Advanced complex calculator
Enter your imaginary numbers
Use a + bi form. Select an operation, then calculate exact components and useful representations.
Example data
These sample inputs show common imaginary-number operations.
| z₁ | Operation | z₂ or setting | Expected result |
|---|---|---|---|
| 3 + 4i | Add | 1 − 2i | 4 + 2i |
| 3 + 4i | Multiply | 1 − 2i | 11 − 2i |
| 3 + 4i | Divide | 1 − 2i | −1 + 2i |
| 3 + 4i | Modulus | — | 5 |
| 1 + i | Power | Exponent 2 | 2i |
| −4 | Square roots | n = 2 | 2i, −2i |
Formula used
How to use this calculator
- Enter the real and imaginary coefficients for z₁.
- Select the operation that matches your problem.
- Provide z₂ for addition, subtraction, multiplication, or division.
- Set an exponent or root index when those options apply.
- Choose degrees or radians for angular answers.
- Select the calculation button and review the result above the form.
- Use the CSV or PDF buttons to save the visible result.
Imaginary numbers in practice
Start with the i unit
Imaginary numbers use the unit i. It satisfies i² = −1. A complex number combines a real part and an imaginary part. Its standard form is a + bi. The value a lies on the real axis. The value b lies on the imaginary axis. This notation keeps both components visible.
Use component rules first
Addition and subtraction are direct. Combine real parts together. Combine imaginary parts together. Multiplication needs the rule i² = −1. That rule changes a negative square into a real contribution. Division uses the conjugate. It removes i from the denominator. The calculator shows the related components and intermediate checks.
Read the Argand plane
Complex numbers can be plotted as points. The horizontal coordinate is the real part. The vertical coordinate is the imaginary part. This picture is called an Argand plane. It makes signs and quadrants easier to inspect. It also helps explain multiplication and roots. The graph displays each input and generated answer.
Move between rectangular and polar forms
Rectangular form uses a + bi. Polar form uses a length and angle. The length is called the modulus. The angle is called the argument. Polar form is especially useful for powers and roots. De Moivre’s rule scales the angle cleanly. The calculator reports both forms when needed.
Check answers with structure
A good result should preserve useful relationships. A conjugate keeps the real component. Its imaginary sign reverses. The modulus never becomes negative. Division fails when the divisor is zero. Roots spread evenly around a circle. These checks catch common sign mistakes. Exports make review and record keeping easier.
Use precise inputs responsibly
Decimals are accepted for both components. Round only after interpreting the answer. Very small values may display as zero. This reduces floating-point clutter. For classroom work, write the exact expression too. For engineering tasks, retain enough decimal places. Always confirm the expected angle unit before using polar results.
Remember that i is not a variable. It is a defined number system extension. Complex arithmetic appears in circuit analysis and signal processing. It also supports control systems and wave models. The same calculation rules remain consistent. Careful notation makes technical communication faster, clearer, and easier to verify. Use exports when sharing completed exercises with tutors or peers.
Frequently asked questions
1. What is an imaginary number?
An imaginary number contains i, where i² equals −1. Examples include 5i and −2.4i. Complex numbers combine a real term with an imaginary term.
2. What does a + bi mean?
It is the standard rectangular form of a complex number. The value a is real. The value b is the coefficient of i. For example, 3 + 4i has real part 3 and imaginary part 4.
3. Can I enter negative imaginary values?
Yes. Enter a negative number in the imaginary coefficient field. For 2 − 7i, use 2 for the real part and −7 for the imaginary part.
4. Why does multiplication use i² = −1?
Expanding two complex numbers creates a term containing i². Replacing i² with −1 converts that term into a real value. This produces the multiplication rule ac − bd for the real component.
5. Why is division by zero invalid?
A complex divisor c + di has magnitude squared c² + d². When both c and d are zero, this denominator is zero. No finite complex quotient exists.
6. What is a complex conjugate?
The conjugate of a + bi is a − bi. It keeps the real part and reverses the imaginary sign. Conjugates are useful when simplifying complex fractions.
7. What is the modulus?
The modulus is the distance from the origin on the Argand plane. For a + bi, it equals √(a² + b²). It is always zero or positive.
8. What is the argument?
The argument is the angle from the positive real axis to the complex-number point. This calculator uses atan2, which identifies the correct quadrant. Results can appear in degrees or radians.
9. How are roots found?
The calculator converts the number to polar form. It divides the argument across n evenly spaced angles. Each root has modulus r^(1/n). All roots are listed and plotted.
10. Can I calculate negative powers?
Yes, provided z₁ is not zero. A negative power represents the reciprocal of the corresponding positive power. Zero cannot have a negative exponent.
11. What downloads are available?
After a successful calculation, you can download result rows as CSV or a compact PDF summary. The files include the operation output and working steps.