Argand diagrams, modulus and argument
Read complex numbers as points and distinguish reflection, negation, translation and multiplication by i.
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Real and imaginary parts give coordinates
On an Argand diagram, z = a + bi is the point (a,b): real part along the horizontal axis and imaginary part along the vertical axis. Both coordinates are real numbers.
The modulus |z| = square root of (a² + b²) is the nonnegative distance from the origin. The argument is a direction angle from the positive real axis, with anticlockwise angles positive. Distance and direction answer different questions: points on the same circle have equal moduli but can have different arguments.
Use radians and the principal-argument convention -π < arg z ≤ π here. For -1 + i, the modulus is square root of 2 and the argument is 3π/4. The zero complex number has modulus zero but no direction argument.
You need coordinate signs, Pythagoras and the reference angles of a right triangle. Start with the point’s position before using an inverse tangent. You will work with Cartesian coordinates throughout.
Choose the quadrant before the angle
Worked example 1
A negative direction angle
Problem
Find the exact modulus and principal argument of z = -2-2 square root of 3 i.
Show full solution
Read the distance and quadrant
Method
The coordinates are (-2,-2 square root of 3), so the point lies below and to the left of the origin.Reason
The modulus uses squared coordinates, while the signs determine the direction.Working
Answer
|z| = square root of ((-2)² + (-2 square root of 3)²) = square root of (4 + 12) = 4.Put the direction in the stated interval
Method
The reference angle is arctan(square root of 3) = π/3.Reason
In the third quadrant the principal direction is clockwise from the positive real axis, between -π and -π/2.Working
Answer
arg z = -π + π/3 = -2π/3.
Quick check
A calculator applied only to b/a = square root of 3 would return π/3, which points into the wrong quadrant. The angle 4π/3 has the right direction but lies outside the requested principal interval. The modulus is four, not its square sixteen and not a negative square root.
Opposite points have the same ratio b/a when a ≠ 0, so that ratio cannot distinguish their directions. If a = 0, the ratio is undefined even though a nonzero point on the imaginary axis still has a direction.
| Point location | Principal argument |
|---|---|
| Positive real axis | 0 |
| Negative real axis | π |
| Positive imaginary axis | π/2 |
| Negative imaginary axis | -π/2 |
| Origin | Undefined |
Check your understanding 1 The same ratio, different directions
For z = 3-3i and w = -3 + 3i, find each modulus and principal argument. Explain why using only the inverse tangent of the coordinate ratio loses information.
Show answer
Both moduli are 3 square root of 2. The arguments are -π/4 for z and 3π/4 for w. Both coordinate ratios equal −1, but the points are in the fourth and second quadrants respectively. The ratio gives a reference line; the two signs choose its direction. Check the interval as well as the quadrant.
Operations have visible geometrical effects
Conjugation reflects in the real axis: a + bi becomes a − bi. Negation turns through 180° about the origin. Adding a fixed complex number translates every point by its real and imaginary components; subtracting two complex numbers gives their directed difference.
Multiplication by i sends a + bi to −b + ai, a 90° anticlockwise rotation with modulus unchanged. For z = 1 + 2i, iz = −2 + i, z̄ = 1 − 2i and −z = −1 − 2i.
Equal axis scales preserve the visible rotation and length comparisons. A stretched display can distort angles even when coordinate labels are correct.
Subtraction also has a direction. If A represents 1 + 2i and B represents -2 + i, the displacement from A to B is
It moves three units left and one unit down, so the distance AB is |-3-i| = square root of 10. Reversing the subtraction reverses the displacement but leaves its modulus, and therefore the distance, unchanged.
Check your understanding 2 Conjugation is not negation
For z = −3 + 4i, find |z|, z̄, −z and iz. State which operation reflects in the real axis.
Show answer
The modulus is 5. The conjugate is −3 − 4i, the negative is 3 − 4i, and iz = −4 − 3i. Conjugation reflects in the real axis; negation turns through 180°.
A direction has a branch convention
Angles differing by a full turn represent the same direction, but a requested principal argument must use the stated interval. A negative real number has principal argument π under this convention; −π represents the same direction but is outside the chosen interval. No argument is assigned to zero.
Conjugation and multiplication by i preserve distance from the origin; translation generally does not. Be careful when a rotation moves a direction across the end of the principal interval. For z = -1 + i, the direction is 3π/4. Its image iz = -1-i has principal argument -3π/4, not 5π/4.
The order of operations matters
Let z = 1 + 2i and u = 2-i. Compare rotating after translation with translating after rotation:
In the first expression the translation vector is rotated too; in the second it is added afterwards. Drawing the two paths from the same starting point makes the difference visible. The second image is the origin, so its modulus is zero and its argument is undefined. Neither answer can be obtained by adding 90° to the original argument while ignoring the translation.
Try it yourself 1
Recover the original point
Problem
A transformation sends z to w = iz bar + (2-i). The image is w = 5 + 3i. Find z, its exact modulus and its principal argument in radians to six decimal places. Describe the geometrical operations in the given transformation.
Write your answer, then compare it with the solution
Hints
Hint 1
Reverse the operation order
The image is known. Undo the last operation first, and keep track of whether the intermediate number is z or its conjugate.
Show full solution
Recover the number before translation and rotation
Method
Subtract 2-i, then divide by i.Reason
The translation was performed last, so it must be undone before the rotation.Working
Answer
iz bar = 3 + 4i, hence z bar = -i(3 + 4i) = 4-3i and z = 4 + 3i.Find distance and direction
Method
The recovered point (4,3) is in the first quadrant.Reason
Its reference angle already lies in the required principal interval.Working
Answer
|z| = square root of (4² + 3²) = 5, and arg z = arctan(3/4) ≈ 0.643501 radians to six decimal places.
Quick check
The original transformation reflects in the real axis, rotates 90° anticlockwise about the origin, then translates two units right and one unit down. Substitution checks the order: i(4-3i) + (2-i) = 5 + 3i. Keep the unrounded inverse-tangent value until the final rounding. Compare your geometrical description yourself; a correct number alone does not verify it.
Before moving on, sketch one of these calculations with equal axis scales. Label the original and image points, and decide whether you are describing a point, a directed displacement or a distance. Then cover the solution and recover the quadrant and branch convention from the original coordinates.
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