Imaginary numbers arise when a number is squared to produce a negative result. They are represented using the symbol (i), such that √-9 = 3i. Imaginary numbers can be used to solve equations involving negative results by including (i) in the calculation. Examples of imaginary numbers include √-9 = 3i, √-25 = 5i, (3+2i)(3-2i) = 13, and (3+2i) + (5-6i) = -5-12i.