The Number e

    This topic assumes no calculus background but rather is intended for someone who is preparing to study calculus in the near future, for example precalculus students. We will discuss what the following means:

    as      ;

and in doing so, introduce the number e as discovered and defined by Jacob Bernoulli (1654-1705) and Leonard Euler (1707-1783), respectively.
    First we start off by mentioning Jacob Bernoulli (1654-1705) and his study of the calculus of exponential functions in 1697. His work can be found in his publication: Principia calculi exponentialium seu percurrentium where he investigates properties of exponential functions by using recently developed methods of calculus. It is interesting that he first recognized the importance of the number e, by studying compound interest problems. Here is what we are talking about, in modern terms:  

Definition (Periodic Compounding Interest) If dollars is invested for years at a nominal interest rate compounded times per year, then the total number of compounded periods is and the interest rate per period is and the future value is or

    Here is a typical example of how this might be used today.

Example (Future Value for Compounding Periodically) Find the future value of the given investment.

(a) Find the future value if $3500 is invested for 6 years at 8% compounded quarterly.

    Solution. The future value is given by the formula where and so we have
    


(b) Find the future value if $3500 is invested for 6 years at 8% compounded monthly.

    Solution. The future value is given by the formula where and so we have
    


This is a slight increase from part (a) when the compounded period is quarterly instead of monthly.

    Let's take this example to an extreme, let's say that the compounding periods are yearly, quarterly, monthly, daily, hourly, by the minute and so on. The following table illustrates what will happen as the compounding interest period grows shorter and shorter.  

    If dollars is invested for years at an interest rate compounded times per year, then the future value is





    Notice that as the compounded period becomes shorter the number of compounding periods become larger, and so it becomes interesting to ask: what happens to as becomes larger and larger? The following computations demonstrates what happens to the expression (in the second column) when (in the first column) grows larger and larger.

1 : 2.00000,00000,00000,00000,00000,00000,00000,0000

2 : 2.25000,00000,00000,00000,00000,00000,00000,0000

3 : 2.37037,03703,70370,37037,03703,70370,37037,0370

4 : 2.44140,62500,00000,00000,00000,00000,00000,0000

5 : 2.48832,00000,00000,00000,00000,00000,00000,0000

19 : 2.65003,43266,40444,90726,32676,12930,00997,5806

20 : 2.65329,77051,44420,13394,54307,65151,97753,9062

    Okay, is not large and in fact it will take larger computations to see what happens. Let's get more to the point by letting take on the values from 100,000 to 500000 in multiples of 100000.

100000 : 2.71826,82371,74489,66803,50648,24426,04644,7974

200000 : 2.71827,50327,85620,91837,63098,81890,95504,5220

300000 : 2.71827,72980,03174,19480,96606,88748,93183,7563

400000 : 2.71827,84306,14546,38779,28903,33439,34222,7012

500000 : 2.71827,91101,82200,28348,65924,37786,13808,0569

1000000 : 2.71828,04693,19376,88381,97997,08454,35639,2752

    Okay, is still not large and in fact it will take several larger computations to see what happens. Let's see what happens when takes on the values from 10,000,000 to 50,000,000 in multiples of 10,000,000.

10000000 : 2.71828,16925,44966,27119,85502,25777,81327,3154

20000000 : 2.71828,17605,02002,63858,19363,73971,19819,6462

30000000 : 2.71828,17831,54349,47868,63465,88074,18718,3812

40000000 : 2.71828,17944,80523,15829,66855,58008,83317,9577

50000000 : 2.71828,18012,76227,44912,14941,54588,53276,1278

    The number e is defined by this limiting process:



and in fact, it is due to Leonard Euler (1707-1783) to use the letter to represent this number. We are now ready to talk about compounding continuously.

Definition (Continuous Compounding Interest)  If dollars is invested for years at an interest rate compounded continuously, then the future value is given by

Example (Future Value for Compounding Continuously) What lump sum do parents need to deposit in an account earning 9%, compounded continuously, so that it will grow to $40,000 for their daughter's college tuition in 18 years?

    Solution. The future value is $40,000 and is given by the formula where and and so we have

Example (Interest for Compounding Continuously) Which investment will earn more money, a $1000 investment for 6 years at 8% compounded annually, or a $1000 investment for 6 years at 8% compounded continuously?  
    
    Solution. The investment that is compounding annually will have future value of where and which is The investment that is compounding continuously will have future value where and which is
    
  

Thus, the investment which is compounding continuously is the better investment.

    Finally, we end this topic with 5 hundred digits of the number e.

2.71828182845904523536028747135266249775724709369995957496696762772407663035354759
45713821785251664274274663919320030599218174135966290435729003342952605956307381
32328627943490763233829880753195251019011573834187930702154089149934884167509244
76146066808226480016847741185374234544243710753907774499206955170276183860626133
13845830007520449338265602976067371132007093287091274437470472306969772093101416
92836819025515108657463772111252389784425056953696770785449969967946864454905987
9316368892300987931

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Cite this as:
The Number E
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/the-number-e.html