Calculator Input
Graph
The plot shows the rectangular point and its vector from the origin.
Formula Used
a = r cos θ
b = r sin θ
z = a + bi
Here, r is the magnitude. The angle θ is the argument.
The value a is the real part. The value b is the imaginary part.
If the angle is entered in degrees, it is converted into radians before using sine and cosine.
How to Use This Calculator
- Enter the magnitude of the complex number.
- Enter the angle value.
- Select degrees or radians.
- Choose precision and zero tolerance settings.
- Press the calculate button.
- Review the rectangular form above the input form.
- Use CSV or PDF export for records.
Example Data Table
| Magnitude r | Angle | Unit | Real part a | Imaginary part b | Rectangular form |
|---|---|---|---|---|---|
| 5 | 53.130102 | Degrees | 3 | 4 | 3 + 4i |
| 10 | 180 | Degrees | -10 | 0 | -10 + 0i |
| 8 | 270 | Degrees | 0 | -8 | 0 - 8i |
| 2 | 0.785398 | Radians | 1.414214 | 1.414214 | 1.414214 + 1.414214i |
Understanding Rectangular Form
A complex number can be written in many useful forms. Rectangular form is the most direct form because it separates the horizontal part and the vertical part. It is written as a plus bi, where a is the real part and b is the imaginary part. This structure is easy to graph, compare, add, subtract, and use inside algebraic work.
Why Conversion Matters
Polar form describes a complex number with a magnitude and an angle. It is excellent for rotation, roots, powers, and signals. Rectangular form is often better for arithmetic and coordinate work. Converting between the two forms helps students see the same value from two viewpoints. It also prevents sign mistakes when the angle moves into another quadrant.
How the Calculator Helps
This calculator accepts magnitude and angle values. It lets you choose degrees or radians. It also gives precision control and a zero tolerance setting. Small floating point values close to zero can be cleaned automatically. This is helpful when a result should be exactly zero, but decimal rounding produces a tiny number instead.
The result panel shows the real part, imaginary part, rectangular expression, normalized angle, quadrant, and a modulus check. The graph plots the point and draws a vector from the origin. This visual check is useful. It shows whether the calculated signs match the intended quadrant.
Using Results in Study
When you solve by hand, first convert the angle into radians if needed. Then multiply the magnitude by cosine for the real part. Multiply the magnitude by sine for the imaginary part. Keep track of the sign of each trigonometric value. Finally, write the answer as a plus bi or a minus bi.
The CSV export is useful for spreadsheets and class records. The PDF export is useful for reports, homework notes, and printed explanations. The example table gives sample conversions that you can compare against your own entries. These features make the page more than a simple converter. It becomes a compact learning tool for polar and rectangular complex number work. For best accuracy, keep the inputs visible until the answer has been checked against the graph and formula steps.
FAQs
1. 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. It places the number on a coordinate plane.
2. What is polar form?
Polar form uses magnitude and angle. It is usually written as r(cos θ + i sin θ). It describes distance from the origin and direction from the positive real axis.
3. How do I convert polar to rectangular form?
Multiply the magnitude by cosine of the angle for the real part. Multiply the magnitude by sine of the angle for the imaginary part. Then write a + bi.
4. Should I use degrees or radians?
Use the unit that matches your problem. Many classroom examples use degrees. Many advanced math, physics, and engineering formulas use radians. The calculator supports both units.
5. Why does a tiny number appear instead of zero?
Computers use decimal approximations for trigonometric values. A result that should be zero may appear as a tiny value. The zero tolerance option cleans those small rounding artifacts.
6. What does the quadrant result mean?
The quadrant tells where the rectangular point lies on the complex plane. It helps confirm the signs of the real and imaginary parts after conversion.
7. Can I use j instead of i?
Yes. Electrical engineering often uses j for the imaginary unit. Mathematics usually uses i. The calculator lets you choose either symbol for the displayed answer.
8. What does modulus check mean?
The modulus check recalculates the distance from the rectangular parts. It should closely match the entered magnitude. Small differences may occur because of rounding precision.