Regular readers know that I am a big fan of Better Explained in which Kalid makes mathematical ideas accessible.

Today’s installment:

Integration is just multiplication when one of the operands is changing.

Most people grok integration as area under a curve but, as Kalid explains, area is just one convenient way of visualizing multiplication…but we don’t need to visualize multiplication as multiplication is already pretty straightforward – in the simplest case, it’s just repeated addition.

Many ideas in maths start out simple like that and then gradually generalize to a more complex idea. In Kalid’s words:

Our understanding of multiplication changed over time:

• With integers (3 x 4), multiplication is repeated addition
• With real numbers (3.12 x sqrt(2)), multiplication is scaling
• With negative numbers (-2.3 x 4.3), multiplication is flipping and scaling
• With complex numbers (3 x 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.

In other words,

Integration is just a better multiplication

or, conversely,

Multiplication is a special case of integration when the values are static.