Exponential to Rectangular Calculator

Calculator

Magnitude r

Angle θ

Angle Unit

Exponential Type

Decimal Precision

Imaginary Symbol

Normalize angle before conversion

Formula Used

For a positive imaginary exponent, the calculator uses Euler’s identity.

r e^(iθ) = r cos(θ) + i r sin(θ)

For a negative imaginary exponent, the imaginary part changes sign.

r e^(-iθ) = r cos(θ) - i r sin(θ)

The rectangular form is written as a + bi. Here, a = r cos(θ) and b = r sin(θ).

How to Use This Calculator

Enter the magnitude value in the magnitude field.
Enter the angle value in degrees or radians.
Select the correct angle unit.
Choose positive or negative exponential form.
Set the number of decimal places.
Choose i or j for the imaginary part.
Press Calculate to view the result above the form.
Use CSV or PDF download for saving the answer.

Example Data Table

Exponential Form	Angle Unit	Rectangular Form	Note
5e^(i30°)	Degrees	4.3301 + 2.5000i	Quadrant I
10e^(i90°)	Degrees	0.0000 + 10.0000i	Positive imaginary axis
8e^(-i45°)	Degrees	5.6569 - 5.6569i	Negative exponent
3e^(iπ)	Radians	-3.0000 + 0.0000i	Negative real axis

Understanding Exponential to Rectangular Conversion

Exponential form is a compact way to describe a complex number. It uses a magnitude and an angle. The magnitude shows distance from zero. The angle shows direction from the positive real axis. A value such as re^(iθ) can be expanded with Euler’s identity. The final rectangular value is written as a + bi.

Why Rectangular Form Matters

Rectangular form is useful because it separates the horizontal part and the vertical part. The real part is easy to compare. The imaginary part is easy to track. Engineers use this format for circuits, waves, filters, and signals. Students use it for algebra checks and graph work. The same conversion also helps when adding complex numbers.

Advanced Input Options

This calculator gives control over practical details. You can enter angles in degrees or radians. You can choose e^(iθ) or e^(-iθ). You can set decimal precision for clean reporting. You can also choose the imaginary symbol. Use i for mathematics. Use j for electrical work.

Accuracy Tips

Accuracy depends on clear input. Use a nonnegative magnitude when you want standard polar meaning. Use radians only when the angle was already measured in radians. Use degrees for common engineering and classroom angles. Very small values may appear as zero after rounding. This avoids noisy results like 0.0000000001.

Calculation Process

The conversion has two main parts. First, the angle is prepared. If degrees are selected, the angle is converted to radians. Second, cosine and sine are applied. The cosine result becomes the real part. The sine result becomes the imaginary part. A negative exponential changes the sign of the imaginary part.

Saving Results

CSV export is helpful for spreadsheets. PDF export is useful for printed notes. The example table shows typical conversions. It also shows how signs change between quadrants. Review the formula section before using the result in a report.

Best Practice

For best results, record the original exponential value with each answer. Keep the selected angle unit beside the result. Keep the chosen precision visible too. These details make your work easier to verify later. They also reduce mistakes when values are shared with another person. The output card appears above the form after submission. This placement helps users read the answer before adjusting advanced settings again quickly.

FAQs

What is exponential form?

Exponential form writes a complex number using magnitude and angle. It often appears as r e^(iθ). The magnitude gives size. The angle gives direction.

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 part.

Which formula does this calculator use?

It uses Euler’s identity. The positive form is r cos(θ) + i r sin(θ). The negative form is r cos(θ) - i r sin(θ).

Should I use degrees or radians?

Use the unit that matches your angle. Classroom and engineering examples often use degrees. Calculus and advanced math problems often use radians.

What does e^(-iθ) change?

It changes the sign of the imaginary part. The real part stays r cos(θ). The imaginary part becomes negative r sin(θ).

Why does zero sometimes appear after rounding?

Some trigonometric results are extremely close to zero. Rounding removes tiny calculator noise and gives a cleaner rectangular answer.

Can I use j instead of i?

Yes. Choose j when working with electrical or engineering notation. Choose i when using standard math notation.

What does angle normalization do?

It rewrites the angle into one standard rotation range before conversion. This keeps large or negative angles easier to read.