Search This Blog

THE BEAUTY OF MATHEMATICS- Collected from The Internet and Various Books to enrich The students and Teachers. SUPPORT with YOUR COMMENTS ...

Friday, July 24, 2009

SEQUENCE OF NUMBERS WITHOUT 8

12345679 x 09 = 111111111
12345679 x 18 = 222222222
12345679 x 27 = 333333333
12345679 x 36 = 444444444
12345679 x 45 = 555555555
12345679 x 54 = 666666666
12345679 x 63 = 777777777
12345679 x 72 = 888888888
12345679 x 81 = 999999999

NUMERIC PALINDROMES WITH 1'S


1 x 1 = 1
11 x 11 = 121
111 x 111 = 12321
1111 x 1111 = 1234321
11111 x 11111 = 123454321
111111 x 111111 = 12345654321
1111111 x 1111111 = 1234567654321
11111111 x 11111111 = 123456787654321
111111111 x 111111111 = 12345678987654321

SEQUENTIAL 8'S WITH 9

9 x 9 + 7 = 88
98 x 9 + 6 = 888
987 x 9 + 5 = 8888
9876 x 9 + 4 = 88888
98765 x 9 + 3 = 888888
987654 x 9 + 2 = 8888888
9876543 x 9 + 1 = 88888888
98765432 x 9 + 0 = 888888888

SEQUENTIAL 1'S WITH 9

1 x 9 + 2 = 11
12 x 9 + 3 = 111
123 x 9 + 4 = 1111
1234 x 9 + 5 = 11111
12345 x 9 + 6 = 111111
123456 x 9 + 7 = 1111111
1234567 x 9 + 8 = 11111111
12345678 x 9 + 9 = 111111111
123456789 x 9 + 10 = 1111111111

SEQUENTIAL INPUT OF NUMBERS WITH 8

1 x 8 + 1 = 9
12 x 8 + 2 = 98
123 x 8 + 3 = 987
1234 x 8 + 4 = 9876
12345 x 8 + 5 = 98765
123456 x 8 + 6 = 987654
1234567 x 8 + 7 = 9876543
12345678 x 8 + 8 = 98765432
123456789 x 8 + 9 = 987654321

Wednesday, July 22, 2009

ORIGIN OF ARABIC NUMERALS






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

Saturday, July 18, 2009

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}

Friday, July 17, 2009

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)




Thursday, July 16, 2009

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

            THE PROOF 1 + 1 = 0

            1+1
            = 1+ sqrt (1)
            = 1+ sqrt [(-1) (-1)]
            = 1+ [sqrt (-1) * sqrt (-1)]
            = 1+ [i * i]
            = 1+ (-1 )
            = 1 -1
            = 0

            MATH INVOLVED IN ELECTRICAL AND ELECTRONIC ENGINEERING



            • There's really a lot of math involved in electrical and electronicengineering. How much you do depends on what area of EE (shorthand for electrical and electronic engineer) you do.





            • For example, there's a lot more abstract math in communication theory and signal processing, and many more very direct calculation differential equations in circuit theory and systems design.





            • Circuit theory at its simplest form is really differential equations, which is basically solving equations involving derivatives, so you need some CALCULUS and ALGEBRA and TRIGONOMETRY are fundamental to understanding it. Every basic circuit element (resistor, capacitor, inductor) has arelated current-voltage relation determined by its impedance. This iswhere COMPLEX NUMBERS come in.




            • If we move on to the theory of "how" electromagnetism works, we haveMaxwell's equations. These pretty much form the basis for EE. They are written in both integral and derivative forms and involve vectors. So, suddenly, we also have VECTOR CALCULUS.




            • If we move to Communication Theory/Information Theory, a mathematician named Claude Shannon developed a mathematical theory to explain various quantities related to how to communicate between devices.Communication Theory is used everywhere, from RADAR, to telephones, to devices within computers. The underlying theory requires at least CALCULUS , some LINEAR ALGEBRA , some MEASURE THEORY, etc.








            • Even wavelets, which have revolutionized signal processing, were discovered by mathematicians early in the 20th century, but not used by engineers until 20 years ago.






            In general, it is not possible to do EE without math.


            Each abstract mathematical theorem somehow finds its use in EE.

            Tuesday, July 14, 2009

            IMPACT OF EIGHT ON INDIA


            • In India most of the people are aware of the inauspiciousness of the number 8 and Saturday. Especially those who are under Sade-Sati (7 ½ years Sani period) or in the period called Sani-Rahu or Rahu-Sani have noticed that trouble usually comes on 8th or 17th or 26th or on Saturdays or when the star is Pushyami, Anuradha or Uttarabhadra.
            • It is generally believed that India is ruled by this number. India became independent on 15-8-1947 which adds up to 35 which adds up to 8. On that day the star was Pushyami – the 8th Star. Using the Koorma Chakra – we find that Delhi the Capital of India is in the Capricorn area that is ruled by Saturn.
            • The Gujarat earthquake of India on 26.01.2001 and the Tsunami of 26.12.2004 made it worse for those who already feared number 8. The Kashmir Earthquake which happened on 08.10.2005 killed about 80,000 people. While earthquakes are common, I noticed that either the death toll was higher or the magnitude of the earthquake was very high whenever such calamities happened on dates under the influence of Saturn. Recently the terror stike on Mumbai hotel happened on 26.11.2008.




            BUDDHA AND EIGHT

            Number 8 is auspicious for all those who look into the suffering of human beings. The Buddha was enlightened when He was 35 (=8) years old. The 8 fold path was His solution to the world of suffering. He attained final Nirvana when He was 80. When we stray away from the right path number 8 becomes inauspicious. 8 is considered a lucky number in China where Buddhism flourishes.

            FASCINATING NUMBER 8


            The twist and turns

            • 8 is the most fascinating and enigmatic single digit number. Feared by many, preferred by a few, this number continues to be the most controversial single digit number in the world of Astrology and Numerology

            • The shape of 8 makes it unique. Nothing happens to this number if you invert it upside down. See it in the mirror and you will see the same number. You can draw a line vertically to split the number to get 2 identical halves, a 3 and a laterally reversed 3. If you cut it horizontally you get once again identical halves but this time they are 2 zero's. 8 resembles symbol of infinity that is familiar to mathematicians if you rotate it 90 degrees.
            • If you write the numbers from 1 to 9 the stroke of the pen always end downwards. Lo and behold!… 8 is the only number that ends upwards
            • Life appears complicated like the loop or the knot in the number 8 but on a closer look you will know that it is just 2 bubbles.
            • Chess which is the most mind absorbing game we have 8 pieces and 8 pawns. The board is a 8 x 8 square.
            There was an error in this gadget