I’ve worked out a formula for e!
This came up when I was looking for an antiderivative, if n isn’t equal to 1:
if n is equal to 1, then it’s suddenly a natural log!
But I’ve realized that if I change it only a tiny bit, it becomes a really famous existing formula for e:
Still impressive that you have worked it out all on your own, Simon!
Simon tried to predict the weight of some of his constructor pieces by calculating their approximate area.
He later also successfully predicted the weight of a binder clip through calculating the ratio of a binder clip to a ruler. Simon did this by observing the center of gravity of the ruler (or center of mass, as “they are very near for objects on Earth because the Earth is very homogenous) as opposed to the center of gravity of the ruler plus binder clip.
“Are you impressed?” – Simon asks, laughingly, and I can see it must be a pun. We are in bed, reading up on Newton’s laws of motion that talk of forces being “impressed” upon bodies.
Simon continues: “Newton’s mechanics says that the speed limit is infinite, which says that matter doesn’t exist, which says that Physics doesn’t exist, which says that Newton’s mechanics doesn’t exist. Newton’s mechanics contradicts itself!”
The book we are reading (17 Equations that Changed the World by Ian Stewart) goes on to describe how in Newton’s laws, calculus peeps out from behind the curtains and how the second law of motion specifies the relation between a body’s position, and the forces that act on it, in the form of a differential equation: second derivative of position = force/mass. To find the position, the book says, we have to solve this equation, defusing the position from its second derivative. “Do you get it?” – I ask, “Because I don’t think I do”. — “I’ll need a piece of paper for this”, – Simon quickly comes back dragging his oversized sketchbook. Then he quickly writes down the differential equation (where the x is the position) to explain to me what the second derivative is. And then he solves it:
The idea comes from a video by Mathologer. Simon sketches a geometric definition of the Euler’s number (e) using integrals. He messed up a little with the integral notation, but corrected it later (after we stopped filming). Please see the photos below: