Calculator Inputs
Formula Used
Rectangular form: z = a + bi
Modulus: r = √(a² + b²)
Argument: θ = atan2(b, a)
Exponential form: z = r eiθ
Power rule: zn = rneinθ
Verification: a = r cos θ and b = r sin θ
How to Use This Calculator
- Enter the real part a from a + bi.
- Enter the imaginary coefficient b. Do not type the letter i.
- Select radians, degrees, or both for the argument.
- Choose the principal angle or a positive angle range.
- Set decimal precision for the final display.
- Enter an optional integer power if you need z raised to n.
- Add batch pairs when several values need conversion.
- Press the calculate button and review the result above the form.
Example Data Table
| Rectangular value | Modulus r | Principal argument | Exponential form |
|---|---|---|---|
| 3 + 4i | 5 | 0.9273 rad | 5ei0.9273 |
| -2 + 2i | 2.8284 | 2.3562 rad | 2.8284ei2.3562 |
| 0 - 5i | 5 | -1.5708 rad | 5e-i1.5708 |
Understanding the Conversion
Rectangular and exponential forms describe the same complex number. Rectangular form writes the value as a + bi. The real part shows horizontal movement. The imaginary part shows vertical movement. Exponential form writes the value as r e^(iθ). It shows distance from the origin and angle from the positive real axis.
This conversion is useful in signals, circuits, vibration, control systems, and phasor work. Multiplication becomes easier in exponential form. You multiply magnitudes. You add angles. Division is also cleaner. You divide magnitudes. You subtract angles.
The modulus r is the length of the complex vector. It is found with the Pythagorean theorem. The argument θ is found with atan2. This function reads both signs. It places the angle in the correct quadrant. That matters when the real or imaginary part is negative.
A zero complex number needs special care. Its modulus is zero. Its angle is undefined. Many tools show zero radians for convenience. This calculator explains that special case. It also keeps the output useful for reports and checks.
Angle units can change the look of the answer. Radians are common in mathematics. Degrees are common in engineering notes and teaching. The same complex value can use either unit. This tool can show radians, degrees, or both. It can also normalize the angle. A principal angle usually sits between -π and π. A positive angle sits between 0 and 2π.
Exponential form helps with powers. De Moivre's rule states that (r e^(iθ))^n = r^n e^(inθ). This avoids repeated rectangular multiplication. The calculator includes a power option. It shows the powered exponential form and a rectangular verification.
The sign of the imaginary part affects the angle direction. A positive imaginary part points upward. A negative imaginary part points downward. The real part decides whether the vector sits on the right or left side. Together, both parts define the quadrant.
Rounding can change the final display. Higher precision gives safer intermediate values. Lower precision gives cleaner published answers. Use more digits while checking. Round only after the final step. This prevents avoidable angle and magnitude errors.
Batch conversion saves time when many complex numbers appear in one task. Each row can hold one real part and one imaginary part. The table then reports each modulus and argument. This is helpful for homework sets, laboratory tables, impedance lists, and signal samples. It also makes mistakes easier to spot. Similar magnitudes and angles should show clear patterns.
For learning, the step view is important. It shows how r and θ are built. That makes the answer easier to audit before using it elsewhere in later calculations too.
This calculator is designed for careful conversion. Enter the real part and imaginary part. Choose angle units and precision. Review the formula steps. Then compare the rectangular verification. The result shows the modulus, argument, exponential expression, polar notation, quadrant, conjugate, and optional power form in one place.
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 coefficient of the imaginary part.
What is exponential form?
Exponential form writes a complex number as r e^(iθ). The value r is the modulus. The value θ is the argument angle.
How is the modulus calculated?
The modulus is calculated with r = √(a² + b²). It is the distance from the origin to the complex point.
How is the argument calculated?
The argument is calculated with atan2(b, a). This method checks signs of both values and places the angle in the correct quadrant.
Why use atan2 instead of arctan?
atan2 avoids quadrant mistakes. A plain arctan ratio can give the same value for different quadrants. atan2 reads both coordinates separately.
What happens when z equals zero?
The modulus is zero. The angle is undefined because the point has no direction from the origin. The calculator states this clearly.
Can I use degrees instead of radians?
Yes. Choose degrees from the angle unit menu. You can also show both units when comparing math and engineering notes.
What is the principal angle?
The principal angle is a standard angle choice, usually shown between -π and π. It gives one consistent representation.
Why does the calculator show a power option?
Exponential form makes powers easier. It uses De Moivre's rule, where magnitudes are raised to n and angles are multiplied by n.
How should batch values be entered?
Enter one pair per line. Use a space, comma, or semicolon between the real part and imaginary coefficient.
How much precision should I choose?
Use enough precision before rounding your final reported answer.