Square root of a complex number

Square Root of a Complex Number

Finding the square root of a complex number extends the familiar concept of square roots from real numbers into the complex plane. Given a complex number
\( z = a + bi \),
where \( a \) and \( b \) are real numbers, and \( i = \sqrt{-1} \), the goal is to find a complex number
\( w = x + yi \)
such that

\( w^2 = z \)

Understanding the Problem

To find \( w \), we need to determine real numbers \( x \) and \( y \) such that
\( (x + yi)^2 = a + bi \) Applying algebraic expansion and using the identity \( i^2 = -1 \) on the left-hand side:
\( (x + yi)^2 = x^2 + 2xyi + (yi)^2 = x^2 - y^2 + 2xyi \) Here, we use \( i^2 = -1 \).

Formulating Equations from Real and Imaginary Components

Two complex numbers are equal only when both their real components and imaginary components are identical. Therefore,

• Real parts: \( x^2 - y^2 = a \)
• Imaginary parts: \( 2xy = b \)

Solving for \( x \) and \( y \)

From the imaginary part,
\( y = \frac{b}{2x} \) (assuming x ≠ 0). Substituting into the real part:
\( x^2 - \left(\frac{b}{2x}\right)^2 = a \) Multiply both sides by \( x^2 \) to clear the denominator:
\( x^4 - \frac{b^2}{4} = a x^2 \) Rearranged:
\( x^4 - a x^2 - \frac{b^2}{4} = 0 \) Let \( t = x^2 \), converting this into a quadratic equation:
\( t^2 - a t - \frac{b^2}{4} = 0 \) Using the quadratic formula:
\( t = \frac{a \pm \sqrt{a^2 + b^2}}{2} \) Since \( t = x^2 \), take the positive root for \( t \) (because \( x^2 \geq 0 \)). Finally,
\( x = \pm \sqrt{t} \) and
\( y = \frac{b}{2x} \)

Properties of Square Roots of Complex Numbers

• Every non-zero complex number has exactly two square roots, which are negatives of each other.
• When \( b = 0 \) and \( a \geq 0 \), the roots reduce to the real square roots of \( a \).
• The square root function is multi-valued in the complex plane, but we typically take the principal root (with positive real part or positive imaginary part) when needed.

Example: Find the Square Roots of \( 3 + 4i \)

Given \( z = 3 + 4i \), with \( a = 3 \) and \( b = 4 \):

1. Compute \( t \):
   \( t = \frac{3 \pm \sqrt{3^2 + 4^2}}{2} = \frac{3 \pm \sqrt{9 + 16}}{2} = \frac{3 \pm 5}{2} \) The two solutions are:
   \( t_1 = 4, \quad t_2 = -1 \) Discard the negative root \( t_2 = -1 \) because \( x^2 \geq 0 \).
2. Calculate \( x \):
   \( x = \pm \sqrt{4} = \pm 2 \)
3. Calculate \( y \):
   Using \( y = \frac{b}{2x} = \frac{4}{2 \times 2} = 1 \) for \( x = 2 \).
4. Consequently, one of the square roots can be expressed as:
   \( w = 2 + i \) and the other is its negative:
   \( -w = -2 - i \)

The square root of a complex number \( a + bi \) can be found by solving the system:

\[ x^2 - y^2 = a \\ 2xy = b \]

This results in a quadratic equation in terms of \( x^2 \), enabling us to solve for both \( x \) and \( y \). The two roots obtained are negatives of each other, completing the process of finding complex square roots.