Modulus & Argument

• Argand representation: A complex number $z=x+iy$ is plotted as the point $(x,y)$, where $x$ is the real coordinate and $y$ is the imaginary coordinate. This turns complex numbers into vectors from the origin, so length and direction become natural quantities. You use this model whenever a question asks for geometry, loci, modulus, or argument.

• Modulus and argument: The modulus $|z|$ is the distance from the origin, and the argument $\arg z$ is the angle from the positive real axis measured anticlockwise. The principal argument is usually restricted to $-\pi<\arg z\le\pi$ so each nonzero complex number has one standard angle. At the origin, direction is not unique, so $\arg(0)$ is undefined.

• Why modulus is a square-root formula: Since $z$ is the point $(x,y)$, modulus follows Euclidean distance and gives $|z|=\sqrt{x^2+y^2}$. The expression is always nonnegative because it is a distance, not a signed quantity. This principle is the base check for any modulus result that appears negative or complex.

• Why argument needs quadrant logic: Trigonometry gives a reference angle from ratios like $\tan\theta=\frac{y}{x}$, but inverse tangent alone does not uniquely determine quadrant. The signs of $x$ and $y$ decide whether the principal angle is acute, obtuse, or negative. This is why a sketch is a reasoning tool, not just a drawing step.

• Polar decomposition principle: Every nonzero complex number can be separated into magnitude and direction as $z=r(\cos\theta+i\sin\theta)$ with $r=|z|$ and $\theta=\arg z$. This works because $(\cos\theta,\sin\theta)$ is the unit direction vector and $r$ scales it to the correct length. The decomposition explains why geometry and algebra stay consistent across forms.

• ### Compute modulus and argument from Cartesian form\n- Step 1: Read coordinates from $z=x+iy$ and sketch the point to identify quadrant. Step 2: Compute modulus with $|z|=\sqrt{x^2+y^2}$, then compute a reference angle using a trig ratio. Step 3: Adjust to principal range $-\pi<\arg z\le\pi$ based on quadrant, and state exact values when possible.

• ### Convert between Cartesian and modulus-argument forms\n- To go from Cartesian to polar, calculate $r=|z|$ and $\theta=\arg z$, then write $z=r(\cos\theta+i\sin\theta)$. To go from polar to Cartesian, use $x=r\cos\theta$ and $y=r\sin\theta$, then form $x+iy$. This two-way method is essential because exam questions often switch form mid-solution.

• ### Use structural checks\n- After finding $r$ and $\theta$, verify by reconstructing $x$ and $y$ from $r\cos\theta$ and $r\sin\theta$. If signs do not match the original quadrant, your angle choice is wrong even if the trig calculation is numerically neat. This check catches most argument errors before final submission.

• Different quantities answer different questions: Modulus answers "how far," while argument answers "which direction," so they are complementary rather than interchangeable. Cartesian form is best for addition and direct coordinate reading, while modulus-argument form is best for geometric interpretation and directional reasoning. Choosing the wrong representation usually creates unnecessary algebra.

• Comparison table for exam decisions: | Feature | Modulus $|z|$ | Argument $\arg z$ | | --- | --- | --- | | Meaning | Distance from origin | Angle from positive real axis | | Typical formula | $\sqrt{x^2+y^2}$ | Quadrant-adjusted trig angle | | Output type | Nonnegative real number | Real angle in radians | | Undefined case | Never for finite $z$ | Undefined at $z=0$ |

Use this distinction to avoid mixing size calculations with angle conventions.

• Principal argument vs general arguments: The principal argument is one chosen representative in a fixed interval, while general arguments differ by integer multiples of $2\pi$. This distinction matters when a question explicitly requests principal value or a full family of angles. Ignoring this instruction can make a mathematically equivalent answer invalid in context.

• Start with a fast sketch: A rough Argand sketch immediately reveals signs, quadrant, and plausible angle range before any trigonometry. This prevents using an inverse trig output in the wrong quadrant, which is one of the most common lost-mark errors. Even when not to scale, the sketch should preserve relative position.

• Use exact values when the geometry is special: If the triangle corresponds to standard angles, keep arguments in exact radians rather than decimal approximations unless rounding is requested. Exact form preserves structure and avoids compounded rounding drift when results are reused. Decimal answers should include clearly stated precision such as significant figures.

• Memorize and apply core identities:\n> Key formulas: $|z|=\sqrt{x^2+y^2}$, $z=r(\cos\theta+i\sin\theta)$, and $-\pi<\arg z\le\pi$ for principal values.\nThese formulas work best when paired with a sign-check from the sketch. Treat formula use and geometric validation as a single workflow, not separate tasks.

• Inverse tangent trap: Students often set $\arg z=\tan^{-1}(y/x)$ directly and stop, forgetting that this may return only a reference angle. The result then has the wrong sign or wrong quadrant despite correct arithmetic. Always adjust using the signs of $x$ and $y$.

• Confusing undefined with zero: It is incorrect to claim $\arg(0)=0$ because the origin has no unique direction. Any angle could point to the same point, so argument is not well-defined there. This conceptual issue is geometric, not computational.

• False linearity with modulus: A frequent mistake is assuming $|z_1+z_2|=|z_1|+|z_2|$ as a rule. In reality only an inequality relation generally holds, so equality is special rather than automatic. Treat modulus as distance in vector geometry, where triangle behavior governs outcomes.