Where does e come from and what does it do?
Suppose you put £1 in a bank. The bank pays 4% interest a year, and this is credited to your account at the end of a year. A little thought shows that at the end of five years an amount of money equal to £
However, if the interest (still at an annual rate of 4%) was "compounded" every quarter, then the amount at the end of five years would be £
If the bank gave an interest rate of 100% annually, then after one year the bank balance would then be £
It’s obvious that compounding more frequently results in more money in the bank. So it is natural to ask whether compounding at every instant in time (that is, continuously) leads to an infinite amount in the bank.
To answer this question we need to evaluate
![\[ \lim _{n \rightarrow \infty } (1 + 1/n)^ n. \]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sHhZ7yFaTKm-WzNZaGDih97MApRP213zF6u7iYWT7EQIQZIB_AGaZnK1A9nzHRN1ks_KiPhMRiAxYzveRQeOv3_Pl2uJfP55P8g_GUSWadG6FQ1n0qb_T6lnoU78y9r2a0FSJLKy4Fnj10ZVcomjtv=s0-d) |
This quantity turns out again to be 
Now 
![\[ e = 1+ \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \dots + \frac{1}{r!} + \dots . \]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uh4EVnbWv_M36XirIM2MMh4W3AvXcKUC1GaUurcYFtFXCgfEVfWYqgbeFhcvbaqJBiy8Qbf9pSLnkPwkq2JD3AZznMDPJ_-i_JK38GIpdCoUpa6xVcrIzvzGokRCKi5MTrGAAeQaJt_cAq8P0FlxZl=s0-d) |
This series is convergent, and evaluating the sum far enough to give no change in the fourth decimal place (this occurs after the seventh term is added) gives an approximation for
It was that great mathematician Leonhard Euler who discovered the number e and calculated its value to 23 decimal places. It is often called Euler's number and, like pi, is a transcendental number (this means it is not the root of any algebraic equation with integer coefficients). Its properties have led to it as a "natural" choice as a logarithmic base, and indeed e is also known as the natural base or Naperian base (after John Napier).
There is the remarkable property that if the function 
![\[ \frac{dy}{dx} = \frac{d(Ae^ x)}{dx} = A\frac{d(e^ x)}{dx} = Ae^ x = y. \]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_v6L1l1qru1uTP5ZvIg8BnxCg_TNP8q5vG-vAGiMDEdUa-aqZ6vqMafuKfqmJ-6jzaJTcTyVmLJoB5INE2hELiafI56K813kF3Mrpl_1iYY21yIwRDaPNmXfaa5p28tAnRXRVR8_Ez0v4igjfoJlsmn=s0-d) |
In fact, it is true that given the equation:
![\[ \frac{dy}{dx} = y, \]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_u7xf__bpFQwZoLflJcJlEP8PrHxaHQ9Pokk4Vi6YC8Un1ZCBeFh8SuqAR9zoab79fKlFSEpdNpcFire2R1kch9QlTc4bzKlUSvNywKABWKSV--at20OMbQrB_RUIrbbeifeHEr9tNYPHFwD4rglUlj=s0-d) |
then a solution for 
It’s also possible to relate the general exponential function, 
![\[ y = b\left(e^ c\right)^ x = b e^{cx}. \]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_voprgzLh_SDkcX38Y9EfvigNe-P3nE2DPOg43w9pQQs-lpX8BpsWegBMQy5rmL3BoZyM1-KtKHfJ9Y3U4ilUPyNAlmmtv-2GGKSj8nzCxhV4McgWyZgwSuI-bKvY4ejfWWinqUErioPoNQo89b2nE=s0-d) |
Differentiating with respect to
![\[ \frac{dy}{dx} = cb e^{cx} = cy, \]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_t1rDsQ2VsLyRnw-T3p8k0LIdGEGGpxFdqwnO-EKiT7HKe9ztltqR7zL14f2oUGEoLAxkuH5FOiGaxFod--L-46VhaclWH4ZxGJoaMaJK-KqQZbHtHltmBZELuSDyYhfi01z3ipDcQuCn4U94i-CeQ=s0-d) |
which tells us that the general exponential relation,
![\[ \frac{dy}{dx} = cy. \]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uWxseGdfhP2qmel0Pl_TXOliVJfU-FxoBwxhZX4a1PMPxIJC_GUgGgPFiTb_FkYomxbDmbGFq_M4D9C0GZAIjxzawZ7VVfzmasn9umw3SfNLqhRvXIn1M1ouu8SBzr_8BFcJHJP_vDQGSfeBAr-o75=s0-d) |
(Actually it happens to be the only type of solution to this important equation.)
It is this equation that emerges naturally when attempting to model various processes.
