From what is the number e derived?

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...

 

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!

 

The original question was posted April 5, 2016. That person has not returned here.

 

Therefore, what? Other people can learn from this.

 

Ok. Do your best.