Number Theory Concepts

Study Notes

Definition: Number theory is the branch of mathematics that deals with the properties and behavior of integers and other whole numbers.
Key concepts:
- Divisibility: a | b if a divides b exactly without leaving a remainder
- Prime numbers: positive integers greater than 1 that are divisible only by 1 and themselves
- Composite numbers: positive integers greater than 1 that are not prime
- Greatest Common Divisor (GCD): the largest number that divides both a and b exactly
- Euclidean algorithm: a method for finding the GCD of two numbers
Theorems:
- Fundamental Theorem of Arithmetic: every positive integer can be expressed as a product of prime numbers in a unique way
- Euclid's Lemma: if p is prime and p | ab, then p | a or p | b

Definition: An algebraic equation is an equation that involves variables and constants, and can be expressed using only addition, subtraction, multiplication, and division, and roots (such as square roots or cube roots).
Types of equations:
- Linear equations: equations of the form ax + by = c, where a, b, and c are constants
- Quadratic equations: equations of the form ax^2 + bx + c = 0, where a, b, and c are constants
- Polynomial equations: equations of the form a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0 = 0, where a_n, ..., a_1, a_0 are constants
Methods for solving equations:
- Factoring: expressing an equation as a product of simpler equations
- Quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a, for solving quadratic equations
- Synthetic division: a method for dividing a polynomial by another polynomial
Applications:
- Solving systems of equations
- Finding roots of polynomials
- Modeling real-world problems

Branch of mathematics focused on integers and their properties
Divisibility: Indicated as a | b; means a divides b without remainder
Prime Numbers: Greater than 1, only divisible by 1 and itself; examples include 2, 3, 5, 7
Composite Numbers: Greater than 1 and not prime; includes numbers like 4, 6, 8
Greatest Common Divisor (GCD): Largest integer that divides two numbers without leaving a remainder
Euclidean Algorithm: A systematic method for calculating the GCD of two integers
Fundamental Theorem of Arithmetic: States every positive integer can be uniquely expressed as a product of prime numbers
Euclid's Lemma: If a prime number p divides the product of two integers ab, then p must divide at least one of a or b

Definition: Involves variables and constants, expressible through basic arithmetic operations and roots
Types:
- Linear Equations: Form ax + by = c; a, b, and c are constants, represents a straight line on a graph
- Quadratic Equations: Form ax² + bx + c = 0; represents a parabolic curve, solutions can be found using the quadratic formula
- Polynomial Equations: General form a_n x^n + a_(n-1) x^(n-1) +...+ a_1 x + a_0 = 0; includes multiple degrees of x
Methods for Solving Equations:
- Factoring: Process of breaking down an expression into simpler components that multiply to the original equation
- Quadratic Formula: x = (-b ± √(b² - 4ac)) / 2a; used for finding roots of quadratic equations
- Synthetic Division: Simplifies the process of dividing a polynomial by a linear factor
Applications:
- Solving systems of equations to find variable values that satisfy multiple conditions
- Finding roots of polynomials, relevant in calculus and function analysis
- Modeling real-world problems in fields like physics, finance, and engineering through algebraic relationships