Cube root
Complex numbers

For complex numbers, the principal cube root is usually defined as the cube root that has the greatest real part, or, equivalently, the cube root whose argument has the least absolute value. It is related to the principal value of the natural logarithm by the formula

If we write x as

where r is a non-negative real number and θ lies in the range

then the principal complex cube root is

This means that in polar coordinates, we are taking the cube root of the radius and dividing the polar angle by three in order to define a cube root. With this definition, the principal cube root of a negative number is a complex number, and for instance √ −8 will not be −2, but rather 1 + i √ 3.

This difficulty can also be solved by considering the cube root as a multivalued function: if we write the original complex number x in three equivalent forms, namely

The principal complex cube roots of these three forms are then respectively

Unless x = 0, these three complex numbers are distinct, even though the three representations of x were equivalent. For example, √ −8 may then be calculated to be −2, 1 + i √ 3, or 1 − i √ 3.

This is related with the concept of monodromy: if one follows by continuity the function cube root along a closed path around zero, after a turn the value of the cube root is multiplied (or divided) by e 2 i π / 3. {\displaystyle e^{2i\pi /3}.}
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