From what is the number e derived?

Discussion in 'Basic Math' started by kmcandy14, Apr 5, 2016.

  1. kmcandy14

    kmcandy14

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    Pi (approximately 3.14159265) is derived from the quotient of 22/7. I was wondering how the number e (approximately 2.71828182818...) was derived. Does anybody know? This may not entirely fit the category of this section, but nowhere else seemed to make sense...
     
    Last edited: Apr 8, 2016
    kmcandy14, Apr 5, 2016
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  2. kmcandy14

    knoppi

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    c16ee0bbc785080c215034407389c3551064a4dc.jpg
    ;)
     
    knoppi, Mar 14, 2020
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  3. kmcandy14

    HallsofIvy

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    No, it is not. The number, "pi", was defined, independently of the approximation 22/7, as the ratio of the circumference of a circle to its diameter.

    To understand "e", and understand how important it is, you have to know some Calculus. In Calculus, we define the "derivative" of a function as the instantaneous rate of change of the function. If the variable changes from x to x+ h then the amount of change in the function value is f(x+ h)- f(x). The rate of change is the change in the function value divided by the change in the variable, (f(x+ h)- f(x))/h. The instantaneous rate of change is the limit of that as h goes to 0.

    Look at, say, f(x)= a^x. f(x+h)= a^(x+ h)= (a^h)(a^x) so the amount of change is (a^h)(a^x)- a^x= a^x(a^h- 1) and the rate of change is a^x(a^h- 1)/h. That is just the original function, a^x, times (a^h- 1)/h. If we take the limit as h goes to 0 we get the original function, a^x, times a constant that depends on a but not on x.

    For example, if a= 2 then (2^(x+ h)- 2^x)/h= (2^x)(2^h- 1)/h. What happens as h "goes to 0"- that is, gets smaller and smaller? If h= 0.01, 2^0.01- 1 is approximately 0.0069555500567... and dividing by h= .01 we have rate of change 0.69555500567... If h= 0.001, 2^0.001- 1 is approximately 0.000693387... and dividing by h= .001 we have rate of change 0.693387... Taking h smaller and smaller we get a number somewhere in the vicinity of 0.69 but surely less than 1. The derivative of 2^x is 2^x itself times a constant less than 1.

    If a= 3 (3^(x+ h)- 3^x)/h= (3^x)(3^h- 1)/h. If h= 0. 01, 3^0.01- 1 is approximately 0.01104669... so (3^0.01- 1)/0.01 is approximately 1.104669.... If h= 0.001, 3^0.001- 1 is approximately 0.0010992.... so (3^0.001- 1)/0.001 is approximately 1.0992.... Taking h smaller and smaller we get a number somewhere in the vicinity of 1.0992 but surely larger than 1. The derivative of 3^x is 3^x times a constant larger than 1.

    So the derivative of a^x is a^x times a constant. If a= 2, that constant is less than 1 and if a= 3, that constant is larger than 1. Somewhere between 2 and 3 there is a value of a such that that constant is exactly equal to 1! We call that number "e". "e" is that number such that the derivative of the function f(x)= e^x is just e^x itself! That is a very useful property so e^x is a very useful function and e is a very important number!
     
    Last edited: Nov 27, 2021
    HallsofIvy, Nov 27, 2021
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  4. kmcandy14

    nycmathguy

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    The original question was posted April 5, 2016. That person has not returned here.
     
    nycmathguy, Nov 27, 2021
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  5. kmcandy14

    HallsofIvy

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    Therefore, what? Other people can learn from this.
     
    HallsofIvy, Dec 4, 2021
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  6. kmcandy14

    nycmathguy

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    Ok. Do your best.
     
    nycmathguy, Dec 4, 2021
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