Number Theory and Algebra Concepts

Number Theory

• Definition: Study of integers and their properties.
• Key Concepts:
  • Prime Numbers: Numbers greater than 1 that have no positive divisors other than 1 and themselves.
  • Composite Numbers: Numbers that have more than two positive divisors.
  • Greatest Common Divisor (GCD): Largest integer that divides two or more integers without leaving a remainder.
  • Least Common Multiple (LCM): Smallest multiple that is exactly divisible by two or more integers.
  • Divisibility Rules: Guidelines to determine if one number is divisible by another (e.g., even numbers are divisible by 2).
  • Modular Arithmetic: A system of arithmetic for integers, where numbers wrap around upon reaching a certain value (modulus).

Algebra

• Definition: Branch of mathematics dealing with symbols and the rules for manipulating those symbols.
• Key Concepts:
  • Variables: Symbols used to represent numbers in expressions or equations.
  • Expressions: Combinations of variables, numbers, and operations (e.g., 3x + 2).
  • Equations: Mathematical statements that assert the equality of two expressions (e.g., 2x + 3 = 7).
  • Functions: Relationships where each input has a single output (e.g., f(x) = 2x + 3).
  • Quadratic Equations: Polynomial equations of the form ax² + bx + c = 0; solutions can be found using factoring, completing the square, or the quadratic formula.
  • Inequalities: Expressions that show the relationship of one quantity being greater than or less than another (e.g., x + 2 > 5).

Triangles

• Definition: A polygon with three edges and three vertices.
• Types of Triangles:
  • By Sides:
    • Equilateral: All sides are equal; all angles are 60°.
    • Isosceles: Two sides are equal; angles opposite the equal sides are equal.
    • Scalene: All sides and angles are different.
  • By Angles:
    • Acute: All angles are less than 90°.
    • Right: One angle is exactly 90°.
    • Obtuse: One angle is greater than 90°.
• Key Properties:
  • Sum of Angles: The sum of the interior angles of a triangle is always 180°.
  • Pythagorean Theorem: In a right triangle, a² + b² = c² (where c is the hypotenuse).
  • Area Calculation: Area = 1/2 × base × height.
  • Perimeter Calculation: Perimeter = sum of the lengths of all sides.

Number Theory

• Study of integers and their properties.
• Prime Numbers: Defined as numbers greater than 1 with no positive divisors other than 1 and themselves.
• Composite Numbers: Defined as numbers that possess more than two positive divisors.
• Greatest Common Divisor (GCD): Identified as the largest integer that divides two or more integers without a remainder, important for simplifying fractions.
• Least Common Multiple (LCM): Recognized as the smallest multiple that is divisible by two or more integers, often used in finding common denominators.
• Divisibility Rules: Set of guidelines to determine whether one number can be evenly divided by another, e.g., all even numbers are divisible by 2.
• Modular Arithmetic: System of arithmetic where numbers wrap around after reaching a specified value or modulus, essential in computer science and cryptography.

Algebra

• Branch of mathematics focused on symbols and their manipulation.
• Variables: Symbols representing unknown numbers in mathematical expressions or equations.
• Expressions: Combinations of variables, constants, and operations, such as 3x + 2, which can be simplified or evaluated.
• Equations: Mathematical statements asserting the equality of two expressions, e.g., 2x + 3 = 7, requiring solutions to find unknowns.
• Functions: Relationships connecting each input to a single output, denoted as f(x) = 2x + 3, illustrating dependency of one variable on another.
• Quadratic Equations: Polynomial equations formatted as ax² + bx + c = 0, solvable by factoring, completing the square, or employing the quadratic formula.
• Inequalities: Mathematical expressions indicating the comparative size of quantities, such as x + 2 > 5, which identifies value ranges rather than exact solutions.

Triangles

• Defined as a polygon consisting of three edges and three vertices.
• Types by Sides:
  • Equilateral: All sides equal, and all angles measure 60°.
  • Isosceles: Two equal sides with equal opposite angles.
  • Scalene: No equal sides or angles, all differing from one another.
• Types by Angles:
  • Acute: All angles are less than 90°.
  • Right: Contains one angle that is exactly 90°.
  • Obtuse: Features one angle greater than 90°.
• Key Properties:
  • Sum of Angles: Interior angles always sum up to 180°.
  • Pythagorean Theorem: In right triangles, the relation a² + b² = c² holds, where c denotes the hypotenuse.
  • Area Calculation: Determined using the formula Area = 1/2 × base × height.
  • Perimeter Calculation: Found by summing the lengths of all three sides.