Number Theory Basics

Numbers

Number Theory

• Definition: Branch of mathematics dealing with integers and their properties.
• Types of Numbers:
  • Natural Numbers: Positive integers (1, 2, 3, ...).
  • Whole Numbers: Natural numbers plus zero (0, 1, 2, ...).
  • Integers: Whole numbers and their negatives (..., -2, -1, 0, 1, 2, ...).
  • Rational Numbers: Numbers that can be expressed as a fraction (p/q, where p and q are integers, q ≠ 0).
  • Irrational Numbers: Numbers that cannot be expressed as a simple fraction (e.g., √2, π).
  • Real Numbers: All rational and irrational numbers.

Arithmetic Operations

• Basic Operations:
  • Addition (+): Combining quantities.
  • Subtraction (−): Finding the difference between quantities.
  • Multiplication (×): Repeated addition of a number.
  • Division (÷): Splitting a quantity into equal parts.
• Properties:
  • Commutative Property: a + b = b + a, a × b = b × a.
  • Associative Property: (a + b) + c = a + (b + c), (a × b) × c = a × (b × c).
  • Distributive Property: a × (b + c) = a × b + a × c.

Data Representation

• Numerical Systems:
  • Binary: Base-2, uses digits 0 and 1.
  • Decimal: Base-10, uses digits 0-9.
  • Octal: Base-8, uses digits 0-7.
  • Hexadecimal: Base-16, uses digits 0-9 and letters A-F.
• Representation of Numbers:
  • Integer Representation: Stored in binary format, using fixed bits.
  • Floating Point Representation: Used for real numbers, allows representation of very large or small values.

Mathematical Proofs

• Definition: Logical argument establishing the truth of a mathematical statement.
• Types of Proofs:
  • Direct Proof: Derived from established facts and definitions.
  • Indirect Proof (Contradiction): Assumes the opposite to show a contradiction.
  • Mathematical Induction: Proves a statement for all natural numbers by first proving it for a base case and then proving it for n + 1 assuming it holds for n.

Prime Numbers

• Definition: Natural numbers greater than 1 that have no positive divisors other than 1 and themselves.
• Properties:
  • The smallest prime number is 2 (the only even prime).
  • All other even numbers are not prime.
  • Fundamental Theorem of Arithmetic: Every integer greater than 1 can be uniquely factored into prime numbers.
• Primality Testing: Algorithms to determine if a number is prime (e.g., Sieve of Eratosthenes, trial division, Miller-Rabin test).
• Applications: Cryptography, computer algorithms, and number patterns.

Number Theory

• Branch of mathematics focused on integers and their properties.
• Natural Numbers: Positive integers starting from 1 (1, 2, 3,...).
• Whole Numbers: Natural numbers including zero (0, 1, 2,...).
• Integers: Comprise whole numbers and their negative counterparts (..., -2, -1, 0, 1, 2,...).
• Rational Numbers: Numbers expressible as a fraction of two integers (p/q; q ≠ 0).
• Irrational Numbers: Cannot be written as simple fractions, examples include √2 and π.
• Real Numbers: Encompass all rational and irrational numbers.

Arithmetic Operations

• Basic Operations: Include addition, subtraction, multiplication, and division.
• Addition (+): Combines quantities.
• Subtraction (−): Calculates the difference between quantities.
• Multiplication (×): Represents repeated addition of a number.
• Division (÷): Divides a quantity into equal parts.
• Commutative Property: The order of addition or multiplication does not affect the result (a + b = b + a, a × b = b × a).
• Associative Property: Grouping of numbers does not change the result for addition or multiplication ((a + b) + c = a + (b + c), (a × b) × c = a × (b × c)).
• Distributive Property: Describes distributing multiplication over addition (a × (b + c) = a × b + a × c).

Data Representation

• Numerical Systems:
  • Binary: Base-2 system, comprising digits 0 and 1.
  • Decimal: Base-10 system, using digits from 0 to 9.
  • Octal: Base-8 system, with digits from 0 to 7.
  • Hexadecimal: Base-16 system, including digits 0-9 and letters A-F.
• Integer Representation: Stored in binary format using a fixed number of bits.
• Floating Point Representation: Used for real numbers to represent very large or small values efficiently.

Mathematical Proofs

• Definition: Logical arguments that verify the truth of mathematical statements.
• Types of Proofs:
  • Direct Proof: Constructs through established facts and definitions.
  • Indirect Proof (Contradiction): Assumes the opposite of what is to be proved to find a contradiction.
  • Mathematical Induction: Proves a statement universally by verifying a base case and then proving for n + 1 based on n.

Prime Numbers

• Definition: Natural numbers greater than 1 with no positive divisors besides 1 and themselves.
• Properties:
  • 2 is the smallest prime number and the only even prime number.
  • All other even numbers are not prime.
  • Fundamental Theorem of Arithmetic states each integer greater than 1 can uniquely be factored into prime numbers.
• Primality Testing: Involves algorithms like the Sieve of Eratosthenes, trial division, and the Miller-Rabin test to identify prime numbers.
• Applications: Vital in cryptography, computer algorithms, and identifying number patterns.