Numbers and Operations Basics

Numbers and Operations

• Number Systems:
  • Natural Numbers (N): 1, 2, 3, ...
  • Whole Numbers (W): 0, 1, 2, 3, ...
  • Integers (Z): ..., -3, -2, -1, 0, 1, 2, 3, ...
  • Rational Numbers (Q): fractions, e.g., 3/4, 22/7
  • Real Numbers (R): including decimals and irrationals, e.g., π, e
• Operations:
  • Addition (+) and Subtraction (-)
  • Multiplication (×) and Division (÷)
  • Order of Operations (PEMDAS/BODMAS):
    • Parentheses
    • Exponents
    • Multiplication and Division
    • Addition and Subtraction

Algebra

• Variables and Expressions:
  • Variables: letters or symbols representing unknowns
  • Expressions: combinations of variables, constants, and operations
• Equations and Inequalities:
  • Equations: statements with equal signs (=)
  • Inequalities: statements with <, >, ≤, ≥, ≠
• Graphing:
  • Cartesian Coordinates (x, y)
  • Linear Equations: y = mx + b
  • Quadratic Equations: y = ax^2 + bx + c

Geometry

• Points, Lines, and Planes:
  • Points: locations in space
  • Lines: sets of points extending infinitely
  • Planes: flat surfaces
• Angles and Shapes:
  • Angles: formed by two rays sharing a vertex
  • Triangles: three-sided shapes
  • Quadrilaterals: four-sided shapes
• Measurement:
  • Perimeter: distance around a shape
  • Area: size of a shape's interior

Calculus

• Limits:
  • Concept of limits: approach to a value
  • Notation: lim x→a f(x) = L
• Derivatives:
  • Rate of change
  • Notation: f'(x) or (d/dx)f(x)
• Integrals:
  • Accumulation of quantities
  • Notation: ∫f(x)dx or ∫[a, b] f(x)dx

Numbers and Operations

• Natural Numbers (N) start from 1 and go on indefinitely.
• Whole Numbers (W) include 0 and all natural numbers.
• Integers (Z) include all whole numbers and their negative counterparts.
• Rational Numbers (Q) are fractions, such as 3/4 or 22/7.
• Real Numbers (R) include decimals and irrationals, like π and e.
• Addition and subtraction are inverse operations.
• Multiplication and division are inverse operations.
• Order of Operations (PEMDAS/BODMAS) follows a specific sequence: parentheses, exponents, multiplication and division, and finally addition and subtraction.

Algebra

• Variables are letters or symbols representing unknown values.
• Expressions are combinations of variables, constants, and operations.
• Equations are statements with equal signs (=), requiring a balance between both sides.
• Inequalities are statements with <, ≤, ≥, or ≠, describing a relationship between both sides.
• Cartesian Coordinates (x, y) are used to graph points on a plane.
• Linear Equations take the form y = mx + b, where m is the slope and b is the y-intercept.
• Quadratic Equations take the form y = ax^2 + bx + c, where a, b, and c are coefficients.

Geometry

• Points are locations in space, denoted by coordinates (x, y, z).
• Lines are sets of points extending infinitely, with a unique slope and intercept.
• Planes are flat surfaces that can be defined by three points or two intersecting lines.
• Angles are formed by two rays sharing a vertex, with measures in degrees or radians.
• Triangles are three-sided shapes with unique properties, such as congruence and similarity.
• Quadrilaterals are four-sided shapes with varying properties, such as rectangles and trapezoids.
• Perimeter is the distance around a shape, while area is the size of its interior.

Calculus

• Limits describe the approach to a value, denoted by lim x→a f(x) = L.
• Derivatives measure the rate of change of a function, notated as f'(x) or (d/dx)f(x).
• Integrals calculate the accumulation of quantities, notated as ∫f(x)dx or ∫[a, b] f(x)dx.