what is Euler's Constant, and how is it derived?

Answer 1

Euler's constant (not to be confused with e) was first defined in 1735 and is now usually denoted by the letter gamma. It is approximately equal to 0.57721566490153286...

It's equal to (lim n –> ∞) ∑ [(1/k) - ln(n)], taken from k = 1 to n.

Strangely, no one has ever been able to prove whether Euler's constant is a rational or irrational number.

Answer 2

Let me give you a bit more information on how you can "visualize" Euler's number.

Take the curve y = 1/x. (Sorry, I can't draw a graph here.) Draw a vertical line from the graph to the x-axis where x = 1. Now move to the right and draw another vertical line at the point where the area under the curve is equal to 1. That is, the area you are calculating will be bounded on the bottom by the x-axis, on the top by the curve y = 1/x, on the left by the first vertical line you drew (at x = 1), and on the right by the second vertical line that you draw.

As it turns out, this value shows up a BUNCH in the sciences, so it has proven to be a very useful value for calculating a wide variety of different things. For that reason, it serves as the base for natural logarithms--because it is a naturally occuring value.

Answer 3

Euler's huge type (e) is, in a large number of texts besides, defined because the reduce as n is going to infty of (a million + a million/n) ^ n The summation notation given above is in straightforward words an artifact from the by-product of e^x being, definite, e^x. yet this isn't Euler's consistent (in many situations wide-spread because the Euler-Mascheroni consistent). This consistent, often denoted through little gamma, looks in a large number of elements, like huge type idea, diagnosis, even differential equations. HTH, Steve

Answer 4

http://en.wikipedia.org/wiki/Euler-Mascheroni_constant