Complex Number Input
Enter a rectangular complex number as a + bi. The tool returns modulus, argument, quadrant, polar notation, powers, roots, and verification values.
Example Data Table
| Rectangular form | Modulus | Principal argument | Polar form |
|---|---|---|---|
| 3 + 4i | 5 | 0.927295 rad | 5 cis(0.927295) |
| -5 + 12i | 13 | 1.965587 rad | 13 cis(1.965587) |
| -7 - 24i | 25 | -1.854590 rad | 25 cis(-1.854590) |
| 0 - 6i | 6 | -1.570796 rad | 6 cis(-1.570796) |
| 8 + 0i | 8 | 0 rad | 8 cis(0) |
Formula Used
For a complex number z = a + bi, the polar form is z = r cis(θ), or z = r(cos θ + i sin θ).
r = √(a² + b²)
θ = atan2(b, a)
The atan2 function is important. It reads both signs. It places the angle in the correct quadrant. De Moivre options use zⁿ = rⁿ cis(nθ). Roots use r^(1/n) cis((θ + 2πk)/n), where k runs from 0 to n - 1.
How to Use This Calculator
- Enter the real part in the first field.
- Enter the imaginary coefficient in the second field.
- Choose radians, degrees, or both angle formats.
- Select the angle range you need.
- Pick a polar notation style.
- Add optional power or root settings.
- Press the calculate button.
- Review the result above the form.
Rectangular and Polar Complex Forms
Complex numbers can be written in two useful ways. Rectangular form uses a real part and an imaginary part. It looks like a plus bi. This form is direct. It shows horizontal and vertical movement on the complex plane.
Polar form tells the same story differently. It gives a distance and an angle. The distance is the modulus. The angle is the argument. Together they locate the same point from the origin.
Why Modulus Matters
The modulus measures how far the complex point is from zero. It is found with the Pythagorean theorem. Square the real part. Square the imaginary coefficient. Add both values. Then take the square root. This gives the radius of the point.
A larger modulus means the point is farther away. A modulus of zero means the point is at the origin. In that special case, the angle is not defined. The calculator warns you when this happens.
Why Argument Matters
The argument shows direction. It is the angle from the positive real axis. Many mistakes happen when the ordinary arctangent is used alone. That method can lose quadrant information. It then chooses the correct quadrant.
You can show the angle as a principal value. That range runs from negative pi to pi. You can also use the positive range from zero to two pi. Both ranges describe the same point. The best choice depends on your class, textbook, or engineering rule.
Working With Powers
Polar form is powerful for exponent work. De Moivre rule multiplies the angle by n. It raises the modulus to n. This makes powers much easier than repeated rectangular multiplication. The calculator also converts the result back to rectangular form for checking.
Negative powers need care. They use reciprocal values. A zero complex number cannot have a negative power. The tool detects that case and shows a clear warning.
Finding Complex Roots
Complex roots are also simpler in polar form. The root modulus is the nth root of r. The root angles are spaced evenly around the circle. Each value of k gives one root. For an nth root, there are n roots unless the original number is zero.
The roots table lists each k value. It shows the root modulus, root angle, and rectangular check. This is useful for homework, signals, circuit work, and algebra review. It also helps you see the geometry behind roots.
Practical Uses
This converter is useful in algebra, trigonometry, electrical analysis, controls, digital signals, and physics. Engineers use polar form for phasors. Students use it for powers and roots. The two forms are equal, but each form makes different tasks easier.
Use rectangular form when adding or subtracting complex numbers. Use polar form when multiplying, dividing, raising powers, or taking roots. Switching forms at the right time saves work. This calculator gives both forms and checks them step by step.
FAQs
What is rectangular form?
Rectangular form writes a complex number as a + bi. The value a is the real part. The value b is the imaginary coefficient. It matches x and y coordinates on the complex plane.
What is polar form?
Polar form writes a complex number with a modulus and an argument. It may appear as r cis(θ), re^(iθ), or r(cos θ + i sin θ). All forms describe the same point.
How is the modulus calculated?
The modulus is the distance from the origin. Use r = √(a² + b²). Square the real part. Square the imaginary coefficient. Add them. Then take the square root.
How is the argument calculated?
The argument is found with atan2(b, a). This function uses both the real and imaginary signs. That helps place the angle in the correct quadrant.
Why use atan2 instead of arctangent?
Plain arctangent can miss quadrant information. Atan2 reads a and b separately. It handles negative values and axis cases better. That makes the angle safer for complex conversion.
Can the calculator show degrees?
Yes. Select the angle display option. You can show radians only, degrees only, or both. The formula uses radians internally, then converts to degrees when requested.
What happens when the complex number is zero?
The modulus becomes zero. The argument is undefined because the origin has no unique direction. The calculator shows a warning and uses zero as a display angle.
What is cis notation?
Cis notation means cos θ + i sin θ. So r cis(θ) means r(cos θ + i sin θ). It is a compact way to write polar complex numbers.
How are powers calculated?
Powers use De Moivre rule. The modulus is raised to n. The angle is multiplied by n. The calculator also gives a rectangular check for the power result.
How are complex roots calculated?
Roots use the nth root of the modulus. Their angles are (θ + 2πk) / n. Each value of k gives a different root around the circle.
Why does the angle range matter?
Different courses and tools use different angle ranges. Principal angles use -π to π. Positive angles use 0 to 2π. Both can represent the same complex number.