Beauty of Mathematics

Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry-- Bertrand Russel.
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Saturday, 2 November 2013

e for eternity

The eternal number ‘e’

“The most beautiful thing you can experience is the mysterious. It is the source of all true art and science.
He ... who can no longer pause to wonder and stand rapt in awe is as good as dead”- Albert Einstein

The above lines can describe the beauty of e to a great extent. It is indeed a mysterious number and is one of the most famous Irrational Numbers. Probably it is the most famous Irrational number after π (pi). In comparison to pi and Golden Ratio it is less famous among the laypersons but its importance and ubiquity is second to none.  

This is one Irrational number which confounds the Students most of the time.( For example how can an Irrational number be considered as base of the ‘Natural’ Logarithm!?). However before doing questions involving ‘e’; if it is understood fully, the questions involving e may become easier.

Numerical value of e can be calculated by the following series:

 e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + ...


Adding the above values we get the approximate value of e as  2.7182818284…

The number e can be defined as:              

It can further be defined in many ways; for example:
(1)   e is a real number such that the value of the derivative (slope of the tangent line) of the function f(x) = ex at the point x = 0 is equal to 1.
(2)   The function ex so defined is called the exponential function, and its inverse is the natural logarithm or logarithm to the base e.

(3)    The natural logarithm of a positive number k can also be defined directly as the area under the curve y = 1/x between x = 1 and x = k, in which case, e is the number whose natural logarithm is 1.

The number e was first studied by the Swiss mathematician Leonhard Euler in the 1720s, although its existence was evident in the work of John Napier, the inventor of logarithms, in 1614. Euler was also the first to use the letter e for it in 1727. As a result, sometimes e is called the Euler Number, or Napier's Constant. It was proven by Euler that "e" is an irrational number.

The value of "e" is found in many mathematical formulas involving a nonlinear increase or decrease such as growth or decay found in:

(1)   Compound interest.
(2)    the statistical bell curve,
(3)    The shape of a hanging cable or a standing arch.
(4)    some problems of probability,
(5)   Counting problems,
(6)   Study of the distribution of prime numbers.
(7)   Ultrasound attenuation in a material.
(8)   The sound energy decays as it moves away from the sound source
(9)    The base of natural logarithms.

Understanding e: The best way to understand what e is, let us take an example of Compound Interest:

A = P {1+r/(k.100)}nk
(Where A is the Amount, P is the Principal or Sum, r is the rate of interest per annum , n is the number of years and k is the number of compounding done per annum Here the meaning of compounding is the act of adding the interest in the Principal for calculation purposes.)

Compound Interest = Amount – Principal.

Let us consider a hypothetical case of a bank account that starts with Re1.00 and pays 100 percent rate of interest per year. If the interest is given once, at the end of the year (meaning compounding is done once a year), the value of the account at year-end will be Re 2.00. What happens if the interest is computed and credited more frequently during the year? Obviously the Amount will increase; but it will not increase in a linear manner. It will increase in an ‘Exponential’ manner involving e.

If the interest is credited twice in the year Amount will be equal to Re1.00×1.52 = Re 2.25 at the end of the year; using the above mentioned formula. Compounding quarterly yields Re 1.00×1.254 = Re 2.4414...and compounding monthly (meaning k equal to 12) yields Re 1.00 × (1+1/12)12 = Re 2.613035...
If there are k compounding intervals, the interest for each interval will be 100%/k and the value at the end of the year will be Re1.00×(1 + 1/k)k.

It can be noticed that this sequence approaches a limit with larger k and, smaller compounding intervals. Compounding weekly (k = 52) yields Re 2.692597..., while compounding daily (k = 365) yields Re 2.714567.. Only two Paise more! 

The limit as k grows large is the number that came to be known as e; with continuous compounding, the account value will reach Re 2.7182818.... 

In general, an account that starts at Re 1 and offers an annual interest rate of r will, after n years, yield Rs.  enr/100 with continuous compounding or compounding done every moment. 

By considering the above example now students can easily appreciate the formula;
       e =2.7182818....