Number Theory - Lesson 1 - Introduction to Number Theory
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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
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Number Theory - Lesson 1 - Introduction to Number Theory
- 1. Discrete Structure I Number Theory FOR-IAN V. SANDOVAL
- 2. INTRODUCTION TO NUMBER THEORY
- 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.
- 10. NUMBER THEORY Have you ever thought about why 1 is “one”, 2 is “two”, 3 is “three”…..?
- 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.
- 12. NUMBER THEORY
- 13. NUMBER THEORY
- 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.