Complex numbers, typically denoted as z, consist of two parts: a real part and an imaginary part. A complex number is expressed in the form $z = a + bi$, where 'a' represents the real component, and 'bi' is the imaginary component. The concept of square roots in complex numbers is intriguing as it involves finding a number which, when squared, returns the original complex number. This is a fundamental skill in various mathematical and engineering fields.

## Cartesian Form of Complex Numbers

**Form:**$z = a + bi$ ($a$ = real, $bi$ = imaginary).**Use:**Easy for arithmetic and finding square roots.**Technique:**Uses algebra and knowledge of imaginary numbers.

## The Square Root of a Complex Number

**Task:**Find $\sqrt{z} = \sqrt{a + bi}$.**Solution:**Find by separating and solving real and imaginary parts after squaring.

## Example Problems

### Problem 1 : Find square roots of $5+12i$.

1. Assume square root is $x + yi$.

2. Square it: $(x + yi)^2 = x^2 + 2xyi - y^2$.

3. Set real parts equal: $x^2 - y^2 = 5$, and imaginary parts equal: $2xy = 12$.

4. Solve Equations

- From $2xy = 12$, get $y = \frac{12}{2x} = \frac{6}{x} $.</li><li>Substitute in$x^2 - \left(\frac{6}{x}\right)^2 = 5 $.</li><li>Solve for$x$, then find$y$.</li></ul><p><strong>Solution:</strong></p>$x = 3, y = 2 → 3 + 2i.$<p></p>$x = -3, y = -2 → -3 - 2i.$<h3>Problem 2: Square Root of$7+24i$</h3><p>1. Assume square root is$x + yi$.</p><p>2. Square it:$(x + yi)^2 = x^2 + 2xyi - y^2$.</p><p>3. Equate real parts:$x^2 - y^2 = 7$, imaginary parts:$2xy = 24$.</p><p>4. Solve Equations</p><ul><li>From$2xy = 24$, get$y = \frac{12}{x}$.</li><li>Substitute in$ x^2 - y^2 = 7: x^2 - (\frac{12}{x})^2 = 7$.</li><li>Solve for$x$, then$y$.</li></ul><p><strong>Solution:</strong></p>$x = 4, y = 3 → 4 + 3i.$<p></p>$x = -4, y = -3 → -4 - 3i.$