Enter Complex Number Values
Use the full expression box, or leave it blank and enter real and imaginary parts separately.
Formula Used
Rectangular form: z = a + bi
Modulus: r = √(a² + b²)
Argument: θ = atan2(b, a)
CIS form: z = r cis(θ) = r[cos(θ) + i sin(θ)]
The atan2 function is used because it reads signs of both a and b. That helps place the angle in the correct quadrant.
How to Use This Calculator
- Enter a complete complex number, or enter the real and imaginary parts.
- Choose degrees or radians for the argument.
- Select positive coterminal angle or signed principal angle.
- Set the decimal precision for rounded answers.
- Press the convert button to view CIS form and steps.
- Use copy, print, or CSV export for saving results.
Example Data Table
| Complex Number | Modulus | Argument | CIS Form |
|---|---|---|---|
| 3 + 4i | 5 | 53.1301° | 5 cis(53.1301°) |
| -3 + 4i | 5 | 126.8699° | 5 cis(126.8699°) |
| -2 - 2i | 2.8284 | 225° | 2.8284 cis(225°) |
| 0 - 6i | 6 | 270° | 6 cis(270°) |
Understanding CIS Form
Complex numbers often start in rectangular form. A value like 3 + 4i shows horizontal and vertical parts. CIS form shows the same value by distance and direction. It uses r cis theta. The letters cis mean cosine plus i sine. This notation links algebra, geometry, and trigonometry in one compact statement.
Why The Conversion Matters
CIS form is useful when angles matter. It makes multiplication easier. Moduli multiply. Arguments add. Division also becomes cleaner. Moduli divide. Arguments subtract. This is why polar work, roots, powers, impedance, and phasor questions often use cis notation. The rectangular form still matters, because it gives exact coordinates. Together, both forms describe one point on the complex plane.
What The Calculator Finds
The calculator finds the modulus first. It squares the real part and imaginary part. Then it adds both squares. The square root gives the distance from the origin. Next, it uses atan2 to find the correct argument. This function checks signs of both parts. That avoids wrong quadrant errors. The answer can be shown in degrees or radians. You can also choose a signed angle or a positive coterminal angle.
Interpreting The Result
The modulus is never negative. It tells how far the point is from zero. The argument tells the rotation from the positive real axis. A complex number in quadrant two has a positive imaginary part and negative real part. A number on an axis has a boundary angle. The zero complex number is special. Its modulus is zero, and its direction is undefined. The calculator explains that case instead of forcing a false angle.
Advanced Checks
Conjugates and reciprocals help verify the output. The conjugate reflects the point across the real axis. Its argument changes sign. The reciprocal has modulus one divided by r. Its argument also changes sign. These checks are useful for homework, electronics, waves, and signal problems. Step notes show the route from a + bi to r cis theta. They also make rounding choices clear. Use more decimals for technical work. Use fewer decimals for quick study.
Good Input Habits
Clean input gives cleaner results. Enter real and imaginary parts with their signs. If you paste a full expression, keep the imaginary symbol at the end. Examples include 5 - 2i, -7 + 3i, and 4i. Select the angle style before calculation. Positive angles are helpful for diagrams. Signed angles are helpful for principal arguments and many textbook answers. Always check whether your course expects degrees or radians.
Using CIS For Powers
Once a number is in cis form, powers become faster. De Moivre's theorem says that the modulus is raised to the power. The angle is multiplied by that power. It also reduces expansion mistakes. Roots work in a way. The modulus root is taken, and several angles are produced. That is why cis form is a bridge between complex arithmetic and standard applications.
FAQs
What is cis form?
CIS form writes a complex number as r cis(θ). It means r[cos(θ) + i sin(θ)]. The value r is the modulus. The value θ is the argument.
How do I convert a + bi to cis form?
Find r with √(a² + b²). Then find θ with atan2(b, a). Write the final answer as r cis(θ), using the required angle unit.
Why is atan2 used?
atan2 checks both the real and imaginary signs. This places the angle in the correct quadrant. A simple arctangent can miss that quadrant information.
Can the calculator use radians?
Yes. Select radians in the angle unit field. The output will show the argument in radians, plus a π multiple for easier checking.
What happens when the complex number is zero?
The modulus is zero. The angle is undefined because the point has no direction from the origin. The calculator explains this special case.
What is a positive coterminal angle?
It is an angle between 0° and 360°, or between 0 and 2π radians. It points in the same direction as the principal angle.
Is cis form the same as polar form?
They are closely related. Polar form often means r∠θ. CIS form expands that angle into cosine and sine notation.
How is the modulus interpreted?
The modulus is the distance from the origin to the complex point. It is always zero or positive. It becomes the scale factor in cis form.
Why does the conjugate matter?
The conjugate helps check symmetry. It keeps the real part the same and changes the imaginary sign. In cis form, the argument changes sign.
Can I paste a full complex number?
Yes. You may enter formats like 3+4i, -2-5i, 7, i, -i, or 4i. Leave the separate fields unchanged if using that box.
How many decimals should I use?
Use two to four decimals for quick study. Use more decimals for engineering, electronics, and numerical work where small rounding differences matter.