How was Eulers constant originally computed?

Question

Im not referring to approximations. Anyone can guess a number and methodologically modify it for accuracy. Im talking a real algorithm or equation of what have you that was used for sole purpose of determining the value of e, not validating it or verifying it. How was the number originally computed? What mathematical background and conceptual basis was necessary for this technique?

e^x
ln x = log_e x

Answer 1

I believe this can be done through calculus.

e was created as a necessity of differential calculus. Exponential functions and logarithmic functions predate... if not historically then at least conceptually... that of calculus and Eulers constant.

It wasnt until people questioned how to find derivatives of exponential and logarithmic functions that Eulers constant appeared. Its properties redefined mathematics... even without a known value attached to it, but merely as a representation of a concept. Properties could be determined through basic math and definition... and definition was expanded as properties were expanded into other aspects of calculus.

From differential calculus sprang integral calculus... and from that Taylors Theorem and Power Series. Those things existed independent of Eulers constant.

But when you apply the function e^x to Taylors theorem and find a power series, you derive a rather simple series of numerical constants whose value can be computed arithmetically. Accuracy was then a matter of how many terms to use.

∞
∑ 1/n! = e
n=0

e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + ...

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The guy below me is correct. I understood what you meant. Technically e is called "Eulers Number". Eulers constant is a separate thing.

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Let me say one thing. I only answered in terms of conceptual development. Dont quote me for historical accuracy. The truth is that e actually predated Taylor expansions, and yet its value was known even then. I only answered respecting the concept tree of understanding necessary to illustrate how e can be computed. If that makes any sense.

Answer 2

Euler's number (e) is, in many texts anyway, defined as the limit as n goes to infty of

(1 + 1/n) ^ n

The summation notation given above is only an artifact from the derivative of e^x being, yes, e^x.

But this is not Euler's constant (commonly known as the Euler-Mascheroni constant). This constant, usually denoted by little gamma, appears in many areas, like number theory, analysis, even differential equations.

HTH,
Steve

Answer 3

I agree with the answer above - I believe it would have been calculated as
the limit as n-> inf of (1+1/n)^n

ie the limiting amount that a sum of money could increase to at a rate of 100% if the compounding frequency was increased.

Perhaps some guy in the 15th or 16th century came up with it - remember they didn't have tv or myspace back then - so people had to fill in their time somehow.

Answer 4

For a very useful number, this one came pretty easily. I know that everybody calls it Euler's constant but Napier used it and so did Leibniz. It's original application was compound interest calculation. No calculus here at all.

If you have Excel it's easy.

(1+(1/n))^n

That's it!

Put this in A1:
=1+(1/B1)^B1
Put a 1 in B1
put in B2
=b1+1

then drag everything down

You'll get close to 2.716

Answer 5

e is calculated by using a remond sum and I believe that e is calculated by using a Taylor series... I don't have a math reference book on me.

Answer 6

Euler's constant isn't e