Calculator Inputs
Formula Used
z = a + bi
r = √(a² + b²)
θ = atan2(b, a)
z = r e^(iθ)
The calculator converts the rectangular complex number into exponential form by finding the modulus and argument. It uses the atan2 function because it preserves quadrant information. This is safer than using a simple arctangent ratio.
How to Use This Calculator
- Enter the real part of the complex number in the first field.
- Enter the imaginary coefficient in the second field.
- Select decimal precision, angle unit, angle range, and notation.
- Press Calculate to see the result above the form.
- Use copy, print, or CSV download for saving the answer.
Example Data Table
| Rectangular Number | Modulus | Argument | Exponential Form |
|---|---|---|---|
| 3 + 4i | 5 | 53.130102° | 5e^(i53.130102°) |
| -5 + 12i | 13 | 112.619865° | 13e^(i112.619865°) |
| -8 - 6i | 10 | -143.130102° | 10e^(i-143.130102°) |
| 7 - 2i | 7.280110 | -15.945396° | 7.280110e^(i-15.945396°) |
Rectangular to Exponential Conversion Explained
A complex number can be written in several useful ways. Rectangular form uses a real part and an imaginary part. It looks like a plus bi. This form is easy for addition, subtraction, and coordinate plotting. Exponential form shows the same value as r e raised to i theta. It highlights size and rotation. That makes it useful in electronics, waves, and control systems.
The value r is called the modulus. It measures the distance from the origin to the point a comma b. The angle theta is called the argument. It measures the rotation from the positive real axis. The calculator finds both values from the entered rectangular parts. It uses atan2, so the correct quadrant is handled automatically.
Exponential form is closely linked with Euler's formula. Euler's formula says e raised to i theta equals cosine theta plus i sine theta. Because of that identity, a complex number can move between rectangular, polar, trigonometric, and exponential forms. The calculator shows a cis form when selected. That form is r times cosine theta plus i sine theta.
Precision matters when answers are used in engineering or homework. Small rounding changes can affect phase values. This tool lets you choose decimal places. It also lets you show the angle in degrees or radians. A normalized range option helps when you need angles from zero to three hundred sixty degrees or from negative one hundred eighty to one hundred eighty degrees.
The zero complex number needs special care. When a and b are both zero, the modulus is zero. Its angle is not unique. Any angle would place the point at the origin. The calculator reports this clearly. It still provides a practical default value for display.
This conversion is common in alternating current analysis. Voltage and current phasors often start in rectangular form. Exponential form makes multiplication, division, powers, and roots faster. Multiplying complex numbers becomes multiplying moduli and adding angles. Dividing them becomes dividing moduli and subtracting angles.
Students can use this tool to check manual steps. Professionals can use it for phase review. The result section shows the modulus, argument, quadrant, and final exponential expression. The step table explains how each number was formed. This makes the answer easier to verify.
Common mistakes come from signs and quadrants. A positive real part and a negative imaginary part sit in quadrant four. A negative real part and a positive imaginary part sit in quadrant two. Simple tangent alone can miss these cases. The atan2 method avoids that issue and returns a dependable angle.
Always enter the sign of the imaginary part carefully. For example, 3 - 4i means a equals 3 and b equals -4. A negative imaginary value changes the quadrant and the angle. Use enough precision for your purpose. Then copy the final expression into your notes, circuit worksheet, or complex number problem.
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 imaginary coefficient. It is useful for plotting points and adding complex numbers.
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. It shows the number as a distance and rotation.
How is the modulus calculated?
The modulus is calculated with r = √(a² + b²). It is the distance from the origin to the point represented by the complex number on the complex plane.
How is the argument calculated?
The argument is calculated with θ = atan2(b, a). This function checks the signs of both parts. It returns an angle in the correct quadrant.
Why is atan2 better than arctan?
A simple arctan ratio can lose quadrant information. The atan2 function uses both real and imaginary signs. That makes the angle more reliable for all quadrants.
Can I use negative imaginary values?
Yes. Enter the imaginary coefficient with its sign. For 5 - 2i, enter 5 for the real part and -2 for the imaginary part.
What happens when both parts are zero?
The modulus becomes zero. The angle is not unique because every angle reaches the origin. The calculator displays a clear note for this special case.
Should I use degrees or radians?
Use degrees when the answer must be easy to read. Use radians for calculus, advanced algebra, signal work, and many engineering formulas.
What is cis notation?
Cis notation means cos θ + i sin θ. It is another way to show the same complex number. It directly follows Euler's formula.
Why does angle range matter?
The same complex number can use equivalent angles. For example, -45° and 315° point in the same direction. Choose the range your course or project needs.
Can I save the calculator result?
Yes. After calculation, you can copy the final expression, print the page, save it as a document, or download the result as a CSV file.