Calculator
Choose a format, enter two complex numbers, then compute the product with optional steps and rounded output.
Formula used
For rectangular form, let z₁ = a + bi and z₂ = c + di. The product is:
For polar form, z = r∠θ. Multiplication follows r₁r₂ for magnitude and θ₁+θ₂ for angle, then the result is also shown in rectangular form.
How to use this calculator
- Select Rectangular or Polar input format.
- Enter values for z₁ and z₂ in the fields provided.
- Pick rounding decimals, and optionally enable Show steps.
- Press Multiply to display results above the form.
- Use Download CSV or Download PDF in the result card.
Example data table
| # | z₁ | z₂ | Product | Magnitude | Angle (deg) |
|---|---|---|---|---|---|
| 1 | 3 + 2i | 1 + 4i | -5 + 14i | 14.866 | 109.654° |
| 2 | -2 + 5i | 4 - 1i | -3 + 22i | 22.204 | 97.765° |
| 3 | 0.5 - 3i | -1.2 + 2.5i | 6.9 + 4.85i | 8.434 | 35.103° |
Rectangular multiplication workflow
Rectangular input treats each number as a pair (a,b). The calculator expands (a+bi)(c+di) and separates parts with ac−bd and ad+bc. Example: (3+2i)(1+4i) gives −5+14i because 3×1−2×4=−5 and 3×4+2×1=14. Another quick check: (−2+5i)(4−i)= (−3)+22i. This mode is ideal when homework is written in a+bi.
Polar representation and angle control
With polar input, the tool converts r∠θ to rectangular via rcosθ + r sinθ·i. Choose degrees or radians for θ, then both units are reported for the result. In polar form, magnitudes multiply and angles add: r=r₁r₂ and θ=θ₁+θ₂. Multiplying by 1∠90° rotates a point by 90° without changing length, while 2∠0° doubles size with no rotation.
Rounding and numerical stability
Rounding affects only displayed outputs. Internally, computations use floating‑point values, so 4–6 decimals usually match textbook answers, while 8–10 helps for precision checks in signals or controls. “Normalize angle” maps the reported argument to (−π, π], reducing jumps near ±180°. If your θ is 190°, normalization shows −170° instead. Normalization never changes the product, only its reported angle.
Interpreting magnitude and argument
Magnitude |z| is √(Re²+Im²), the distance from the origin, and it equals the polar radius. The argument uses atan2(Im,Re) to preserve the correct quadrant, even when Re is 0. Products satisfy |z₁z₂|=|z₁||z₂| and arg(z₁z₂)=arg(z₁)+arg(z₂). If |z₁|=3 and |z₂|=0.5, expect |z₁z₂|=1.5 before checking components.
Graphing the result on the complex plane
The Plotly graph draws vectors for z₁, z₂, and z₁·z₂ from (0,0). If both inputs lie on the unit circle (|z|=1), the product also lies there, showing pure rotation. If one factor has magnitude 2, the product length doubles relative to the other factor. Equal axes keep angles visually honest, and grid lines help estimate coordinates to within about 0.1 units.
Exportable reporting for study and review
After a calculation, download a CSV row or a one‑page PDF summary. Exports include inputs, rectangular product, polar forms, magnitude, angles, and a timestamp. This supports lab notes, tutoring sheets, and consistent checking across exercises. Exports always use the most recent result, so compute and archive in one flow. For practice sets, you can paste the CSV into a spreadsheet and compare expected versus computed results. That makes grading faster and keeps your working neat across many problems today.
FAQs
Can I multiply using polar inputs only?
Yes. Select Polar, enter r and θ for both numbers, choose degrees or radians, then multiply. The calculator converts to rectangular internally and also reports the product back in polar form for verification.
What does “Normalize angle” change?
It changes only the displayed angle, mapping θ to (−π, π]. This removes wraparound confusion like 190° becoming −170°. The rectangular product remains identical.
Why do my answers differ slightly from a textbook?
Most textbooks round at specific steps. This tool computes with floating‑point values and rounds at the end. Increase decimals to compare, or enable steps to see intermediate terms.
Does the graph affect the calculation?
No. The Plotly graph is a visual aid. It plots z₁, z₂, and z₁·z₂ on the complex plane after the computation is finished.
What is included in CSV and PDF exports?
Exports include the two inputs, the rectangular product, polar representations in degrees and radians, magnitude, angles, and a timestamp. Exports always use the most recent computed result.
How should I enter i or j in inputs?
Enter only numeric parts. For rectangular form, type the real part in a and the imaginary coefficient in b. For example, 3−2i means a=3 and b=−2.