This document provides an introduction to number theory, including:
- Number theory is the study of integers and their properties
- It discusses the origins and early developments of number theory in places like Mesopotamia, India, Greece, and Alexandria
- It defines different types of numbers like natural numbers, integers, rational numbers, irrational numbers, and describes properties like prime and composite numbers
- It discusses applications of number theory like public key cryptography and error-correcting codes
3. LEARNING OBJECTIVES
• Introduce Number Theory
• Discover how arithmetic started
• Recognize the applications of Number Theory
• Compare and contrast the types of numbers
4. NUMBER THEORY
➢ Number theory or, in older usage, arithmetic is a branch of pure
mathematics devoted primarily to the study of the integers.
➢ It is sometimes called "The Queen of Mathematics“
➢ The word "arithmetic" is used by the general public to mean
"elementary calculations“ (+, -, *, /).
➢ It has also acquired other meanings in computer science, as
floating point arithmetic.
➢ Particularly in the study prime numbers as well the properties of
objects made out of the integers or defined as generalization of the
integers. .
5. NUMBER THEORY
➢ The first historical
find of an
arithmetical
nature is a
fragment of a
table: the broken
clay tablet
Plimpton 322
(Larsa,
Mesopotamia,
ca.1800 BCE)
6. NUMBER THEORY
➢ It contains a list of "Pythagorean triples", i.e., integers such that.
7. NUMBER THEORY
➢ Pythagorean mystics gave great importance to the odd and the
even.
➢ The discovery that √ 2 is irrational is credited to the early
Pythagoreans (pre-Theodorus )
➢ By revealing (in modern terms) that numbers could be irrational,
this discovery seems to have provoked the first foundational crisis
in mathematical history;
➢ Its proof sometimes credited to Hippasus
Hippasus
8. NUMBER THEORY
➢ Āryabhaṭa (476–550 CE) showed that pairs of
simultaneous agreement n ≡ a 1 mod m 1
could be solved by a method he called
pulveriser
➢ this is a procedure close to the Euclidean
algorithm, which was probably discovered
independently in India. Āryabhaṭa
➢ Āryabhaṭa seems to have had in mind
applications to astronomical calculations.
9. NUMBER THEORY
➢ lived in the third century, that is about 500
years after Euclid
Diophantus of Alexandria
➢ Six out of the thirteen books of
Diophantus's ”Arithmetica” survived in the
original Greek and four more books
survived in an Arabic translation
➢ ”Arithmetica” is a collection of worked-out
problems where the task is to find out
rational solutions to a system of polynomial
➢ equations or algebraic equations.
11. NUMBER THEORY
➢ The numbers we write are made up of algorithms, (1, 2, 3, 4, etc)
called arabic algorithms, to distinguish them from the roman
algorithms (I; II; III; IV; etc.)
➢ The Arabs popularize these algorithms, but their origin goes back
to the Phenecian merchants that used them to count and do their
commercial countability.
14. TYPES OF NUMBERS
➢ Counting Numbers
- positive whole numbers excluding zero or {1,2,3,4, 5…} also
called natural numbers
➢ Whole Numbers
- positive integers including zero or {0,1,2,3,4, 5…}.
➢ Integers
- numbers formed by the natural numbers including 0 together with
the negatives of the non-zero natural numbers or {…,-3,-2,-
1,0,1,2,3,…}
15. TYPES OF NUMBERS
➢ Rational Numbers
- numbers that can be written as fraction and whose numerator and
denominators are integers provided that the denominator is not equal
to 0
- it can also be written in decimal form as terminating decimal or as an
infinite repeating decimal
- Some examples of rational numbers are
➢ Real Numbers
- numbers compromised all rational and irrational numbers
➢ Imaginary Numbers
- the square root of negative one
- Any real number times I is an imaginary number some examples are
i,4i, -6.3i.
16. TYPES OF NUMBERS
➢ Complex Numbers
- the combination of real numbers and imaginary number (non-real
numbers) some examples are
➢ Odd Numbers
- a number when divided by 2 contains a remainder of 1.
- Mathematically, n is odd if there are exist a number k, such that
n=2k+1 where k is an integer.
➢ Even Numbers
- a number divisible by 2
- Mathematically n is even if there exist a number k, such that
n=2k where k is integer.
17. TYPES OF NUMBERS
➢ Prime Numbers or A Prime
- a natural numbers greater than 1 that has no positive divisors other
than 1 and itself, some example are 2,3,5,11.
➢ Composite Numbers
- a positive integer which has a positive divisor other than 1 or itself
- in other words any positive integer greater than 1 that is not a prime
number
- some examples are 4, 6, 8, 9, 10, etc.
➢ Perfect Numbers
- a positive integer that is equal to the sum of its proper positive
divisors, that is, the sum of its positive divisors excluding the number
itself
- Some examples are 6, 28, 496, 8128, 33550336
18. TYPES OF NUMBERS
➢ In symbols
R – real numbers
Q – rational numbers
N – natural numbers or counting numbers
W – whole numbers
Z – integers
Z – positive integers
Z – negative integers
19. NUMBER THEORY APPLICATION
➢ Number theory can be used to find out some of the important
divisibility tests, whether a given integer n is divisible by an integer
m
20. PUBLIC KEY CRYPTOGRAPHY
➢ Everybody has a key that encrypts and a separate key that
decrypts
➢ They are not interchangeable!
➢ The encryption key is made public
➢ The decryption key is kept private
➢ Public key cryptography goals
- Key generation should be relatively easy
- Encryption should be easy
- Decryption should be easy (with the right key
- Cracking should be very hard
21. PUBLIC KEY CRYPTOGRAPHY
➢ Number Theory for Digital Cash
- The whole of encryption works due to number theory. As a
result, security of transactions is ensured.
- If it were not for number theory, your money will not be safe in
your bank, information about you could be accessed by anyone.
➢ Error-Correcting Code
- is an algorithm for expressing a sequence of numbers such that
any errors which are introduced can be detected and corrected
(within certain limitations) based on the remaining numbers.
22. PUBLIC KEY CRYPTOGRAPHY
➢ Encrypting and Decrypting RSA messages
- Formula is c = me mod n
➢ Quantum computers
- A quantum computer could (in principle) factor n in reasonable
time
• This would make RSA obsolete!
• Shown (in principle) by Peter Shor in 1993
• You would need a new (quantum) encryption algorithm to
encrypt your messages
23. References
• Arefin, S. (2016). Number Theory. Retrieved from https://www.slideshare.net/SamsilArefin2/number-theory-70169905
• Aslam, A. (2016). Discrete Mathematics and Its Application. Retrieved from https://www.slideshare.net/AdilAslam4/number-
theory-in-discrete-mathematics
• Levin, O. (2019). Discrete Mathematics: An Open Introduction 3rd Edition. Colorado: School of Mathematics Science University
of Colorado.