The expression $\sqrt{-1}$ is not defined within the set of real numbers because the square root of a negative number is not a real number. However, in the field of complex numbers, $\sqrt{-1}$ is denoted as the imaginary unit $i$, which is defined by the property:

$i^2 = -1$

Thus:

$\sqrt{-1} = i$

This concept is fundamental in complex number theory and is used to extend the real number system to the complex plane.

Would you like a detailed explanation of how complex numbers work or an example application? Here are some follow-up questions to consider:

1. What are complex numbers, and how are they represented?
2. How is the imaginary unit $i$ applied in solving quadratic equations?
3. Can you explain the polar form of complex numbers?
4. How are complex conjugates used in division of complex numbers?
5. What are some real-world applications of complex numbers?

Tip: Always remember that the square of $i$, $i^2$, simplifies to $-1$, which is a key property in complex number arithmetic.