Calculator Input
Enter each complex number in rectangular form. Use positive or negative imaginary parts.
Complex Plane Graph
The graph shows the two input vectors and their product on the complex plane.
Formula Used
For two complex numbers:
z₁ = a + bi
z₂ = c + di
Their product is:
z₁z₂ = (ac - bd) + (ad + bc)i
Magnitude is:
|z| = √(real² + imaginary²)
Argument is:
θ = atan2(imaginary, real)
How to Use This Calculator
- Enter the real and imaginary parts of the first number.
- Enter the real and imaginary parts of the second number.
- Select decimal places for rounded output.
- Choose degrees or radians for polar angles.
- Use conjugate options when needed.
- Press the multiply button.
- Read the result above the form.
- Download the result as CSV or PDF.
Example Data Table
| First Number | Second Number | Real Calculation | Imaginary Calculation | Product |
|---|---|---|---|---|
| 3 + 2i | 4 - 5i | 3×4 - 2×(-5) = 22 | 3×(-5) + 2×4 = -7 | 22 - 7i |
| 1 + 6i | 2 + 3i | 1×2 - 6×3 = -16 | 1×3 + 6×2 = 15 | -16 + 15i |
| -2 + 4i | 5 - i | -2×5 - 4×(-1) = -6 | -2×(-1) + 4×5 = 22 | -6 + 22i |
Understanding Complex Number Multiplication
What This Calculator Does
Complex number multiplication combines two values with real and imaginary parts. This calculator handles the full rectangular form. It also gives polar details. You can check magnitude, angle, product, and steps. That makes the tool useful for algebra, signals, physics, and engineering lessons.
Why the Real Part Changes
The real part is not found by only multiplying real values. The imaginary values also affect it. This happens because i² equals -1. So the term bd becomes negative in a special way. The final real part is ac minus bd.
Why the Imaginary Part Changes
The imaginary part comes from the cross products. These are ad and bc. Both terms contain one real factor and one imaginary factor. Their sum becomes the coefficient of i. This is why signs must be entered carefully.
Rectangular and Polar Views
Rectangular form shows the answer as x plus yi. Polar form shows magnitude and direction. Multiplication has a useful polar rule. Magnitudes multiply. Angles add. The calculator shows both views. This helps users verify the same answer from two methods.
Practical Use
Students can use the tool for homework checks. Teachers can use it for quick examples. Engineers can inspect phase and amplitude changes. The graph gives a visual view of each vector. Export buttons help save results for reports. The conjugate options support extra practice and advanced review.
FAQs
1. What is a complex number?
A complex number has a real part and an imaginary part. It is usually written as a + bi, where i is the square root of -1.
2. How are complex numbers multiplied?
Multiply them like binomials. Then replace i² with -1. The final result becomes (ac - bd) + (ad + bc)i.
3. What does the conjugate option do?
The conjugate changes the sign of the imaginary part. For example, 4 + 3i becomes 4 - 3i before multiplication.
4. Why does the calculator show polar form?
Polar form shows magnitude and angle. It is helpful because complex multiplication multiplies magnitudes and adds angles.
5. Can I use negative imaginary values?
Yes. Enter a negative number in the imaginary field. The calculator handles signs automatically in every step and result.
6. What does the graph represent?
The graph places each complex number on the complex plane. The horizontal axis is real. The vertical axis is imaginary.
7. Is the CSV file editable?
Yes. The CSV file can be opened in spreadsheet tools. It includes inputs, product, magnitude, angle, and polar output.
8. Is this calculator suitable for advanced study?
Yes. It includes rectangular output, polar output, conjugate options, graphing, exports, formulas, and step-by-step verification.