Mathematics Basics: Arithmetic, Geometry, Calculus

Arithmetic

• Definition: The branch of mathematics dealing with basic operations: addition, subtraction, multiplication, and division.
• Operations:
  • Addition (+)
  • Subtraction (−)
  • Multiplication (×)
  • Division (÷)
• Order of Operations: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right) – often abbreviated as PEMDAS.

Geometry

• Definition: The study of shapes, sizes, and properties of space.
• Key Concepts:
  • Points, Lines, Angles: Basic elements of geometry.
  • Shapes:
    • 2D Shapes: Circles, triangles, squares, rectangles.
    • 3D Shapes: Cubes, spheres, cylinders, cones.
• Theorems:
  • Pythagorean Theorem: a² + b² = c² for right triangles.
  • Properties of triangles: Sum of interior angles equals 180°.

Calculus

• Definition: The study of change and motion; divided into differential and integral calculus.
• Key Concepts:
  • Differential Calculus: Focuses on rates of change (derivatives).
  • Integral Calculus: Focuses on accumulation of quantities (integrals).
• Fundamental Theorem of Calculus: Links differentiation and integration, stating that integration can be reversed by differentiation.

Fractions

• Definition: A way to represent a part of a whole, expressed as a/b where 'a' is the numerator and 'b' is the denominator.
• Types:
  • Proper fractions: Numerator < Denominator.
  • Improper fractions: Numerator ≥ Denominator.
  • Mixed numbers: A whole number combined with a proper fraction.
• Operations:
  • Addition/Subtraction: Common denominators needed.
  • Multiplication: Multiply numerators and denominators.
  • Division: Invert the second fraction and multiply.

Perimeter

• Definition: The total distance around a shape.
• Formulas:
  • Rectangle: P = 2(l + w)
  • Square: P = 4s (where s = side length)
  • Triangle: P = a + b + c (sum of all sides)
  • Circle: P (Circumference) = 2πr (where r = radius)

Area

• Definition: The amount of space inside a shape.
• Formulas:
  • Rectangle: A = l × w
  • Square: A = s²
  • Triangle: A = (1/2) × base × height
  • Circle: A = πr² (where r = radius)
• Units: Typically measured in square units (e.g., cm², m²).

Arithmetic

• Focuses on fundamental operations: addition, subtraction, multiplication, division.
• Key operations are represented by symbols:
  • Addition (+), Subtraction (−), Multiplication (×), Division (÷).
• Order of operations is essential for accurate calculations, summarized as PEMDAS: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).

Geometry

• Investigates shapes, sizes, and the properties of space.
• Basic elements include points, lines, and angles, essential for understanding geometry.
• Shapes categorized as:
  • 2D Shapes: Circles, triangles, squares, rectangles.
  • 3D Shapes: Cubes, spheres, cylinders, cones.
• Important theorems include:
  • Pythagorean Theorem: a² + b² = c² applies to right triangles.
  • Triangle property: Sum of interior angles is always 180°.

Calculus

• Studies change and motion, split into differential and integral calculus.
• Differential Calculus: Analyzes rates of change through derivatives.
• Integral Calculus: Concerned with accumulation of quantities represented by integrals.
• The Fundamental Theorem of Calculus bridges differentiation and integration, indicating they can reverse each other.

Fractions

• Represents parts of a whole in the format a/b, where 'a' is the numerator and 'b' is the denominator.
• Types of fractions include:
  • Proper fractions: Numerator less than denominator.
  • Improper fractions: Numerator equal to or greater than denominator.
  • Mixed numbers: A whole number plus a proper fraction.
• Operations with fractions require:
  • Addition/Subtraction: Must have a common denominator.
  • Multiplication: Multiply numerators and denominators directly.
  • Division: Invert the second fraction before multiplication.

Perimeter

• Defined as the total distance around a geometric shape.
• Perimeter formulas by shape:
  • Rectangle: P = 2(l + w)
  • Square: P = 4s (s = side length)
  • Triangle: P = a + b + c (sum of all three sides)
  • Circle: P (Circumference) = 2πr (r = radius).

Area

• Represents the total space contained within a shape.
• Area formulas by shape:
  • Rectangle: A = l × w
  • Square: A = s²
  • Triangle: A = (1/2) × base × height
  • Circle: A = πr² (r = radius).
• Area is measured in square units, commonly in cm² or m².