Euler’s Constant

“e”. All of us has come across “e”. What is it?

It is the 5th alphabet and 2nd vowel in the English language. It’s what we say when we show someone our teeth. But, Mathematicians recognise it as the Euler’s constant. Standing alongside other important mathematical constants like π , i, Φ , sqrt{2}, etc, this constant, irrational number has a value of 2.718281828459045235……

Most of the mathematical constants are geometrical. For example, π is the ratio of the circumference of a circle to its diameter, sqrt{2} is the length of the hypotenuse of a right angled triangle whose legs measure unity. But, “e” is a constant that is not defined by geometry or any shape. It is based on growth or rate of change. But how?

Let’s go back to the 17th century, when Jacob Bernoulli was working on compound interest, that is, gaining interest on your money.

Suppose, you are a part of a bank, a very generous bank. Let’s say you gave the bank ₹1 and the bank gives an interest of 100% per year. (A very generous bank indeed). So now, towards the end of the year, you will have ₹2. So, if you gain a 50% interest per 6 months, will you end up with the same amount, ₹2? Or more than that? or less than that? Let’s calculate and see, shall we?

Well, this shows that if you take 50% interest per 6 months, it will help you gain more than having an interest of 100% per annum. What about if you take 1/12th interest every month?

Then, it will be,

If 1/52th of interest is given per week, your final amount would be,

What about 1/365th interest each day, then your amount towards the end of the year after giving ₹1 to the bank would be,

You can similarly calculate the amount of money you get in every hour, every minute, every second or even every millisecond!

So, what do you observe? The value is calculated as n increases using the general formula, as

So, you can notice that as the value of n increases, the value is coming closer and closer to a certain value. This is the value of “e”.

But, Jacob Bernoulli did not calculate the value of the constant. He just knew that it’s value would be somewhere between 2 and 3. It was Euler who finally calculated this constant and proved that it was irrational. He used a formula to calculate the value, not

But another formula. He used the following formula.

This is a continued fraction. You can say that as it goes on forever, there is a pattern to this fraction, 2,1,2,1,1,4,1,1,6,1,1,……So, if it goes on forever, then, it is an irrational fraction. If it would have terminated, it would have been rational as you can write it as a fraction. Thus, this proves that “e” is an irrational constant.

To calculate the value of “e”, Euler used a different formula. That is,

“e”, is the natural language of growth, it is the natural language of calculus. Why?

The figure given above shows the graph of e^x. Now, the speciality of an e^x graph is that, if you take any point on the graph, the value of that point is e^x, the gradient at that point is e^x and the area under the graph from that point onwards to -∞ is also e^x. Thus, when you integrate or differentiate e^x, you get e^x itself. This constant “e”, forms a very strong tool in calculus.

The Euler’s constant “e” is also known to bring some of the big constants in math together in one formula, that is, root of -1, which is i, π, 1 and 0. This is also many times termed as the most beautiful equation in Math:

I will write more about this equation in an upcoming article.