From difference to differential equations

Now imagine that your bank changes its regime of interests. It shortens the time intervals at which interests are compounded. Instead of annually compounding, it accredits your gains from interest twice a year.

This changes your wealth. Your deposit now is compounded with the half of 3% in the first half year, which gives € 101.5 that stand ready to be compounded in the second half of the first year again with 1.5%. This increases your deposit to € 103.023.- after the first year, in comparison to € 103.- with annual compounding. After ten years your deposit now will grow to approximately € 134,69.- instead of 134,39.

If the bank should further shorten intervals, say to quarter years, your account will grow to € 134,84. And if the bank should decide to compound daily, your deposit would hold € 134,98.- after ten years.

These differences might not seem dramatic. But they give reason to consider continual compounding, that is, an accrediting in time steps that approach a limit duration of zero.

To see what this implies, we repeat the above calculations with 1 Euro initial deposit, 100% interest and just one year of aggregated depositing.

If the account is annually compounded this will yield a deposit of € 2, one from the initial deposit plus one from the 100% interest.

With biannual compounding this will yield € 2,25

With quarter-annual compounding this will yield € 2,44141

With daily compounding this will yield € 2,71457

If we plot this sequence with much more interim steps and up to a compounding rate of 10000 times per year, like in the graph below in the interval of 2.714 to 2.719, we see that it approaches a limit at about 2.71828 (red line). This is the famous infinite and aperiodic Euler-number e, named after the mathematician Leonard Euler.


Aggregated compound calculation of interests, with initial deposit = 1, interest = 100%/year, and compounding steps ranging from 1 to 10000. The red line indicates e = 2.71828, the Euler-number.

This number hence simulates a (near) continuous compounding of interests in the above example, dependent on the amount of decimal places considered. We might say, it assumes that we observe the development of our data continuously.

However, it can’t be sufficiently stressed that the continuum is just simulated, that it is an abstraction dependent on the decimal places of e considered. As it should be obvious, no computer, no matter how powerful, will be able to consider an infinite amount of decimal places of e. That’s why the Euler-number and the differential-equation to which it provides an essential foundation is an abstraction. On philosophical grounds, considering the discreteness of digital computation, one might ask, if this kind of calculation is much more than a leftover from 19th century, that is, from a time when mathematicians firmly believed in the continuum of our world and tried to do justice to it with mathematical means.

In the section on growth processes, we will see that e comes in handy for calculating dynamics in terms of differential equations. Unfortunately however, its use is rather limited, since we will not find a solution for all differential equations.