**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

**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

**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**

**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**

**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

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

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. **
**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

**1****+1**

**= 1+ sqrt (1)**

= 1+ sqrt [(-1) (-1)]

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

= 1+ [i * i]

= 1+ (-1 )

= 1 -1

= 0

- 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**.

- 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.**

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. **
**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.