In our journey of learning mathematics, we must have across a mathematical constant called “e”. It is approximately equal to 2.71828.
Unfortunately, most of us memorized the value and its usage without bothering about the concept behind it. That’s because mostly mathematics is taught in a way, where we try to explain concepts by their technical features without really explaining what it is and why it is used in the first place. Let us explore very basic insights about this so called “e”.
The natural exponential “e” explained in simple terms
In simple terms, “e” is manifestation of continuous growth in any naturally and continuously growing phenomenon.
Now what is this continuous growth and how do we deal it mathematically? To answer this question, we need to start with the other type of growth which we understand more easily, i.e. discrete growth.
Discrete growth can be visualized as step by step growth at the end of each period. If something doubles after every period; that is, at 100% growth per period
- 1 becomes 2 at the end of 1st period;
- 2 become 4 at the end of 2nd period;
- 4 become 8 at the end of 3rd period; and so on…
This is basically 2^n where n is the number of periods. Remember, in the case of discrete growth nothing happens till we don’t reach the end of the period, then suddenly it doubles. It’s more like growth in step by step fashion depicted in the diagram below. If you’re just interested in knowing how much time it takes to double your money at certain rate of interest, you will love the Rule of 72.
Discrete Growth vs Continuous Growth
In real life, the growth of anything like money, population, bacteria, plants, etc. is more smooth i.e. it is continuous process rather than discrete.
Given the following formula for compounding, we get discrete results -
A = P (1 + r/n) ^ nt
- A = Is the future value of investment including interest
- P = Principal initial investment or the present value of the investment
- r = yearly rate
- n = number of compounding periods
- t = no. of years
Assuming, the initial amount is Re.1, what happens when the yearly rate of growth is 100% and t is 1 year?
The above compounding formula gives us,
Calculation credit: Purplemath
Now, think how we can, mathematically shift from discrete stairs like growth curve to continuous smooth growth curve?
From the above table and graph, you know intuitively that higher the value of n, smoother the curve will become. So continuous growth is possible when the growth happens over infinitely small periods of time. Thus to make it continuous n is assumed to be “x” which tends to infinity.
Thus, you have got a new number “e” which denotes continuous growth over 1 period in which growth is taken as 100%.
Different Growth Rates and Time Periods
Now the question in your mind must be what if the growth rate is not 100% per period but something else. Fair question.
Remember, “e” by itself expresses the continuous growth in something at the rate of 100% over 1 time period, say 1 year. At 100% growth r becomes 1 (as you know 100% = 1) and for single time period t = 1.
Essentially, continuous compounding is expressed as e^x or it can be further expanded as ert where r is the rate of interest and t is time period.
The number “e” is about continuous growth. For all practical purposes, we can use e^x, where x allows us to merge time and rate. 5 years at 100% growth is the same as 1 year at 500% growth, when continuously compounded. Intuitively, e^x means, how much growth do I get after after x units of time and 100% continuous growth.
So if rate is 20% and time is 3 years, the resultant growth at continuous compounding can be captured using e^rt = e^(0.2*3) = e^0.6. That’s like saying 100% growth rate for 0.6 time years
Hope now “e” has started making more sense to you. Yes or No, please let me know in comments below.