Calculator Inputs
Argand Diagram
Computed Table
| Label | Real | Imag | Modulus |z| | Arg (deg) | Arg (rad) | Form |
|---|---|---|---|---|---|---|
| Submit to compute values. | ||||||
Example Data Table
| Label | Real | Imag | Expected |z| | Expected θ (deg) |
|---|---|---|---|---|
| A | 3 | 4 | 5 | 53.1301 |
| B | -2 | 1 | 2.2361 | 153.4349 |
| C | 1 | -3 | 3.1623 | 288.4349 |
Formula Used
- z = a + bi maps to point (a, b) on the plane.
- Modulus: |z| = √(a² + b²).
- Argument: arg(z) = atan2(b, a) (shown in degrees and radians).
- Polar form: z = r(cosθ + i sinθ), where r=|z|, θ=arg(z).
- Add/Sub: (a+bi) ± (c+di) = (a±c) + (b±d)i.
- Multiply: (a+bi)(c+di) = (ac−bd) + (ad+bc)i.
- Divide: (a+bi)/(c+di) = ((ac+bd) + (bc−ad)i)/(c²+d²).
- De Moivre: (r(cosθ+i sinθ))^n = r^n(cos(nθ)+i sin(nθ)).
- n-th roots: r^(1/n)(cos((θ+360k)/n)+i sin((θ+360k)/n)), k=0..n−1.
How to Use
- Enter Z1 real and imaginary parts.
- Optionally enter Z2 for two-number operations.
- Select an operation such as multiply, polar conversion, powers, or roots.
- Add extra points in the batch box to compare several numbers.
- Press Submit to display the result above the form.
- Review the diagram and computed table, then export CSV or PDF.
Article
Complex Plane Mapping
An Argand diagram represents each complex number z = a + bi as a point (a, b). The horizontal axis stores the real part, while the vertical axis stores the imaginary part. Plotting multiple values reveals clusters, symmetry, and how algebraic operations move points around the plane. In practice, engineers treat points as vectors, so arrow length and direction provide immediate intuition.
Modulus as Distance
The modulus |z| equals the Euclidean distance from the origin to the point. Using |z| = √(a² + b²), you can compare magnitudes quickly, rank signals by strength, or measure error vectors. Circles centered at the origin correspond to constant modulus. When you plot many points, concentric rings often highlight quantization steps or measurement noise.
Argument as Direction
The argument θ describes direction from the positive real axis to the point. This calculator reports θ using atan2(b, a) and normalizes angles to 0–360 degrees for consistent charting. Rays from the origin illustrate constant arguments and phase relationships. Small changes in θ can indicate phase drift, while stable θ across samples suggests coherent rotation.
Operations as Geometry
Addition translates points by vector rules: Z1 + Z2 adds real parts and imaginary parts separately. Subtraction forms the displacement from Z2 to Z1. Multiplication rotates and scales in polar form, while division scales and rotates in the opposite sense when Z2 is nonzero. The plotted result makes these transforms visible, helping you catch sign errors or swapped components.
Polar Conversion and De Moivre
Converting to polar uses z = r(cosθ + i sinθ). Powers follow De Moivre: z^n = r^n(cos(nθ) + i sin(nθ)). The calculator plots the powered result so you can see how repeated rotation and scaling changes position. For large n, values may grow rapidly when r>1, so reading r and θ together improves numerical interpretation.
Roots and Pattern Recognition
The n-th roots distribute evenly around a circle of radius r^(1/n). Each root uses θ_k = (θ + 360k)/n. Visualizing these roots helps verify algebra, design phasor sets, and confirm expected symmetry, especially when n is large. Because the roots are equally spaced, missing or duplicated points usually signals an incorrect angle step or an unintended rounding choice. In quick classroom checks.