Complex Number Square Root Calculator

Complex Number Square Roots

What are Complex Number Square Roots?

The square root of a complex number is another complex number which, when multiplied by itself, gives the original complex number. Every non-zero complex number has two square roots.

Formula

For a complex number \(z = a + bi\) in rectangular form, its square roots are given by:

\[ \sqrt{a + bi} = \pm \left(\sqrt{\frac{r + a}{2}} + i \cdot \text{sign}(b) \cdot \sqrt{\frac{r - a}{2}}\right) \]

Where:

• \(r = \sqrt{a^2 + b^2}\) is the magnitude of the complex number
• \(\text{sign}(b)\) is the sign of \(b\) (+1 if \(b \geq 0\), -1 if \(b < 0\))

Calculation Steps

Let's calculate \(\sqrt{3 + 4i}\):

1. Calculate \(r\):
   \(r = \sqrt{3^2 + 4^2} = \sqrt{25} = 5\)
2. Calculate the real part:
   \(\sqrt{\frac{r + a}{2}} = \sqrt{\frac{5 + 3}{2}} = \sqrt{4} = 2\)
3. Calculate the imaginary part:
   \(\sqrt{\frac{r - a}{2}} = \sqrt{\frac{5 - 3}{2}} = \sqrt{1} = 1\)
4. Combine the results:
   \(\sqrt{3 + 4i} = \pm(2 + i)\)

Example and Visual Representation

Let's visualize \(\sqrt{3 + 4i}\):

This visual representation shows:

• The original complex number 3+4i in blue
• The two square roots: 2+i in red and -2-i in green