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! + ...
1/0!
|
=
|
1/1
|
=
|
1.0000000000000000000000000
|
1/1!
|
=
|
1/1
|
=
|
1.0000000000000000000000000
|
1/2!
|
=
|
1/2
|
=
|
0.5000000000000000000000000
|
1/3!
|
=
|
1/6
|
=
|
0.1666666666666666666666667
|
1/4!
|
=
|
1/24
|
=
|
0.0416666666666666666666667
|
1/5!
|
=
|
1/120
|
=
|
0.0083333333333333333333333
|
1/6!
|
=
|
1/720
|
=
|
0.0013888888888888888888889
|
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....
