On this page What it means Formula Visual explanation Worked example When to be careful Understand the subject
What is Complex Root? Complex numbers extend the real number line with an imaginary direction, allowing calculations that do not remain on the real axis.
Use it to keep both real and imaginary components visible instead of forcing an invalid real-only result.
See the structure
What the calculation is doing imaginary • (2, 3i) real → zₖ = |z|^(1/n)[cos((arg z+2πk)/n)+i sin((arg z+2πk)/n)].
Worked interpretation
Read the result in context Enter the example values shown in the fields to verify the stated relationship.
Interpret with care
Important boundary The real and imaginary parts are separate components; complex roots and arguments can have several valid values.
Use the Mathematics collection to move between connected concepts without duplicating the calculation.
Quick guide
How to use this calculator Enter values using the notation shown beside each field. Choose a calculation mode when the tool offers more than one interpretation. Read the primary answer first, then inspect the supporting details. Calculation method
Apply the stated definition zₖ = |z|^(1/n)[cos((arg z+2πk)/n)+i sin((arg z+2πk)/n)].
The calculator validates the mathematical domain before returning a result and does not replace undefined states with zero.
Worked example
Worked example Enter the example values shown in the fields to verify the stated relationship.
zₖ = |z|^(1/n)[cos((arg z+2πk)/n)+i sin((arg z+2πk)/n)].
Supported inputs
Precision and limits Finite, bounded input Numeric inputs use finite JavaScript floating-point values with magnitude limits. Results are normally displayed to 12 significant digits; outputs labelled approximate are not exact identities.
Notation Enter ordinary finite decimals. Polynomial fields use coefficients in descending powers, separated by commas; for example 1, −3, 2 means x² − 3x + 2.
