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Thought is only a flash between two long nights, but this flash is everything.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
If you read John Conway and Richard Guy’s entertaining romp, The Book of Numbers , you will notice something odd on page 25. That is unless you are British, and then you probably will not notice anything at all. The curiosity is that they refer to e, the transcendental number 2.71828…, as “Napier’s constant.” Some might ask, Didn’t Euler discover e? In fact isn’t the “e” for Euler? Well, in truth, the answers to these questions are “not exactly” and “no.” That, like any other good mathematics problem, leads us to two new questions: “How much credit should Euler get for the discovery of e?” and “What role did Napier play in the discovery of Euler’s number?”
After π, e is probably the most well known mathematical constant. Everyone uses e to denote 2.718...; in fact, it is often referred to as Euler’s number. However, some people, notably the Brits, refer to it as Napier’s constant. How can something so universally known have two different names? Leonhard Euler is traditionally credited with naming this constant, and thus often assumed to be its discoverer. How and why mathematical objects get named after people, while others do not, is often a quirk of history. In our case, the story of e does not start with Euler; it actually ends with him. The trail back to the first appearance of what we call Euler’s number is filled with many interesting people and sidetracks along the way. Please join me on a light-hearted trip back through time to when e was young.
Table Of Contents
Shell-Gellasch, Amy, "Napier's e," Loci (December 2008), DOI: 10.4169/loci003209
rational approximation of an irrational number
students find it really curious, and more intriguing when i tell them that (1+(1/n))^n in the limit of n tending to infinity is an irrational number 'e'. Here n is a positive integer. They find it really difficult to believe that a rational expression on the left hand side tends to an irrational number. How do I convince them ?
Geometrical introduction of Napier's e
Usually Ludolph's number is introduced geometrically as the "perimeter to diameter number" of any curve of constant distance to a center-point. So why shouldn't we introduce Napier's number in a more or less similar way as the "tangent-point-height to base-point-height number" of any curve of constant relative growth ? Here we refer to the following property: Chose an arbitrary point T on an arbitrary growth curve (exponential curve) g with base line (asymptote) b. Let B be the point of intersection of b and the tangent line at T. Let B' be the point of intersection of g and the perpendicular to b containing B; and let T' be the point of intersection of b and the perpendicular to b containing T. It can be proved that line segment T'T measured with BB' gives Napier's number. I would like to learn about publications in which this geometrical approach is mentioned.