Advanced Complex Rectangular Form Calculator

Calculator Inputs

Use rectangular or polar entries for each complex number. Choose the operation, precision, and output angle unit.

Z1 Input Mode

Z1 Real Part

Z1 Imaginary Part

Z1 Modulus

Z1 Angle

Z1 Angle Unit

Z2 Input Mode

Z2 Real Part

Z2 Imaginary Part

Z2 Modulus

Z2 Angle

Z2 Angle Unit

Operation

Exponent n

Root Count n

Decimal Precision

Display Angle Unit

Example Data Table

These examples illustrate common rectangular form calculations and expected outputs.

Case	Input	Operation	Output
Basic analysis	z = 3 + 4i	Analyze	Modulus = 5, argument ≈ 53.13°
Addition	z1 = 2 + 5i, z2 = 4 - 3i	z1 + z2	6 + 2i
Multiplication	z1 = 1 + 2i, z2 = 3 + 4i	z1 × z2	-5 + 10i
Power	z = 1 + i, n = 3	z^n	-2 + 2i
Roots	z = 1 + 0i, n = 4	n-th roots	1, i, -1, -i

Formula Used

Rectangular form writes a complex number as z = a + bi, where a is the real part and b is the imaginary part.

Modulus: |z| = √(a² + b²) Argument: θ = atan2(b, a) Polar to rectangular: a = r cos(θ), b = r sin(θ) Addition: (a + bi) + (c + di) = (a + c) + (b + d)i Subtraction: (a + bi) - (c + di) = (a - c) + (b - d)i Multiplication: (a + bi)(c + di) = (ac - bd) + (ad + bc)i Division: (a + bi)/(c + di) = [(ac + bd) + (bc - ad)i] / (c² + d²) Power: zⁿ = rⁿ [cos(nθ) + i sin(nθ)] Roots: z_k = r^(1/n) [cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)]

The calculator uses these equations to move between forms, perform operations, and display results in precise rectangular notation.

How to Use This Calculator

Select rectangular or polar mode for each complex number.
Enter real and imaginary values, or modulus and angle.
Choose the operation you want to perform.
Set exponent or root count when required.
Pick decimal precision and preferred angle display unit.
Press Calculate Complex Form to see the result.
Review the result summary, root table, and Argand plot.
Use the CSV or PDF buttons to export your output.

Frequently Asked Questions

1. What is rectangular form in complex numbers?

Rectangular form expresses a complex number as a + bi. The value a is the real part, and b is the imaginary part. This format is ideal for addition, subtraction, and direct coordinate plotting.

2. Can I enter polar values instead of rectangular values?

Yes. Each input number can be entered in rectangular or polar form. The calculator converts polar values into rectangular form first, then performs the selected operation and shows both formats in the result.

3. What does the modulus represent?

The modulus is the distance of the complex number from the origin on the Argand plane. It is calculated using √(a² + b²). Larger modulus values mean the point lies farther from the center.

4. Why is the argument important?

The argument gives the direction angle of the complex number from the positive real axis. It is especially useful for polar conversion, powers, roots, phasor work, and geometric interpretation of rotation.

5. What happens if I divide by zero?

Division by a complex number with zero real and imaginary parts is undefined. The calculator blocks that case and displays an error message instead of returning a misleading result.

6. How are roots displayed?

When you choose the roots option, the calculator computes every distinct n-th root of the primary number. It lists each root in rectangular form and also gives its modulus and argument.

7. Why does the graph help?

The graph places complex numbers on the Argand plane, so you can see direction, magnitude, symmetry, and the spread of roots. That visual check helps confirm whether the result looks mathematically reasonable.

8. When should I use rectangular form instead of polar form?

Rectangular form is usually best for addition and subtraction. Polar form is often easier for multiplication, division, powers, and roots. This calculator supports both, so you can work with the most convenient method.