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