This Presentation about Numbers.. and their Theory

1. SWARRNIM SCIENCE
COLLEGE, SSIU
PRESENTED BY:
AKASH SHRIVAS
1753003004
SUBMITTED TO:
Prof. Richa Raval

2. What is number?
Have you taught that why 1 is one, 2 is two, 3 is three….?
What is NUMBER THEORY?
INTRODUCTION
Fundamental theorem of arithmetic
Greatest common divisor
Least common multiple
Modular arithmetic
APPLICATIONS
Cryptography
Computer arithmetic

3. WHAT IS NUMBER?
An arithmetical value, expressed by a
word, symbol, and figure, and used in
counting and making calculation.

5. The number we write are
made up of algorithms,
(1,2,3,4,5,…)called arabic
algorithms; to distinguish
them from the roman
algorithms.
But what is the actual
logic that exist in the
arabic algorithms?

6. There are angles which
made these number……

8. WHAT IS NUMBER THEORY?
The branch of mathematics that deal with the properties
and relationships of numbers, especially the positive
number

9. INTRODUCTION
NUMBER THEORY is the new version of
“ARITHMETIC”.
Number theory is all about integer and their
properties.
NUMBER THOERY is familiar include primes,
composites, multiples, factors, number properties,
etc.
The basic principal of NUMBER THEORY:
Greatest common divisor
Least common multiples &
Modular arithmetic

10. THE FUNDAMENTAL THEOREM OF ARITHMETIC
 Every positive integer can be written uniquely as
the product of primes, where the prime factor are
written in order of increasing size.

11. PRIME
•A positive integer p greater than 1 is called
prime. If the only positive factors of p are 1 and
p.
COMPOSITE
•A positive integer that is greater than 1 and is not
prime is called composite.

12. EXAMPLE
 14 = 7 * 2
 15 = 3 * 5
 48 = 2*2*2*2*3 = 24*3
 100 = 2*2*5*5 = 22*52

13. GREATEST COMMON DIVISOR
Let a & b be two integer, not both zero
The largest integer d such that d|a & d|a is called the
Greatest common divisor of a & b.
The greatest common divisor of a & b is denoted by
gcd(a,b).
Example:
What is gcd(48,72)?
•The positive common integer of 48 and 72 are
1,2,3,4,6,8,12,16 and 24, so gcd(48,72)=24

14. LEAST COMMON MULTIPLE
The least common multiple of the positive integer a
&b is the smallest positive integer that is divisible by
both a & b.
We donate least common multiple of a & b by
lcm(a,b).
Example:
•Lcm(3,7)=21
•Lcm(4,6)=12
•Lcm(5,10)=10

15. MODULAR ARITHMETIC
Let a be an integer and m be a positive
integer.
We donate by a mod m the reminder when a
is divided by m.
Example:
•9 mod 4=1
•9 mod 3=0
•6 mod 3=2

16. APPLICATION
Public key cryptography
such as RSA algorithm
Computer arithmetic.
Error correcting
code

17. CRYPTOGRAPHY
 “IT IS DIFFICULT (EVEN FOR A COMPUTER) TO
FACTOR A BIG NUMBER INTO TWO PRIMES”.
13 × 19 = 247
247
13 19

18. COMPUTER ARITHMETIC
HELLO
0805121215 79

19. Secret key = 3897647858937
HELLO
0805121215 79
Secret key
3897647858937×
ENCRYPTED MESSAGE

20. ENCRYPTED MESSAGE
SECRET KEY
080512121579
=
=
HELLO
Msg= Secret key × HELLO
Msg2=Secret key × HOW ARE YOU?
Secret key = GCD(Msg,Msg2)

21. Carl friedrich Gauss, a great mathematician,
once remarked
that “ Mathematics is the queen of science”
but “NUMBER THEORY is queen of
mathematics”.

22. Regards:
Akash shrivas