Polar form — r(cos θ + i sin θ) = r·cis θ: Question bank

Before we start, a quick reminder of the words we will lean on (so no symbol is used unexplained):

modulus $r = ∣ z ∣$ — the length of the arrow from the origin to the point $z$ , always $\geq 0$ .
argument $θ = ar g z$ — the direction of that arrow, an angle measured anticlockwise from the positive real axis.
principal argument — the one value of the argument chosen in the half-open range $(- π, π]$ , so every arrow gets exactly one "official" angle.
==cis $θ$ == — shorthand for $cos θ + i sin θ$ (a sum, not a product).

True or false — justify

TRUE or FALSE: The modulus

r

of a complex number can be negative if the imaginary part is negative.

False.

r = a^{2} + b^{2}

squares both parts, so it is never negative; a "backwards" arrow is recorded by adding

π

to the angle, not by making the length negative.

TRUE or FALSE:

cis θ

means

cos θ \times i \times sin θ

False. It is a sum:

cos θ + i sin θ

. Read the letters as "cos is summed with

i sin

", a plus in the middle.

TRUE or FALSE: Every non-zero complex number has exactly one argument.

False. It has infinitely many, all differing by full turns of

2 π

; we single out one principal value in

(- π, π]

so each arrow has one official label.

TRUE or FALSE: The number

0

has a well-defined argument.

False. The zero arrow has length

0

and points in no direction, so

ar g 0

is undefined — polar form is stated only for non-zero

z

TRUE or FALSE: Two complex numbers with the same modulus are equal.

False. Same length does not mean same direction; e.g.

1

and

i

both have modulus

1

but point differently. You need both

r

and

θ

to match.

TRUE or FALSE: If two complex numbers are equal, their principal arguments must be equal.

True. Equal arrows point the same way, and the principal argument is a unique label in

(- π, π]

, so the labels coincide.

TRUE or FALSE: Multiplying a complex number by another of modulus

1

leaves its length unchanged.

True. Lengths multiply, and multiplying by

1

does nothing to length; only the angle shifts. This is a pure rotation.

TRUE or FALSE:

ar g (z_{1} z_{2}) = ar g z_{1} + ar g z_{2}

is always exactly true for principal arguments.

False. The sum of angles is right, but the total may fall outside

(- π, π]

, so you sometimes add or subtract

2 π

to get back the principal value.

TRUE or FALSE: A purely real positive number has argument

0

True. It lies on the positive real axis, so the arrow points along it and makes zero angle.

TRUE or FALSE: The point

- 1

has principal argument

π

, not

- π

True. The principal range is

(- π, π]

— the right end is included, the left is not — so the arrow along the negative real axis is labelled

+ π

TRUE or FALSE: Writing

z = - 2 (cos θ + i sin θ)

is a valid polar form.

False. Polar form requires the leading modulus

r \geq 0

. A negative in front must be absorbed into the angle:

- 2 cis θ = 2 cis (θ + π)

Spot the error

Spot the error: "For

z = - 1 + i

ar g z = tan^{- 1} (\frac{1}{- 1}) = tan^{- 1} (- 1) = - \frac{π}{4}

The

tan^{- 1}

key is blind to quadrant.

(- 1, 1)

is in Quadrant II, but

- \frac{π}{4}

points into Quadrant IV. Correct:

θ = π - \frac{π}{4} = \frac{3 π}{4}

Spot the error: "For

z = 3 - 4 i

, the modulus is

r = 3^{2} - 4^{2} = - 7

Modulus adds the squares:

3^{2} + 4^{2} = 25 = 5

. Squaring already kills the minus sign of

b = - 4

, so there is no negative under the root.

Spot the error: "

z = r cis θ

\overset{z}{ˉ}

(the conjugate)

= r cis θ

too, since

r

is unchanged."

The conjugate flips the sign of the imaginary part, i.e. reflects the arrow across the real axis, so its angle negates:

\overset{z}{ˉ} = r cis (- θ)

. Only the modulus stays.

Spot the error: "To multiply

2 cis \frac{π}{6}

3 cis \frac{π}{4}

, add the moduli and multiply the angles."

Backwards. You multiply the moduli (

2 \times 3 = 6

) and add the arguments (

\frac{π}{6} + \frac{π}{4}

). Length scales, directions accumulate.

Spot the error: "

4 cis \frac{3 π}{4}

converts to

a = 4 cos \frac{3 π}{4} = 4 \cdot \frac{1}{2}

In Quadrant II the cosine is negative:

cos \frac{3 π}{4} = - \frac{1}{2}

, so

a = - 22

. Forgetting the sign of cos/sin off the reference angle is the classic slip.

Spot the error: "The reference angle for

z = - 1 + i 3

tan^{- 1} (\frac{3}{- 1})

The reference angle uses the magnitude

\frac{b}{a} = 3

, giving

tan^{- 1} 3 = \frac{π}{3}

. Signs are handled afterwards by the quadrant rule, not inside the reference angle.

Spot the error: "

z = cis \frac{5 π}{2}

is already in principal form."

\frac{5 π}{2}

is more than a full turn and lies outside

(- π, π]

. Subtract

2 π

\frac{5 π}{2} - 2 π = \frac{π}{2}

, so it equals

cis \frac{π}{2} = i

Why questions

Why do we bother converting to polar form at all?

Because multiplying and dividing arrows becomes trivial — multiply lengths, add angles — instead of the messy FOIL expansion needed in Cartesian form.

Why must we "fix the quadrant" after using

tan^{- 1} (b / a)

Because

tan

repeats every

π

(180°): points that are exactly opposite each other give the same ratio

b / a

, so

tan^{- 1}

cannot tell them apart on its own.

Why is the principal argument range chosen as

(- π, π]

rather than

[0, 2 π)

Both are valid conventions, but

(- π, π]

keeps the numbers small and symmetric about the real axis, so a conjugate simply flips the sign of the angle. It is the standard used by Modulus and Argument.

Why does multiplying by

i

rotate a point by

90°

Because

i = cis \frac{π}{2}

, and multiplying by a cis of angle

\frac{π}{2}

adds

\frac{π}{2}

to the argument while keeping the length (modulus

1

) fixed — a pure quarter-turn anticlockwise.

Why does the product of two cis terms collapse to a single cis of the summed angle?

Expanding

(cos A + i sin A) (cos B + i sin B)

and using

cos (A + B) = cos A cos B - sin A sin B

and

sin (A + B) = sin A cos B + cos A sin B

reassembles it as

cos (A + B) + i sin (A + B)

. This is exactly the seed of De Moivre's Theorem.

Why is

r \geq 0

enough, given that arrows point in every direction?

Because all directional information is carried by the angle

θ

, which ranges over a full turn. Length only measures "how far", so it never needs a sign.

Why does Euler's formula

e^{i θ} = cos θ + i sin θ

make polar form feel like an exponent?

Because multiplying

e^{i α} e^{i β} = e^{i (α + β)}

automatically adds the angles — the "add the arguments" rule is just the exponent-addition law in disguise.

Edge cases

Edge case: What is the polar form of a positive real number like

z = 5

r = 5

θ = 0

z = 5 cis 0

. The arrow lies flat along the positive real axis.

Edge case: What is

ar g (- i)

as a principal value?

- \frac{π}{2}

. The arrow points straight down;

- \frac{π}{2}

(not

\frac{3 π}{2}

) is the value inside

(- π, π]

Edge case: What happens to

θ

for a point exactly on the negative real axis, e.g.

- 3

θ = π

(included in the principal range),

r = 3

, so

- 3 = 3 cis π

. Its neighbour

- π

is not used because the left endpoint is excluded.

Edge case: What is the argument of a point on the positive imaginary axis, e.g.

2 i

θ = \frac{π}{2}

r = 2

2 i = 2 cis \frac{π}{2}

. Here

a = 0

, so

tan^{- 1} (b / a)

is undefined — you must read the direction geometrically.

Edge case: If

a = 0

and

b < 0

, why can't you compute

θ

from

tan^{- 1} (b / a)

Division by

a = 0

is undefined, so the formula breaks. Read it straight off the picture: the arrow points down, so

θ = - \frac{π}{2}

Edge case: What is the modulus of any

cis θ

, and why does this matter for roots?

It is always

1

, since

cos^{2} θ + sin^{2} θ = 1

. This is why cis terms act as pure rotations and why

n

-th roots all sit on a circle of the same radius.

Connections

Argand Diagram — the plane where every trap here is visualised.
Modulus and Argument — the two ingredients being probed.
Multiplication and Division of Complex Numbers — the "multiply lengths, add angles" claims.
De Moivre's Theorem — the power rule built on the product rule.
Euler's Formula — the exponent view of "add the arguments".
Roots of Complex Numbers — uses the unit-modulus fact of cis.

Question bank — Polar form — r(cos θ + i sin θ) = r·cis θ

True or false — justify

Spot the error

Why questions

Edge cases

Connections

Concept Map