## Square root of a complex number

Let be a complex number such that where x and y are real numbers. Then,

or

or

On equating real and imaginary parts, we get

Now,

or

or since

From the above, we get

and

which in turn implies

and

If b is positive, then by the relation , x and y are of the same sign. Hence,

If b is negative, then by the relation , x and y are of different signs. Hence,

.

*Note: When you have to actually, find the square root of a particular complex number or even a complex expression, carry out the above steps and don’t just mug up the formula and try to substitute! There are a thousands of such derivations in math, with fancy formulae, so it is better to gain a deep understanding of the proofs rather than mug up techniques or tips or tricks !!*

**Try this homework now:**

- Find the square root of
- Find all possible values of
- Solve for z:

Have fun the complex way 🙂

Nalin Pithwa

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