Worked examples — Argand plane — geometric representation

This page assumes only the parent note the parent. Every symbol used there — the point $P(x,y)$, the modulus $r=|z|=\sqrt{x^2+y^2}$, the argument $\theta$, the reference angle $\alpha ={\tan }^{-1}|y/x|$ — is used exactly as defined there.

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The scenario matrix

Cell	Situation	What's tricky	Example
A	Quadrant I $(+,+)$	none — the "friendly" case	Ex 1
B	Quadrant II $(-,+)$	naive ${\tan }^{-1}$ gives wrong angle	Ex 2
C	Quadrant III $(-,-)$	${\tan }^{-1}$ gives right ratio, wrong sign	Ex 3
D	Quadrant IV $(+,-)$	angle is negative	Ex 4
E	On an axis (a leg is $0$)	${\tan }^{-1}(y/0)$ undefined / $y=0$	Ex 5
F	Zero $z=0$	modulus $0$, argument undefined	Ex 6
G	Limiting behaviour ($y\to 0$, $x\to 0$)	angle sweeps toward $0$ or $\pm \pi /2$	Ex 7
H	Word problem (real-world)	translate a physical direction into $z$	Ex 8
I	Locus (geometry from algebra)	equation → shape	Ex 9
J	Exam twist (product / rotation)	combine argument-adds-under-multiply	Ex 10

Figure s01 (below) is the compass for this whole page: it shades each quadrant with the correction it needs and marks the axis cells. Keep glancing back at it — every example lands somewhere on this picture.

Caption s01. The four quadrants each carry their own sign/formula for $\arg z$; the axes (Cell E) and the origin (Cell F) are the borderlines where the formula must be replaced by a picture.

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Cell A — Quadrant I (the friendly case)

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Cell B — Quadrant II (naive formula fails)

Figure s02 (below) shows exactly this: the magenta arrow to $(-3,4)$, the orange arc that measures $\arg z$ from the positive real axis, and the small navy arc that is the reference angle $\alpha$ against the negative axis. Compare the two arcs — their sum is $\pi$.

Caption s02. Cell B — the argument (orange arc) is measured from $+x$; the calculator's raw answer only ever finds the reference angle $\alpha$ (navy arc).

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Cell C — Quadrant III (right ratio, wrong sign)

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Cell D — Quadrant IV (negative angle, no $\pi$ needed)

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Cell E — On an axis (a leg is zero)

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Cell F — The degenerate case $z=0$

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Cell G — Limiting behaviour

Figure s03 (below) plots $\arg z={\tan }^{-1}(y)$ against $y$ for Step 1–3: the orange dots mark the two numeric checkpoints and the violet dashed line is the $\frac{\pi }{2}$ ceiling the curve never crosses.

Caption s03. Cell G — as $y$ grows the argument climbs but flattens toward the horizontal asymptote $\pi /2$; it only equals $\pi /2$ in the true axis case (Cell E).

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Cell H — Real-world word problem

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Cell I — Locus (geometry out of algebra)

Figure s04 (below) draws that circle: violet centre at $(1,-2)$, an orange radius arrow of length $4$, and the navy test point $(1,2)$ sitting on the rim (matching the Verify line).

Caption s04. Cell I — the equation $|z-(1-2i)|=4$ is the circle of radius $4$ centred at $(1,-2)$.

See Loci in the Complex Plane for perpendicular-bisector and ray loci.

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Cell J — Exam twist (multiplication adds arguments)

This is the seed of De Moivre's Theorem — repeatedly adding the same argument.

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Recall Scenario self-test (hide answers)

For each, name the cell and the fix used.

$z=-3+4i$ ::: Cell B, Q II, $\arg =\pi -\alpha$. $z=-5$ ::: Cell E, negative real axis, $\arg =\pi$. $z=0$ ::: Cell F, argument undefined. $\arg (z_1{z}_2)$ for two Q I numbers ::: Cell J, add the arguments. $|z-z_0|=r$ ::: Cell I, circle centre $z_0$ radius $r$.

Connections

• Argand plane — geometric representation (parent)
• Complex Numbers — Modulus and Argument
• Polar Form of Complex Numbers
• Multiplication as Rotation and Scaling
• De Moivre's Theorem
• Vectors — Position Vectors and Addition
• Loci in the Complex Plane

Scenario map