Enter Polar Complex Number Values
Use z = r(cos θ + i sin θ). Magnitude must be zero or greater.
Example Conversion Data
| Magnitude | Angle | Unit | Rectangular result |
|---|---|---|---|
| 5 | 53.130102 | Degrees | 3.0000 + 4.0000i |
| 10 | 30 | Degrees | 8.6603 + 5.0000i |
| 4 | 180 | Degrees | −4.0000 + 0.0000i |
| 2 | 4.712389 | Radians | 0.0000 − 2.0000i |
Formula Used
Polar form expresses a complex number as z = r(cos θ + i sin θ).
Real part, a = r × cos(θ)
Imaginary coefficient, b = r × sin(θ)
Rectangular form, z = a + bi
When you select degrees, the angle is converted to radians before trigonometric calculation. Equivalent angles produce the same real and imaginary components.
How to Use This Calculator
- Enter the non-negative magnitude of the complex number.
- Enter the angle shown in your original polar value.
- Select degrees or radians before submitting the form.
- Choose the number of decimal places for the displayed result.
- Select a display preference and optionally add a reference label.
- Press the conversion button and review the result above the form.
- Use the CSV button or print option when you need a record.
Understanding Polar and Rectangular Complex Forms
Why Two Forms Matter
A complex number can be written in rectangular form as a plus bi. The letter a is the real part. The letter b is the imaginary coefficient. Polar form describes the same value differently. It uses a magnitude and an angle. The magnitude measures distance from the origin. The angle measures rotation from the positive real axis. Both forms are useful. Rectangular form helps addition and subtraction. Polar form helps multiplication, division, powers, and roots. They also reveal geometric meaning clearly.
This calculator changes polar values into rectangular coordinates. Enter the magnitude first. Then enter the angle. Choose degrees or radians carefully. The calculator applies cosine to find the real component. It applies sine to find the imaginary component. The result appears as a plus bi. A negative imaginary component is shown with a minus sign. The page also reports normalized angle details. These details help you check equivalent angles and confirm your entry. Use the answer with confidence during later work.
Reading Magnitude and Angles
Magnitude is normally zero or positive. A magnitude of zero represents the origin. Its angle does not change the final rectangular answer. A positive magnitude creates a point away from the origin. Angles may be positive or negative. They may also exceed one full turn. The calculator normalizes them for reference. This does not alter the complex number. For example, 450 degrees equals 90 degrees. Negative 270 degrees also equals 90 degrees. Equivalent angles reach the same point. This rule makes angle interpretation much easier.
Degree mode is convenient for geometry classes and hand calculations. Radian mode is common in calculus, engineering, and programming. One complete turn equals 360 degrees. The same turn equals two pi radians. Enter decimal angles when needed. For example, 37.5 degrees is valid. So is 1.2 radians. Do not mix the units. A correct magnitude with the wrong angle unit gives a different complex number. Always review the selected unit before calculating. Label your notes with the chosen unit.
Rounding and Practical Checks
The rectangular components can be rounded for display. Choose a precision that fits your work. Two or three decimal places suit quick estimates. Higher precision helps technical tasks. Rounding affects only the shown values. The calculator keeps the internal calculation accurate before formatting. Very small values may display as zero. This often happens near 90, 180, 270, or 360 degrees. It is caused by floating point arithmetic. Treat tiny residuals as practical zeros when appropriate.
Use the output to verify homework, circuit calculations, signal analysis, and coordinate conversions. Copy the real and imaginary components into later steps. The conjugate changes only the imaginary sign. Its value is useful in division and magnitude calculations. The calculator includes the conjugate for convenience. You can also download a comma separated record. Printing the result allows a browser PDF save. Recheck source data before sharing a result. Clear units and sensible rounding prevent most conversion mistakes.
Frequently Asked Questions
What is rectangular form for a complex number?
Rectangular form writes a complex number as a + bi. The value a is the real part. The value b is the imaginary coefficient. It also matches the coordinate pair (a, b) on the complex plane.
Which values does the calculator need?
Enter the magnitude r, the angle θ, and its unit. You can also choose display precision, a preferred output style, and an optional reference label for your saved result.
Should I use degrees or radians?
Use the unit supplied with your source value. Geometry exercises often use degrees. Calculus, engineering, and software work often use radians. The same numeric angle can represent different directions in different units.
Can polar magnitude be negative?
This calculator requires a non-negative magnitude. Standard polar form uses r greater than or equal to zero. A negative radial value can be rewritten with a positive magnitude and an angle shifted by 180 degrees or pi radians.
Why does a value near zero appear?
Computers store decimal values with finite precision. Trigonometric results near axes can leave extremely small residual numbers. The calculator treats values smaller than half of the selected display unit as zero.
How is a 450 degree angle handled?
A 450 degree angle ends at the same position as 90 degrees. The calculator uses the supplied angle for calculation and also reports a normalized angle between 0 and less than 360 degrees.
Does zero magnitude have a meaningful angle?
When r equals zero, the result is always 0 + 0i. Any angle gives the same point at the origin. The angle can still be retained as a descriptive input, but it does not change the result.
Can I enter decimal angles?
Yes. The calculator accepts decimal values for both degrees and radians. Use enough decimal places in the input when you need a precise result, then select the display precision you prefer.
What is the conjugate shown in the result?
The conjugate keeps the real part unchanged and reverses the imaginary sign. For example, the conjugate of 3 + 4i is 3 − 4i. Conjugates are useful in complex division.
Does rounding change the actual calculation?
Rounding changes only the displayed text. The conversion is calculated before display formatting. Choose more decimal places when a later calculation needs greater detail or when you compare close values.
What do the CSV and print options provide?
The CSV file contains the input, rectangular result, components, conjugate, and normalized angles. The print option opens your browser print interface, where you can choose a printer or save a PDF.