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.

Explore the fundamental concepts of number theory, including the different types of numbers such as natural, whole, integers, rational, irrational, and real numbers. Additionally, discover the basic arithmetic operations and their properties that form the core of mathematical understanding.