Decimal to Polar Number Calculator

Formula used

For a decimal complex number z = x + yi, polar form is based on radius and angle.

r = √(x² + y²)

θ = atan2(y, x)

z = r(cos θ + i sin θ)

z = r cis(θ)

z = re^(iθ)

The two argument arctangent helps place the angle in the correct quadrant.

How to use this calculator

Enter the real decimal value in the first field. Enter the imaginary decimal value in the second field. Select the number of decimal places. Choose the angle style and output form. Press Calculate. The answer appears above the form and below the header. Use the CSV or PDF buttons to save the result.

Example data table

Decimal to polar number guide

What this conversion means

A decimal to polar number calculator changes rectangular values into polar form. It treats the first value as the real part. It treats the second value as the imaginary part. The tool then finds the distance from the origin and the direction angle.

Why polar form matters

This conversion is common in algebra, trigonometry, engineering, signal work, and complex number study. Rectangular form is useful for adding numbers. Polar form is useful for multiplication, division, powers, and rotations. It also makes phase and magnitude easier to read.

Advanced output details

The calculator accepts positive, negative, and zero decimal values. It can show the angle in degrees or radians. It also reports the quadrant, normalized angle, principal angle, slope, modulus squared, and conjugate polar angle. These extra outputs help you check signs and avoid quadrant errors.

Radius and angle behavior

The radius is always nonnegative. It measures how far the point is from zero. The angle depends on both decimal inputs. A simple arctangent can fail when the real part is negative. This tool uses the two argument arctangent method. That method places the angle in the correct quadrant.

Precision and exports

Precision control is important. Many polar answers contain long decimal values. You can choose how many decimal places appear. The calculator stores the unrounded values internally. It only rounds the display and export output.

Checking examples

Use the example table to compare common coordinates. Points on the axes are special. The angle may be zero, ninety degrees, one hundred eighty degrees, or minus ninety degrees. The origin has radius zero. Its angle is usually treated as undefined or zero by convention.

Saving your result

CSV export is helpful for spreadsheets. PDF export is useful for reports and classroom notes. Both exports use the submitted values and the calculated outputs. They make it easier to save work or share results.

Angle convention reminder

Always confirm your required angle convention. Some classes use angles from zero to three hundred sixty degrees. Others use negative to positive one hundred eighty degrees. The calculator shows both styles for better checking.

Input accuracy

For best results, enter measured values with enough decimal places. Avoid rounding too early. Small changes can move the angle noticeably near the axes. Copy the formatted result when you need a clean final answer.

FAQs

What is a decimal to polar number calculator?

It converts rectangular decimal values into polar form. The real part and imaginary part become a radius and an angle. This helps describe magnitude and direction clearly.

What inputs do I need?

You need a real decimal value and an imaginary decimal value. Together they form a complex number written as x + yi.

Why does the calculator use atan2?

atan2 checks both the real and imaginary parts. It places the angle in the correct quadrant. This avoids common arctangent mistakes.

Can the angle be negative?

Yes. A signed angle can be negative. The same direction can also be shown as a normalized angle between zero and three hundred sixty degrees.

What happens at the origin?

When both decimal values are zero, the radius is zero. The angle is usually undefined, though some tools display zero by convention.

What is CIS form?

CIS form means cosine plus i sine. It writes a complex number as r cis θ. This is a compact polar notation.

Is polar form useful for multiplication?

Yes. Polar form makes multiplication easier. You multiply radii and add angles. This is simpler than expanding rectangular parts.

Can I export my result?

Yes. Use the CSV button for spreadsheet work. Use the PDF button for a report style file with the main calculated values.