Our understanding of multiplication changed over time:
- With integers (3 × 4), multiplication is repeated addition
- With real numbers (3.12 x sqrt(2)), multiplication is scaling
- With negative numbers (-2.3 * 4.3), multiplication is flipping and scaling
- With complex numbers (3 * 3i), multiplication is rotating and scaling
We’re evolving towards a general notion of “applying” one number to another, and the properties we apply (repeated counting, scaling, flipping or rotating) can vary. Integration is another step along this path.
credit above goes to Better Explained
I need some more examples on this topic and to flush out the ideas above (scaling, flipping, rotating)