Math Class: Rational Numbers and Exponents

Numbers and Operations

• Rational Numbers: A rational number is a number that can be expressed as the ratio of two integers, i.e., p/q, where p and q are integers and q ≠ 0.
  • Examples: 3/4, 22/7, 1/2
  • Properties:
    • Commutative property: a/b = b/a
    • Associative property: (a/b) × (c/d) = a/c × b/d
    • Distributive property: a/b + c/d = (a+c)/(b+d)
• Exponents and Powers: An exponent is a small number that is raised to a power to indicate the number of times a base number should be multiplied by itself.
  • Laws of Exponents:
    • Product of powers: a^m × a^n = a^(m+n)
    • Power of a power: (a^m)^n = a^(m×n)
    • Quotient of powers: a^m ÷ a^n = a^(m-n)
• Squares and Square Roots: A square of a number is the result of multiplying the number by itself.
  • Properties:
    • Square of a sum: (a+b)^2 = a^2 + 2ab + b^2
    • Square of a difference: (a-b)^2 = a^2 - 2ab + b^2
    • Square root of a number: √a = b, where b^2 = a

Algebra

• Linear Equations: An equation in which the highest power of the variable is 1.
  • General form: ax + by = c, where a, b, and c are constants
  • Solution: x = (c - by) / a
• Quadratic Equations: An equation in which the highest power of the variable is 2.
  • General form: ax^2 + bx + c = 0, where a, b, and c are constants
  • Solution: x = (-b ± √(b^2 - 4ac)) / 2a

Geometry

• Understanding Shapes: A shape is a self-contained area with a fixed boundary.
  • Types of shapes:
    • 2D shapes: triangles, quadrilaterals, polygons, circles
    • 3D shapes: cubes, cuboids, spheres, cones
• Properties of Shapes: Properties of shapes include:
  • Angles: acute, obtuse, right, straight, reflex
  • Sides: number of sides, length of sides
  • Vertices: number of vertices, properties of vertices

Mensuration

• Perimeter and Area: Perimeter is the distance around a shape, while area is the amount of space inside a shape.
  • Formulas:
    • Perimeter of a rectangle: 2(l + b)
    • Area of a rectangle: l × b
    • Perimeter of a triangle: a + b + c
    • Area of a triangle: (b × h) / 2
• Volume and Capacity: Volume is the amount of space inside a 3D shape, while capacity is the amount of liquid a container can hold.
  • Formulas:
    • Volume of a cube: s^3
    • Volume of a cuboid: l × b × h
    • Capacity of a container: volume of liquid / volume of container

Numbers and Operations

• Rational Numbers: A number that can be expressed as the ratio of two integers, p/q, where p and q are integers and q ≠ 0.
• Examples of rational numbers: 3/4, 22/7, 1/2
• Properties of rational numbers:
  • Commutative property: a/b = b/a
  • Associative property: (a/b) × (c/d) = a/c × b/d
  • Distributive property: a/b + c/d = (a+c)/(b+d)

Exponents and Powers

• An exponent is a small number that is raised to a power to indicate the number of times a base number should be multiplied by itself.
• Laws of Exponents:
  • Product of powers: a^m × a^n = a^(m+n)
  • Power of a power: (a^m)^n = a^(m×n)
  • Quotient of powers: a^m ÷ a^n = a^(m-n)

Squares and Square Roots

• A square of a number is the result of multiplying the number by itself.
• Properties of squares:
  • Square of a sum: (a+b)^2 = a^2 + 2ab + b^2
  • Square of a difference: (a-b)^2 = a^2 - 2ab + b^2
• Square root of a number: √a = b, where b^2 = a

Algebra

• Linear Equations: An equation in which the highest power of the variable is 1.
• General form of linear equations: ax + by = c, where a, b, and c are constants
• Solution of linear equations: x = (c - by) / a

Quadratic Equations

• Quadratic Equations: An equation in which the highest power of the variable is 2.
• General form of quadratic equations: ax^2 + bx + c = 0, where a, b, and c are constants
• Solution of quadratic equations: x = (-b ± √(b^2 - 4ac)) / 2a

Geometry

• Understanding Shapes: A shape is a self-contained area with a fixed boundary.
• Types of shapes:
  • 2D shapes: triangles, quadrilaterals, polygons, circles
  • 3D shapes: cubes, cuboids, spheres, cones
• Properties of Shapes: Properties of shapes include:
  • Angles: acute, obtuse, right, straight, reflex
  • Sides: number of sides, length of sides
  • Vertices: number of vertices, properties of vertices

Mensuration

• Perimeter and Area: Perimeter is the distance around a shape, while area is the amount of space inside a shape.
• Formulas for perimeter and area:
  • Perimeter of a rectangle: 2(l + b)
  • Area of a rectangle: l × b
  • Perimeter of a triangle: a + b + c
  • Area of a triangle: (b × h) / 2

Volume and Capacity

• Volume and Capacity: Volume is the amount of space inside a 3D shape, while capacity is the amount of liquid a container can hold.
• Formulas for volume and capacity:
  • Volume of a cube: s^3
  • Volume of a cuboid: l × b × h
  • Capacity of a container: volume of liquid / volume of container