A big part of math is about transformations and structure. For a wide audience in a short time, you can give an inkling about this in a short time along the following lines:
"Numbers'' are really abstract things, not just "quantities''. They can correspond to transformations --- for example, dilations and translations of the line. Show how this corresponds to multiplication and addition. Negative numbers are flips. This explains "negative times negative is positive," and shows $x^2=1$ has two solutions.
Solving $x^2=-1$ corresponds to answering the question "What can you do twice to get a flip?" Likely as not someone will think of rotating 90 degrees. Dilations, translations and rotations of the plane are complex numbers.
"What about rotations in 3D?" Demonstrate they don't always commute. A lot of math is about understanding the rules and concepts that govern much more general transformations of complicated kinds of data.