ORIGIN OF ARABIC NUMERALS

This chart shows the origin of Arabic numerals, which were defined according to the number of angles .

REGULAR POLYGONS LISTED BY NO. OF SIDES

enagon or monogon {1}


Digon {2}


Triangle {3}


Quadrilaterals {4}


Pentagon {5}


Hexagon {6}


Heptagon {7}


Octagon {8}


Nonagon {9}


Decagon {10}


Hendecagon {11}


Dodecagon {12}


Tridecagon {13}


Tetradecagon {14}


Pentadecagon{15}


Hexadecagon{16}


Heptadecagon{17}


Octadecagon{18}


Enneadecagon{19}


Icosagon{20}

1729 - TAXICAB NUMBER

1729 is smallest taxicab number , i.e., the smallest number representable in two ways as a sum of cubes. It is given by

1729 = 12³ + 1³

1729 = 10³ + 9³

The number derives its name from the following story:

G. H. Hardy told about Ramanujan. I remember once going to see him when he was ill . I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."

1729 is the second taxicab number (the first is 2= 1^3 + 1^3). The number was also found in one of Ramanujan's notebooks dated years before the incident.

"Every positive integer is one of Ramanujan's personal friends."—J.E. Littlewood, on hearing of the taxicab incident

More details are available on the attached link.
Source(s):
http://en.wikipedia.org/wiki/1729_(number)

ONE EQUALS ZERO

Consider the following integral:
INTEGRAL (1/x) dx
Perform integration by parts:
let u = 1/x , dv = dx du = -1/x2 dx , v = x
Then obtain:
INTEGRAL (1/x) dx = (1/x)*x - INTEGRAL x (-1/x2) dx
= 1 + INTEGRAL (1/x) dx
which implies that 0 = 1.

USES OF LAPLACE TRANSFORMS

 PIERRE SIMON LAPLACE

• The Laplace transform is a simple way of converting functions in one domain to functions of another domain.

Here's an example :
Suppose we have a function of time, such as cos(w*t). With the Laplace transform, we can convert this to a function of frequency, which is

cos(w*t) ----L{}-----> w / (s^2 + w^2)

• This is useful for a very simple reason: it makes solving differential equations much easier.

• The development of the logarithm was considered the most important development in studying astronomy. In much the same way, the Laplace transform makes it much easier to solve differential equations. 

• Since the Laplace transform of a derivative becomes a multiple of the domain variable, the Laplace transform turns a complicated n-th order differential equation to a corresponding nth degree polynomial. Since polynomials are much easier to solve, we would rather deal with them. This occurs all the time.

• In brief, the Laplace transform is really just a shortcut for complex calculations. It may seem troublesome, but it bypasses some of the most difficult mathematics.  

• Laplace transform is a technique mainly utilized in engineering purposes for system modeling in which a large differential equation must be solved.

• The Laplace transform can also be used to solve differential equations and is used extensively in electrical engineering.

• Laplace Transform is used in electrical circuits for the analysis of linear time-invariant systems