Use polar forms, arguments and complex multiplication geometry

A nonzero complex number has a length and direction

Write z=r cis θ=r(cos θ+i sin θ), with r=|z|>0. The real component is r cos θ and the imaginary component is r sin θ. Angles here are in radians. Every θ+2kπ, for integer k, gives the same direction

Choose one principal argument convention

Use −π<Arg(z)≤π. The negative real axis has principal argument π, not −π. Zero has modulus zero but no argument. Writing 0 cis θ does not give zero a direction

Enter the coefficient of π in Arg(−3)

What are |0| and Arg(0)?

Use the quadrant as well as the tangent ratio

For z=−1+√3 i, the modulus is 2. The point is in quadrant II and its reference angle is π/3, so Arg(z)=2π/3. The ratio y/x alone cannot choose between opposite directions

Enter the coefficient of π in Arg(−1+√3 i)

A negative outside coefficient is not a modulus

For z=−2 cis(π/3), expand to get −1−√3 i. Make the radius positive by adding π to the angle, then remove a full turn: z=2 cis(−2π/3)

Find the real component of −2 cis(π/3)

For −2 cis(π/3), enter the coefficient of π in its principal argument

Multiplication adds directions and multiplies lengths

For nonzero z=r cis α and w=s cis β, zw=rs cis(α+β). An angle sum need not already be principal. If either factor is zero, the product is zero and has no argument

For z=2 cis(3π/4) and w=3 cis(2π/3), find |zw|

For z=2 cis(3π/4) and w=3 cis(2π/3), enter the coefficient of π in the principal argument of zw

Division subtracts directions

For w≠0, |z/w|=|z|/|w|. For nonzero z as well, subtract an argument of w from one of z, then normalise. A zero numerator gives zero; a zero divisor is undefined

For z=2 cis(3π/4), w=3 cis(2π/3), find |z/w|

For z=2 cis(3π/4) and w=3 cis(2π/3), enter the coefficient of π in the principal argument of z/w.

Must Arg(zw)=Arg(z)+Arg(w) hold as a literal equality for nonzero z,w?

Read multiplication as a plane transformation

A fixed nonzero multiplier w=s cis β rotates the whole plane by β modulo 2π and dilates distances from the origin by s>0. The origin is fixed. Multiplication by zero collapses the plane to the origin and is not an invertible rotation/dilation

The map z↦2iz sends 1+i to −2+2i. What is its dilation scale?

Enter the principal rotation coefficient of π for multiplication by 2i

Integer powers repeat the multiplication rule

For z≠0 and integer n, (r cis θ)ⁿ=rⁿ cis(nθ). Negative powers require a reciprocal radius, and n=0 gives 1. For z=0 only positive powers are used here; 0⁰ and negative powers of zero are excluded

Worked reciprocal power

For z=√3+i, |z|=2 and Arg(z)=π/6. Thus z⁻²=(1/4) cis(−π/3)=1/8−(√3/8)i. Squaring z and rationalising its reciprocal gives the same value

For z=√3+i, enter the modulus of z⁻²

For z=√3+i, enter the coefficient of π in the principal argument of z⁻²

Prove the multiplication and power rules, ungraded

Use sine/cosine addition identities to prove cis α cis β=cis(α+β). Then prove the integer-power rule for positive, zero and negative integers, stating the zero-base restrictions. This proof is ungraded

Model: addition identities and induction

Expansion gives real part cos α cos β−sin α sin β and imaginary part sin α cos β+cos α sin β, hence cis(α+β). Induct on positive n: multiply cis(nθ) by cis θ. For n=0 and nonzero base the result is 1. Since cis θ cis(−θ)=1, reciprocals prove negative powers. The usual real exponent laws give rⁿ. Positive powers of zero are zero; the other zero-base cases are excluded

Modulus identities preserve nonnegative lengths

Conjugation leaves modulus unchanged. For all z,w, |zw|=|z||w|. For w≠0, |z/w|=|z|/|w|. Also |z+w|≤|z|+|w|; equality need not hold, because components can cancel

Prove the modulus identities, ungraded

Prove the conjugation, product and quotient identities, then justify the triangle inequality and its equality case. Include zero cases and the nonzero divisor restriction. These arguments are ungraded

Model: conjugation and the triangle inequality

Conjugation reverses the sign of one coordinate, so it preserves a²+b². Also |zw|²=(zw)conjugate(zw)=(z conjugate z)(w conjugate w)=|z|²|w|². Take nonnegative square roots. Applying the product rule to (z/w)w=z and dividing by |w|>0 proves the quotient identity. Finally |z+w|²=|z|²+|w|²+2Re(z conjugate w)≤(|z|+|w|)² because Re(u)≤|u|. Equality occurs if either number is zero or the two nonzero numbers have the same direction

Sketch and test the transformation, ungraded

Sketch the map z↦2iz for 1+i, −2 and 0 with equal axis scales. Use complex-mode technology to check Cartesian products, moduli and principal arguments. Explain why a different displayed argument may still represent the same nonzero number. This construction and technology task is ungraded

Model: compare components and directions

The images are −2+2i, −4i and 0. Every distance from the origin doubles and every nonzero direction turns π/2 anticlockwise. Use component multiplication to verify the images. A device may use [0,2π) instead of (−π,π]; compare arguments modulo 2π and retain undefined argument for zero. Numeric checks illustrate the map; the general multiplication proof establishes it for all inputs