Calculator Inputs
Use z₁ for single-number tasks. Use both z₁ and z₂ for addition, subtraction, multiplication, and division.
Example Data Table
These examples use z₁ = 3 + 4i and z₂ = 1 - 2i.
| Operation | Expression | Sample Output |
|---|---|---|
| Addition | (3 + 4i) + (1 - 2i) | 4 + 2i |
| Subtraction | (3 + 4i) - (1 - 2i) | 2 + 6i |
| Multiplication | (3 + 4i)(1 - 2i) | 11 - 2i |
| Division | (3 + 4i) / (1 - 2i) | -1 + 2i |
| Modulus of z₁ | |3 + 4i| | 5 |
| Conjugate of z₁ | conj(3 + 4i) | 3 - 4i |
| Square of z₁ | (3 + 4i)² | -7 + 24i |
| Polar Form of z₁ | 3 + 4i | 5(cos(0.927295218) + i sin(0.927295218)) |
Formula Used
Core Arithmetic
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)/(c² + d²)) + ((bc - ad)/(c² + d²))i
Magnitude and Angle
Modulus: |a + bi| = √(a² + b²)
Argument: θ = atan2(b, a)
Conjugate: conj(a + bi) = a - bi
Reciprocal: 1/(a + bi) = a/(a² + b²) - (b/(a² + b²))i
Powers
Square: (a + bi)² = (a² - b²) + 2abi
Cube: (a + bi)³ = (a³ - 3ab²) + (3a²b - b³)i
De Moivre: (r(cosθ + i sinθ))ⁿ = rⁿ(cos(nθ) + i sin(nθ))
Roots and Forms
n-th Roots: r^(1/n)[cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)]
Polar Form: z = r(cosθ + i sinθ)
Exponential Form: z = r e^(iθ)
How to Use This Calculator
- Enter the real and imaginary parts for z₁.
- Enter z₂ only when using addition, subtraction, multiplication, or division.
- Select the operation from the dropdown menu.
- Provide an exponent for integer powers or a root value for n-th roots.
- Click Calculate Now to display the result above the form.
- Review rectangular, polar, exponential, modulus, and argument outputs where available.
- Use the Argand diagram to visualize the point locations on the complex plane.
- Download the current result as CSV or PDF for study notes or reporting.
Frequently Asked Questions
1) What is an imaginary number?
An imaginary number is a real coefficient multiplied by i, where i² = -1. A complex number combines a real part and an imaginary part, usually written as a + bi.
2) What is the difference between imaginary and complex numbers?
An imaginary number has no real part, such as 5i. A complex number can include both parts, such as 3 + 5i. Purely real numbers are also complex numbers with zero imaginary part.
3) Why are modulus and argument important?
They convert rectangular values into magnitude-angle form. That makes powers, roots, rotations, phasors, and signal analysis easier to interpret and compute with standard trigonometric relationships.
4) Can this calculator handle powers and roots?
Yes. It can compute integer powers of z₁ and generate all n-th roots of z₁. The roots are listed individually and plotted on the complex plane.
5) Why does division fail when z₂ is zero?
Division by zero is undefined for complex numbers just as it is for real numbers. If both the real and imaginary parts of z₂ are zero, the denominator has zero magnitude.
6) What does the conjugate do?
The conjugate changes the sign of the imaginary part. It reflects the point across the real axis and is useful in rationalizing denominators, computing magnitudes, and simplifying products.
7) What does the graph show?
The graph shows inputs and results on the Argand plane. The horizontal axis is the real part, and the vertical axis is the imaginary part. Roots appear as multiple plotted points.
8) Are angles shown in degrees or radians?
Both are provided. Radians are standard for formulas and advanced math work, while degrees are often easier to read during quick checks and classroom interpretation.