TOPIC – 1) FINDING VALUE OF PIE 2) PYTHAGORAS

THEOREM 3) CRYPTOGRAPHY

DR. S&S.S GANDHY GOVERMENT ENGINEERING

COLLAGE

PREPARED BYCHELANI PRASHANT VADHER VIJAY PATIL NIKHIL

1] FINDING VALUE OF PIEINFORMATION ABOUT PIENAMEDEFINITIONAPPROXCIMATE VALUEACTIVITY TO FIND PIE

INFORMATION ABOUT PIEThe number π is a mathematical constant, the ratio

of a circle's circumference to its diameter, commonly approximated as 3.14159. It has been represented by the Greek letter "π" since the mid-18th century, though it is also sometimes spelled out as "pi“.

Being an irrational number, π cannot be expressed exactly as a fraction. Still, fractions such as 22/7 and other rational numbers are commonly used to approximate π.

Because its definition relates to the circle, π is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses or spheres

NAMEThe symbol used by mathematicians to

represent the ratio of a circle's circumference to its diameter is the lowercase Greek letter π, sometimes spelled out as pi, and derived from the first letter of the Greek word perimetros, meaning circumference. In English, π is pronounced as "pie“. In mathematical use, the lowercase letter π (or π in sans-serif font) is distinguished from its capital counterpart Π, which denotes a product of a sequence.

The choice of the symbol π is discussed in the section Adoption of the symbol π.

DEFINITIONπ is commonly defined as the ratio of a circle's

circumference C to its diameter d.π = C d The ratio C/d is constant, regardless of

the circle's size. For example, if a circle has twice the diameter of another circle it will also have twice the circumference, preserving the ratio C/d. This definition of π implicitly makes use of flat (Euclidean) geometry; although the notion of a circle can be extended to any curved (non-Euclidean) geometry, these new circles will no longer satisfy the formula π = C/d.

Here, the circumference of a circle is the arc length around the perimeter of the circle, a quantity which can be formally defined independently of geometry using limits, a concept in calculus.For example, one may compute directly the arc length of the top half of the unit circle given in Cartesian coordinates by x2 + y2 = 1, as the integral.

APPROXIMATE VALUE

Some approximations of pi include:

Integers: 3Fractions: Approximate

fractions include (in order of increasing accuracy) 22/7, 333/106, 355/113, 52163/16604, 103993/33102, and 245850922/78256779

ACTIVITY TO FIND PIE

Step 1 Draw a circle on your card. The

exact size doesn't matter, but let's use a radius of 5 cm.

Use your protractor to divide the circle up into twelve equal sectors.

What is the angle for each sector? That's easy – just divide 360° (one complete turn) by 12:

360° / 12 = 30°So each of the angles must be

30°

Step 2Divide just one of the sectors into two equal parts – that's 15° for each sector.

You now have thirteen sectors – number them 1 to 13:

Step 3Cut out the thirteen

sectors using the scissors:

Step4Rearrange the 13 sectors like this

(you can glue them onto a piece of paper):

Step 5Its height is the circle's radius: just look at sectors 1 and

13 above. When they are in the circle they are "radius" high.Its width (actually one "bumpy" edge), is half of the curved

parts around the circle ... in other words it is about half the circumference of the original circle. We know that:

Circumference = 2 × π × radiusAnd so the width is:Half the Circumference = π × radiusWith a radius of 5 cm, the rectangle should be:5 cm highabout 5π cm wide

Step 6Measure the actual length of your

"rectangle" as accurately as you can using your ruler.

Divide by the radius (5 cm) to get an approximation for π

Put your answer here:“”RACTANGLE WIDTH” DIVIDE BY 5 CM.

=  π15.7 3.14

2) PYTHAGOREAN THEOREM

Pythagoras was a Greek mathematician and a philosopher, but was best known for his Pythagorean Theorem.

He was born around 572 B.C. on the island of Samos.

For about 22 years, Pythagoras spent time traveling though Egypt and Babylonia to educate himself.

At about 530 B.C., he settled in a Greek town in southern Italy called Crotona.

Pythagoras formed a brotherhood that was an exclusive society devoted to moral, political and social life. This society was known as Pythagorean

INFORMATIONThe sum of the areas of the two squares

on the legs (a and b) equals the area of the square on the hypotenuse (c).

In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle.

It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The theorem can be written as an equation relating the lengths of the sides a, b and c, often called the "Pythagorean equation":

DEFINITIONThe sum of the squares of each leg

of a right angled triangle equals to the square of the hypotenuse

a² + b² = c²

MANY PROOFS OF PYTHAGOREN THEOREMProof using similar trianglesEuclid's proofProofs by dissection and rearrangement

Einstein's proof by dissection without rearrangement

Algebraic proofsProof using differentials

CRYPTOGRAPHY

Cryptography is the practice and study of techniques for secure communication in the presence of third parties called adversaries.

Modern cryptography exists at the intersection of the disciplines of mathematics, computer science, and electrical engineering. Applications of cryptography include ATM cards,

computer passwords, and electronic commerce

Modern cryptography is heavily based on mathematical theory and computer science practice; cryptographic algorithms are designed around computational hardness assumptions, making such algorithms hard to break in practice by any adversary

ENCRYPTION = It is the process of converting ordinary information (called plaintext) into unintelligible text (called ciphertext).

DECRYPTION = It is the process of converting unintelligible text (called ciphertext) into ordinary information (called plaintext).

TYPES OF CRYPTOSYSTEMS1) SYMMETRIC SYSTEM Symmetric-key cryptography, where a

single key is used for encryption and decryption

Symmetric-key cryptography refers to encryption methods in which both the sender and receiver share the same key

2) ASYMMETRIC SYSTEM Asymmetric systems use a public-key

cryptosystems, the public key may be freely distributed, while its paired private key must remain secret. The public key is used for encryption, while the private or secret key is used for decryption.

Use of asymmetric systems enhances the security of communication. Examples of asymmetric systems include RSA and ECC

CIPHER = It is a pair of algorithms that create the encryption and the reversing decryption.

The detailed operation of a cipher is controlled both by the algorithm and in each instance by a "key"

TYPES OF CIPHERS1] TRANSPOSITION CIPHERS = which

rearrange the order of letters in a message.

E.X= Plain text - 'hello world’. Cipher text - 'ehlol owrdl' 2] SUBSTITUTION CIPHERS = which

systematically replace letters or groups of letters with other letters or groups of letters

E.X= Plain text - 'fly at once' Cipher text - 'gmz bu podf'

3] CAESAR CIPHER in which each letter in the

plaintext was replaced by a letter some fixed number of positions further down the alphabet.

Alphabet shift ciphers are believed to have been used by Julius Caesar over 2,000 years ago.[5] This is an example with k=3. In other words, the letters in the alphabet are shifted three in one direction to encrypt and three in the other direction to decrypt.

HISTORY OF CRYPTOGRAPHY

German Lorenz cipher machine, used in World War II to encrypt very-high-level general staff messages