I learned the following trick for memorizing the value of e from my colleague, Gene White. It never fails to impress calculus students (given a wide enough definition of “impress”).
Start by carefully looking at this picture:
That’s a 20 dollar bill, so memorize “2” and put down the decimal point.
The picture on the bill is of Andrew Jackson. He was our seventh President, so put a “7” after the decimal point to get 2.7.
Jackson was elected in 1828, so put down “1828” next. Since there’s a 2 in front of the decimal place, put “1828” a second time. We’re now up to 2.718281828.
Now look at the red square over Jackson’s face. The diagonal creates two congruent right triangles with angle measures 45, 90, and 45. So, add on 459045 to get 2.718281828459045. And that’s e to 15 places.
I’m open to suggestions on how to memorize more of the digits.