Math Basics: Number Systems and Algebra

Number Systems

• Natural Numbers: 1, 2, 3, ... (counting numbers)
• Whole Numbers: 0, 1, 2, 3, ... (including zero)
• Integers: ..., -3, -2, -1, 0, 1, 2, 3, ... (including negative numbers)
• Rational Numbers: fractions, e.g. 3/4, 22/7
• Irrational Numbers: non-repeating decimals, e.g. π, e
• Real Numbers: all rational and irrational numbers

Algebra

• Variables: letters or symbols representing unknown values
• Constants: numbers
• Expressions: combinations of variables, constants, and operations
• Equations: statements with equal signs, e.g. 2x + 3 = 5
• Inequalities: statements with greater than, less than, or equal to signs

Geometry

• Points: locations in space, represented by coordinates (x, y)
• Lines: sets of points extending infinitely in two directions
• Rays: sets of points extending infinitely in one direction
• Angles: formed by two rays sharing a common endpoint
• Shapes: polygons, circles, triangles, etc.

Calculus

• Limits: the behavior of a function as the input approaches a certain value
• Derivatives: rates of change of functions with respect to the input
• Integrals: accumulation of rates of change over a given interval
• Applications: optimization, physics, engineering, economics

Statistics

• Descriptive Statistics: summarizing and describing data
  • Mean: average value
  • Median: middle value
  • Mode: most frequent value
• Inferential Statistics: making conclusions based on sample data
  • Hypothesis Testing: testing claims about a population
  • Confidence Intervals: estimating population parameters

Other Topics

• Probability: the study of chance and uncertainty
• Graph Theory: the study of graphs and their properties
• Trigonometry: the study of triangles and their relationships

Number Systems

• Natural numbers are counting numbers starting from 1
• Whole numbers include 0 and all natural numbers
• Integers include negative numbers, 0, and all positive whole numbers
• Rational numbers are fractions that can be expressed as the ratio of two integers
• Irrational numbers are non-repeating decimals that cannot be expressed as a ratio of integers
• Real numbers include all rational and irrational numbers

Algebra

• Variables are letters or symbols that represent unknown values or quantities
• Constants are numbers that do not change value
• Expressions are combinations of variables, constants, and mathematical operations
• Equations are statements with equal signs that express the equality of two expressions
• Inequalities are statements with greater than, less than, or equal to signs that compare two expressions

Geometry

• Points are locations in space that can be represented by coordinates (x, y)
• Lines are sets of points that extend infinitely in two directions
• Rays are sets of points that extend infinitely in one direction
• Angles are formed by two rays sharing a common endpoint called the vertex
• Shapes include polygons, circles, triangles, and other geometric figures

Calculus

• Limits describe the behavior of a function as the input approaches a certain value
• Derivatives measure the rate of change of a function with respect to its input
• Integrals calculate the accumulation of rates of change over a given interval
• Calculus has many applications in optimization, physics, engineering, and economics

Statistics

• Descriptive statistics summarize and describe data
• Mean is the average value of a dataset
• Median is the middle value of a dataset when it is arranged in order
• Mode is the most frequent value in a dataset
• Inferential statistics make conclusions about a population based on sample data
• Hypothesis testing involves testing claims about a population
• Confidence intervals estimate population parameters based on sample data

Other Topics

• Probability is the study of chance and uncertainty
• Graph theory is the study of graphs and their properties
• Trigonometry is the study of triangles and their relationships