Lesson How to take a root of a complex number

How to take a root of a complex number

Let n be a positive integer >= 2. The n-th root of a complex number w=a+bi is the complex number z=c+di such as .
Operation of extracting the root of the complex number is the inverse of raising a complex number to a power.
In the lesson Raising a complex number to an integer power in this module we derived De Moivre's formula



for the complex number presented in the trigonometric form, where r is its modulus and is its argument.
Based on this formula, one can expect that the n-th root of the complex number is equal to .
This is true that the complex number , raised to degree n, is equal to .
But this number is not unique: there are other complex numbers that have the same property.

Namely, numbers , k=1, 2, ...,n-1 are n-1 other distinct complex numbers that raised to degree n produce the same number .

This follows directly from the De Moivre's formula.

Thus, the n-th root of the complex number has n distinct values , k = 0, 1, ...,n-1.
They all have the same modulus (positive real value) and n distinct argument values , k = 0, 1, ...,n-1.
All n-roots are located on the circle of radius in the complex plane.
Also, since the argument of each successive n-th root is apart the previous argument value, you see that n-th roots are equally spaced in this circle.

Examples

1) Square root of the complex number 1 (actually, this is the real number) has two values: 1 and -1.       Surely, you know it well from your experience with real numbers (even with integer numbers). In the frame of explanations given above, the number 1 has the modulus and the argument ,, ,.... Taking the square root of 1, you have the modulus (positive value) and two argument values and . This gives two values of the square root of the complex number 1: 1 and -1, exactly as for real numbers. Figure 1 at the right illustrates how -1 arises as a result of manipulating the argument.

Figure 1. Square roots     of the complex number 1

2) Square root of the complex number -1 (of the negative unit) has two values: i and -i. It was explained in the lesson Complex numbers and arithmetical operations in this module, and                   it is how the complex number i was introduced. In the complex plane, the number -1 has the modulus and the argument ,, .... Taking the square root of -1, you have the modulus (positive value) and two argument values and . This gives two values of the square root of the complex number -1: i and -i, exactly as expected. Figure 2 at the right illustrates how i and -i arise as a result of manipulating the argument.

Figure 2. Square roots     of the complex number -1

3) Cube roots of a complex number 1. In the plane of complex numbers, 1 has the modulus and the argument values ,, ,.... Taking the cube root of the complex number 1, you have the modulus (positive value) and                  three argument values , and . This gives three values of the cube root of the complex number 1: 1, and . Figure 3 shows complex (real) number 1 in red and two other roots in blue. Note that these tree points are vertices of the equilateral triangle.

Figure 3. Cube roots     of the complex number 1

4) 4-th roots of a complex number 1. In the plane of complex numbers, 1 has the modulus and the argument values ,, , ,.... Taking the 4-th root of the complex number 1, you have the modulus (positive value) and                  four argument values , , and . This gives four values of the 4-th root of the complex number 1: 1, , and . Figure 4 shows complex (real) number 1 in red and three other roots in blue. These four points are vertices of the square.

Figure 4. 4-th roots     of the complex number 1

5) 5-th roots of a complex number 1. In the complex plane 1 has the modulus and the argument values ,, , , ,.... Taking the 5-th root of the complex number 1, you have the modulus (positive value) and                  five argument values , , , and . This gives five values of the 5-th root of the complex number 1: 1, , , , and . Figure 5 shows complex (real) number 1 in red and four other roots in blue. These five points are vertices of the regular pentagon.

Figure 5. 5-th roots     of the complex number 1

6) 6-th roots of a complex number 1. In the complex plane 1 has the modulus and the argument values ,, , , , ,.... Taking the 6-th root of the complex number 1, you have the modulus (positive value) and                  six argument values , , , , and . This gives six values of the 6-th root of the complex number 1: 1, , , , and . Figure 6 shows complex (real) number 1 in red and five other roots in blue. These six points are vertices of the regular hexagon.

Figure 6. 6-th roots     of the complex number 1

Summary
Let w be a complex number and n be a positive integer n>=2. Then w has n distinct n-th roots

, k = 0, 1, ...,n-1.

They all have the same modulus (positive real value) and n distinct argument values , k = 0, 1, ...,n-1.



For your convenience, below is the list of my relevant lessons on complex numbers in this site in the logical order. They all are under the current topic Complex numbers in the section Algebra II.
Complex numbers and arithmetical operations over them
Complex plane
Addition and subtraction of complex numbers in complex plane
Multiplication and division of complex numbers in complex plane
Raising a complex number to an integer power
How to take a root of a complex number                                                                (this lesson)
Solution of the quadratic equation with real coefficients on complex domain
How to take a square root of a complex number
Solution of the quadratic equation with complex coefficients on complex domain