3 maths major topics 1)finding value of pie 2)pythagorus theorem 3)Cryptography

1. TOPIC – 1) FINDING VALUE OF PIE
2) PYTHAGORAS
THEOREM
3) CRYPTOGRAPHY
DR. S&S.S GANDHY GOVERMENT ENGINEERING
COLLAGE

2. PREPARED BY
CHELANI PRASHANT
VADHER VIJAY
PATIL NIKHIL

3. 1] FINDING VALUE OF PIE
 INFORMATION ABOUT PIE
 NAME
 DEFINITION
 APPROXCIMATE VALUE
 ACTIVITY TO FIND PIE

4. INFORMATION ABOUT PIE
 The 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

5. NAME
 The 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.

6. DEFINITION
 π 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

7. APPROXIMATE VALUE
 Some approximations of
pi include:
 Integers: 3
 Fractions: Approximate
fractions include (in order
of increasing accuracy)
22/7, 333/106, 355/113,
52163/16604,
103993/33102, and
245850922/78256779

8. ACTIVITY TO FINDPIE
 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°

9. Step 2
 Divide 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:

10. Step 3
 Cut out the thirteen
sectors using the
scissors:

11. Step4
 Rearrange the 13 sectors like this (you
can glue them onto a piece of paper):

12.  Step 5
 Its 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 × radius
 And so the width is:
 Half the Circumference = π × radius
 With a radius of 5 cm, the rectangle should be:
 5 cm high
 about 5π cm wide

13. Step 6
 Measure 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.
= π
15.7 3.14

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

15. INFORMATION
 The 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":

16. DEFINITION
 The sum of the squares of each leg of a
right angled triangle equals to the
square of the hypotenuse
a² + b² = c²

17. MANY PROOFS OF PYTHAGORENTHEOREM
 Proof using similar triangles
 Euclid's proof
 Proofs by dissection and
rearrangement
 Einstein's proof by dissection without
rearrangement
 Algebraic proofs
 Proof using differentials

18. 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

19. 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).

20. TYPES OF CRYPTOSYSTEMS
1) 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

21. 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

22.  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"

23. TYPES OF CIPHERS
1] 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'

24. 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.

25. HISTORY OF CRYPTOGRAPHY
 German Lorenz cipher machine, used in World
War II to encrypt very-high-level general staff
messages