A cute proof that makes e natural

For the full article covering many properties of e , including history and comparison with existing methods of teaching: PDF from arXiv. A video explanation will be posted here shortly. This webpage pulls out the part of the article which uses Pre-Calculus language to explain what is so natural about e , while intuitively connecting the following two important properties: The slope of the tangent line to y = e x at the point ( x , e x ) is just e x . (In Calculus language: e x is its own derivative.) at the point is just . (In Calculus language: is its own derivative.) The expression ( 1 + 1 n ) n approaches e as n grows. Key conceptual starting point Geometrically, there really is only one exponential function curve shape, because all exponential function curves y = a x (with positive real bases a ) are just horizontal stretches of each other. This is exactly like how all ellipses are just stretches of each other (and for the same reason). For example, y = 8 x , stretched horizo ... Read full article.