• one main angle
• r = |z|
• z = r (cos theta + i sin theta)

How the ideas connect

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)

Deeper understanding

Go beyond memorising Argand Plane and Polar Form. Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

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)

Common mistakes

Exam focus

Core Idea

1. FIND r AND theta

2. PRINCIPAL ARG

Worked Example

Exam focus

Quick revision

• r = |z|
• z = r (cos theta + i sin theta)
• sqrt(2) cis pi/4
• quadrant
• 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 argum
• 1. FIND r AND theta: r = sqrt(x^2+y^2). tan theta = yx, then pick the quadrant from the signs of x and y. Do not blindly

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

For long answers on Argand Plane and Polar Form: define clearly → explain the main parts → give one example → state one exam caution.