Why do we have imaginary numbers anyway?

The answer is simple. The imaginary unit i allows us to find solutions to many equations that do not have real number solutions.
This may seem weird, but it is actually very common for equations to be unsolvable in one number system but solvable in another, more general number system.
Here are some examples with which you might be more familiar.
  • With only the counting numbers, we can’t solve ; we need the integers for this!
  • With only the integers, we can’t solve ; we need the rational numbers for this!
  • With only the rational numbers, we can’t solve . Enter the irrational numbers and the real number system!
And so, with only the real numbers, we can’t solve . We need the imaginary numbers for this!
As you continue to study mathematics, you will begin to see the importance of these numbers.
Source: https://www.khanacademy.org/