We know, for example, the constant $\pi$ is the perimeter of a circle with diameter $1$ unit. In the similar manner how would we explain the constant $e$. I have searched a lot for it. But I couldn't comprehend it in a practical way. Can anybody help me to understand the constant $e$?
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$ e^x $ is its own derivative, which makes it quite significant in calculus. – The Chaz 2.0 Nov 06 '13 at 04:07
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1http://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/ – cygorx Nov 06 '13 at 04:13
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At 100% annual interest rate, compounded continuously, an initial deposit of $1 will grow to $e in one year. – bof Nov 06 '13 at 04:54
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For someone who hasn't taken calculus, understanding what $e$ is can be difficult. The reason is that the ways in which we define $e$ use calculus concepts. For example, we can define it as the sum of a certain infinite series, as the limit of a certain expression (see below), or we can give a derivative-based definition (which The Chaz 2.0 mentioned in his comment). There is no easy-to-picture geometric interpretation of $e$ like there is for $\pi$. But once you understand the calculus concepts we use to define $e$, you'll have a better understanding of what it is.
If you do know some calculus, perhaps easiest way to think of $e$ is as the limit $$e = \lim_{n \to \infty} \big(1 + \frac{1}{n}\big)^n$$
The examination of this limit is how $e$ was first discovered.
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