Complex Number Calculator
Calculation History
| # | Operation | Input | Result | Label |
|---|---|---|---|---|
| No calculations yet. | ||||
Example Data Table
| First number | Second number | Operation | Expected result |
|---|---|---|---|
| 3 + 4i | 1 - 2i | Add | 4 + 2i |
| 3 + 4i | 1 - 2i | Multiply | 11 - 2i |
| 3 + 4i | 1 - 2i | Divide | -1 + 2i |
| 3 + 4i | Not needed | Modulus | 5 |
Formula Used
Standard form: z = a + bi, where a is the real part and b is the imaginary part.
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)
Modulus: |z| = √(a² + b²)
Argument: θ = atan2(b, a)
Polar form: z = r(cos θ + i sin θ)
Conjugate: conj(a + bi) = a - bi
Power: zⁿ = rⁿ(cos nθ + i sin nθ)
Roots: z^(1/n) = r^(1/n)[cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)]
How to Use This Calculator
- Enter the real and imaginary parts of the first complex number.
- Enter the second number when the selected operation needs it.
- Choose the operation, such as add, divide, polar form, roots, or power.
- Set decimal places and choose degrees or radians for angle output.
- Press the calculate button. The answer appears above the form.
- Use CSV or PDF export to save your latest calculation history.
Article: Simplifying Complex Numbers
Understanding the Main Idea
A complex number combines a real part and an imaginary part. It is usually written as a + bi. The value a shows the horizontal position on the complex plane. The value b shows the vertical position. Simplifying a complex number means writing it in a clean and useful form. This may include combining like parts, removing a complex denominator, finding a conjugate, or converting the number to polar notation.
Why Rectangular Form Matters
Rectangular form is best for addition and subtraction. You only combine real parts with real parts. You combine imaginary parts with imaginary parts. Multiplication uses the rule i² = -1. This rule turns part of the product into a real value. Division uses the conjugate of the denominator. That step removes the imaginary part from the denominator and gives a simpler quotient.
Why Polar Form Matters
Polar form uses a distance and an angle. The distance is called the modulus. The angle is called the argument. This form is powerful when working with powers and roots. De Moivre’s rule lets you raise a complex number to a power by raising the modulus and multiplying the angle. Roots are found by spreading angles evenly around the complex plane. This makes polar form helpful in algebra, signals, electronics, and advanced geometry.
Using Results Carefully
A calculator can reduce errors, but you should still review each step. Check signs, especially during multiplication and division. A negative imaginary part can change the whole result. Also choose a suitable number of decimal places. Exact values are useful in theory. Rounded values are useful in reports. When studying, compare rectangular, polar, modulus, argument, and conjugate outputs. Seeing the same number in several forms builds better understanding and supports faster problem solving.
FAQs
What is a complex number?
A complex number has a real part and an imaginary part. It is commonly written as a + bi, where i represents the square root of -1.
What does simplifying a complex number mean?
It means writing the number in a cleaner form. This can include combining terms, removing an imaginary denominator, finding polar form, or reducing powers of i.
How do I add complex numbers?
Add real parts together. Then add imaginary parts together. For example, (3 + 4i) + (1 - 2i) becomes 4 + 2i.
How do I multiply complex numbers?
Multiply like binomials. Then replace i² with -1. After that, combine real and imaginary terms into standard form.
Why is the conjugate important?
The conjugate changes the sign of the imaginary part. It is often used to simplify division and remove complex numbers from denominators.
What is the modulus?
The modulus is the distance from zero on the complex plane. For a + bi, it equals the square root of a² + b².
What is polar form?
Polar form writes a complex number using its modulus and argument. It is useful for powers, roots, rotations, and trigonometric interpretation.
Can this calculator find roots?
Yes. Enter the complex number, choose roots, and set the degree. The calculator returns all principal roots with step details.