My experience of mental math at the most common and basic is teasing apart the numbers into components and adding/multiplying/dividing their sums. Sometimes tha

My experience of mental math at the most common and basic is teasing apart the numbers into components and adding/multiplying/dividing their sums. Sometimes that's like the math confusion meme where it's "floating in front of my eyes" (mentally, internally) in the "void", sometimes I'm literally imagining in my minds eye doing the math at a desk on notebook paper with all the feedback that comes with it. Sometimes I use a mental TI-84/89 to do the math, or this one calculator I had when I was a kid. It's usually paper though. Many times I apply a synesthetic association to a number, be it textural, color, attitude, or "shape". However, I also have the ability to mathematically evaluate an equation and graph it on a mental visualization of a graph. I've been able to do that in 2d and 3d for a range of quadratic, cubic, and quartic graphs since high school. Also asymptotes, hyperbolas, sinusoidal functions. You know those youtube videos where they do the CGI animations to show a rolling cylinder tracing out a sine wave? I've been able to do that in my mind since I was a teen. In high school I would do calculus rate of change functions in my head with a visualization like a laser going from perpendicular to a wall to parallel with the wall, and how the speed of the point at the end of the beam on the wall approaches light speed as the beam approaches approaches parallel, or the shrinking surface area of a liquid in a vertical conical funnel. The funnel i imagined is yellow with an equilateral cone and a long neck that tapers to a half inch and has a worn, wrinkled blue white and red sticker from being run through the dishwasher a few times. It's injection molded and not shaved/sanded. Excessive detail for a feature of a math problem, but it's there anyway. For me a spinning cow can have horns or be a holstein heifer or an angus steer or a dairy calf, or a buffalo. Could be a chik fil a cow. Or the one from Barnyard...right, Barnyard? the Dreamworks movie... I can go back and read pages of books and documents that I have already read in my mind, which means (with effort) I can retread any sheet music I've ever played, or redo the math that I did in calculus and chemistry and physics classes. For a long time it completely ruined re-reading books for me because I would know exactly what was coming and I didn't need to turn the page. I used the sheet music thing to teach myself guitar based on what I learned on trumpet through recall of arpeggios and and scales. Notes are also synesthetic and I can make them dance off the page in my head.