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
  • Complex Numbers (C): including imaginary numbers, e.g. 3 + 4i

• Basic Operations:

  • Addition: a + b
  • Subtraction: a - b
  • Multiplication: a × b or a · b
  • Division: a ÷ b or a / b
  • Exponentiation: a^b
  • Roots: √a, ∛a, ...

Algebra

• Variables and Expressions:

  • Variables: letters or symbols representing unknown values, e.g. x, y
  • Constants: numbers, e.g. 2, 5
  • Algebraic Expressions: combinations of variables and constants, e.g. 2x + 3, x^2 - 4

• Equations and Inequalities:

  • Linear Equations: ax + by = c, where a, b, and c are constants
  • Quadratic Equations: ax^2 + bx + c = 0, where a, b, and c are constants
  • Inequalities: statements involving >, <, ≥, or ≤, e.g. 2x + 3 > 5

Geometry

• Points, Lines, and Planes:

  • Points: locations in space, represented by coordinates (x, y, z)
  • Lines: sets of points extending infinitely in two directions
  • Planes: flat surfaces extending infinitely in all directions

• Angles and Measurements:

  • Angles: formed by two rays sharing a common endpoint, measured in degrees
  • Perimeter: distance around a shape, e.g. triangle, rectangle
  • Area: size of a shape, e.g. triangle, circle

Calculus

• Limits:

  • Concept of a limit: approaching a value, but not necessarily reaching it
  • Rules of limits: sum, product, chain rule, etc.

• Derivatives:

  • Definition: rate of change of a function with respect to a variable
  • Rules of differentiation: power rule, product rule, quotient rule, etc.

• Integrals:

  • Definition: accumulation of a function over a given interval
  • Rules of integration: substitution, integration by parts, etc.

Number Systems

• Natural Numbers (N) include numbers from 1 to infinity
• Whole Numbers (W) include 0 and all positive integers
• Integers (Z) include all positive and negative whole numbers, including 0
• Rational Numbers (Q) are fractions, such as 3/4 or 22/7
• Real Numbers (R) include decimals and irrationals, such as π or e
• Complex Numbers (C) include imaginary numbers, such as 3 + 4i

Basic Operations

• Addition is represented by the symbol '+'
• Subtraction is represented by the symbol '-'
• Multiplication can be represented by '×' or '·'
• Division can be represented by '÷' or '/'
• Exponentiation is represented by '^'
• Roots can be represented by '√' or '∛'

Algebra

Variables and Expressions

• Variables are letters or symbols representing unknown values
• Constants are numbers, such as 2 or 5
• Algebraic Expressions combine variables and constants, such as 2x + 3 or x^2 - 4

Equations and Inequalities

• Linear Equations are of the form ax + by = c, where a, b, and c are constants
• Quadratic Equations are of the form ax^2 + bx + c = 0, where a, b, and c are constants
• Inequalities involve statements with >, <, ≥, or ≤

Geometry

Points, Lines, and Planes

• Points are locations in space, represented by coordinates (x, y, z)
• Lines are sets of points extending infinitely in two directions
• Planes are flat surfaces extending infinitely in all directions

Angles and Measurements

• Angles are formed by two rays sharing a common endpoint, measured in degrees
• Perimeter is the distance around a shape
• Area is the size of a shape

Calculus

Limits

• A limit is a value approached but not necessarily reached
• There are rules of limits, including sum, product, and chain rule

Derivatives

• A derivative is the rate of change of a function with respect to a variable
• There are rules of differentiation, including power rule, product rule, and quotient rule

Integrals

• An integral is the accumulation of a function over a given interval
• There are rules of integration, including substitution and integration by parts