Significance of Euler's Number

Math and Science is filled with various constants, each one with its own importance. One such constant of special significance is the Euler’s number denoted as “e”. It is named after Swiss mathematician Leonhard Euler. In this blog we will explore the reason behind its importance in Math.

Let us consider a population of bacteria in a petri dish. Let us assume that the population initially begins from a single bacterium and its population doubles every single day. After the first day you will have two bacteria, the next day four and then eight and so on. The population of the bacteria on any day can be given by :

                        
 where x is the number of days. This is actually called exponential growth model.

To further analyse growth of bacterial population, we can find rate of growth. The change of population between day 2 and 3 will be 4 (8 − 4). For day 3 and 4 is 8 (16 − 8). We can see that growth rate changes with time. This gives change in population between two days. So, for a person with no knowledge in calculus it will be easy to assume that rate of change is also 

But we know that this is not true. To find the instantaneous rate of change in the population let us assume an infinitesimally small quantity of time dt. Then the population change can be given by

By the property of exponentials this can be written as

value of the second term i.e., the term in the bracket will approach a constant value k which in this case is 0.6931….So, for a random exponential function 

the instantaneous change can be given as,

For a=3, k=1.0986….; a=4, k=1.3863…. and so on.

We can see that as value of ‘a’ increases k also increases and at some value of a between a=2 and a=3, the value of k becomes 1. For that number the rate of change of the function ‘y’ is same as the value of the function at that point. The number that satisfies this condition is defined as the Euler’s number. Euler’s number is irrational and is equal to 2.7183 (precision up to 4 digits). Also, the value of k is nothing but ln(a)  i.e., log to the base ‘e’ of ‘a’. To simplify this, it is nothing but the value at which

Euler’s number finds application in variety of fields. In economics and finance, it is useful in the concept of compound interests, growth of investments over time. It is used in population modelling. It emerges in the field of physics when describing radioactive decay, the charging and discharging of electrical circuits, and the behaviour of waves and oscillations.