What is significance of eulers number

[rishi kesh]

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why is 'e' so important number. e^x is said to be natural language of growth. Why isn't 2^x a 100% growth.Can anyone explain me the difference between both of these and also bit more about number e which known as famous constant.please explain clearly. I will appreciate it.

 

[scottdave]

I'm on my phone, so I won't go into detail. One thing special about e^x compared to other exponentials is the slope of the function is the value of the function. I would like to point you to this Mathologer video.
. "e to the pi i for dummies"

 

[scottdave]

I'm on my phone, so I won't go into detail. One thing special about e^x compared to other exponentials is the slope of the function is the value of the function. I would like to point you to this Mathologer video.
. "e to the pi i for dummies"

The video ia about 15 minutes, but he discusses where e comes from in the first few minutes.

 

[FactChecker]

why is 'e' so important number. e^x is said to be natural language of growth. Why isn't 2^x a 100% growth.Can anyone explain me the difference between both of these and also bit more about number e which known as famous constant.please explain clearly. I will appreciate it.

It is the only number, a, where d/dx (a^x) = a^x. Any other number requires an "e-related" multiplier (like d/dx (2^x) = ln(2) ⋅ 2^x). So it is the simplest exponential function when it comes to derivatives. Furthermore, the others (like 2^x = e^ln(2)⋅x) are very easy to write in terms of e, so e^x is the most basic exponential function.

Likewise, its inverse function, ln(x), has a very simple derivative, 1/x, whereas derivatives of logarithms to any other base require an "e-related" multiplier.

As the link in @scottdave 's post indicates, there are a lot of nice things about e^z in the complex plane.