The real numbers are not closed under taking square roots. You can't find real numbers that are square roots of negative numbers.

Back in AD 50, a Greek mathematician, Heron of Alexandria, decided this was a bad thing, but didn't get very far with solving the problem. It wasn't until the 16th century that mathematicians gave this problem any serious attention. In the late 16th century, Rafael Bombelli came up with a notation for \(\sqrt{-1}\), the imaginary number that we call \(i\).

So, \(i\) is the square root of \(-1\).

The number system formed by adding \(i\) to the real numbers is called the complex number system. It includes the square roots of all the negative numbers. The complex numbers were originally called the "imaginary" numbers and viewed with great suspicion.