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Loci: Convergence
Napier's e
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- rational approximation of an irrational number
- Geometrical introduction of Napier's e
- Second geometrical introduction of Napier's e
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thread #1:
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 ?
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thread #2:
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.
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thread #3:
Second geometrical introduction of Napier's e
An exponential curve in polar coordinates is a logarithmic spiral. For all points on such a spiral curve, the angle between the direction of the curve and the direction to the center is the same. Choosing 45 degrees for this constant angle, the logarithmic curve has the following property: moving (outward) along the curve corresponding to an increase of the polar angel by ONE RADIAN, the factor by which the distance to the center increases is NAPIER’S CONSTANT.
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