e-pic day

Today, the Second Day of July in the year 2018, is a good day. No, an epic day if you write the date this way: 2/7/18. Yes, my erudite readers, the date does remind you of the mathematical constant often known as Euler’s number, e=2.718… . Around the world and in some local schools, 14th of March, is celebrated annually as Pi Day because the mathematical constant pi, usually denoted by the Greek letter [tex]\pi[/tex], equals 3.14… . Readers will no doubt recall having learnt in schools that [tex]\pi[/tex] is the ratio of the circumference of a circle to its diameter and appears in formulas like circumference, [tex]c=2\pi r[/tex] and the area enclosed by the circle, [tex]A=\pi r^2[/tex]. At this point, readers are urged to refrain from using the tired pun “this looks like Greek to me.” Pi Day is fairly well known, see for example the website www.piday.org, and we have another 256 days to prepare for the next Pi Day, so let us focus on [tex]\pi[/tex]’s less illustrious cousin e.
It is highly unlikely that many of us can afford to wait a hundred years for the date 2/7/18 to reappear, so let us make the most of this once in a lifetime opportunity to celebrate e. First, let us familiarize ourselves with e. Like [tex]\pi[/tex], it is also an irrational number. This does not mean that e behave unreasonably but rather, it means that e, as well as [tex]\pi[/tex], cannot be expressed as a ratio of integers. For [tex]\pi[/tex] in particular, this can potentially cause much confusion since primary school children are taught in mathematics classes to use the ratio 22/7 for [tex]\pi[/tex], which actually contradicts the very definition of an irrational number when that concept is introduced in secondary schools. It is generally problematic to define an object by what it is not. When children asks “what is a cat?” Please do not tell them it is not a dog. So perhaps a better way to describe an irrational number is this: if you write any irrational number in decimal form, the digits goes on indefinitely without any repeating patterns.
Now, you may ask where does e come from? The following example should be of interest. (Pun definitely intended.) Just last month, the US Federal Reserve raised interests rate which means that it is time to think about putting our hard earned money into banks. To simplify calculations, let us assume that our friendly neighbourhood bank promises us a staggering 100% interest per annum. So if we put a deposit of $1000, we will expect in one year’s time to get $1000 in interest, giving us a handsome sum of $2000. Suppose a competing bank claims they can do it better. They still pay the same interest rate per annum but they pay out interest every half a year. How much more would that be? After six months, your original $1000 earn $500 in interest. Things then get better for you because for the next six months, you should be earning interest based on the principal amount of $1500 and not $1000. Now 50% of $1500 equals $750, so you will earn a total of $2250 after one year if the bank pays interest every six months, albeit at the same annual rate. You earn an extra $250. This is the power of compound interest. What if a third bank offers the same annual rate with quarterly payouts? After three months, you get $1250 in total. In the next three months, you get 25% of $1250 which is $312.50 giving a total $1562.50. The calculations is getting complicated but you should persevere on since you are getting rich. Let us tabulate this. The interest rate is 100% or 1 times of the principal. So if the payout is every quarter, the interest rate is 25% or 0.25 = ¼ times of the principal.

Period &Principal (\$) &Interest (\$) &Total Amount (\$) \\
Q1 & 1000 & 250 & 1000*(1+1/4) = 1250\\
Q2 & 1250 & 312.5 & 1250*(1+1/4) = 1562.5\\
Q3 & 1562.5 & 390.625 & 1562.5*(1+1/4) =1953.125\\
Q4 & 1952.125 & 488.28125 & 1953.125*(1+1/4) = 2411.40625

So we observe that even though the interest rate remains constant, by paying out twice as frequently, we increase our wealth from $2000 to $2250, and further to $2411.41 if the pay out is four times a year. A natural question would be how much more can we earn if the pay out is more frequent, say every month, week, day or hour! Before you start dreaming about what you can do with your new found wealth, you should know that even if a crazy bank does promise to pay you interest every hour or every minute, it will not quite break the bank. There is a limit to how much you will get. We can observe from the pattern in our previous tabulation that if the bank pays out [tex]n[/tex] times a year, the interest rate is [tex]1/n[/tex]. You can also expect to be paid a total of [tex]n[/tex] times and the final amount can be computed as:
[tex]1000(1+1/n) (1+1/n) … (1+1/n) = 1000 (1+1/n)^n[/tex]
The value of [tex](1+1/n)^n[/tex] gets larger as [tex]n[/tex] gets larger but there is a limit to how much it grows. It can never get beyond a special number, and … yes, you have guessed it, that number is e. Mathematically, as [tex]n[/tex] gets indefinitely large, the value [tex](1+1/n)^n[/tex] approaches e, i.e.
[tex]\displaystyle \lim_{n \rightarrow \infty} (1+1/n)^n = e.[/tex]
So no matter how frequent the bank pays interest, at 100% per annum you can never get more than $2719 in a year. If a bank is willingly to pay out 4822 times a year, then with compounded interest you will earn $2718.00 at the end of the year.

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