Number Systems in Mathematics

Number Systems

• Natural Numbers: 1, 2, 3, ... (positive integers)
• Whole Numbers: 0, 1, 2, 3, ... (non-negative integers)
• Integers: ..., -3, -2, -1, 0, 1, 2, 3, ... (positive and negative whole numbers)
• Rational Numbers: fractions, e.g. 3/4, -2/3 (can be expressed as a finite decimal)
• Irrational Numbers: non-repeating, non-terminating decimals, e.g. π, e
• Real Numbers: union of rational and irrational numbers
• Complex Numbers: numbers with real and imaginary parts, e.g. 3 + 4i

Algebra

• Equations: statements with equal signs, e.g. 2x + 3 = 5
• Inequalities: statements with >, <, ≥, ≤, e.g. 2x - 3 > 2
• Functions: relations between inputs (x) and outputs (y), e.g. f(x) = 2x^2 + 1
• Graphs: visual representations of functions on a coordinate plane

Geometry

• Points: locations in space, represented by coordinates (x, y, z)
• Lines: sets of points extending infinitely in two directions
• Angles: measures of rotation between two lines or planes
• Shapes: polygons (e.g. triangles, quadrilaterals), circles, and 3D objects (e.g. spheres, cubes)
• Measurements: perimeter, area, volume, and surface area of shapes

Calculus

• Limits: values that functions approach as input values change
• Derivatives: rates of change of functions, e.g. velocity and acceleration
• Integrals: accumulation of rates of change, e.g. area under curves

Statistics and Probability

• Data: collections of information, often numerical
• Probability: likelihood of events, measured between 0 and 1
• Probability Distributions: patterns of probability, e.g. normal, binomial
• Statistical Analysis: methods for interpreting and summarizing data, e.g. mean, median, mode, standard deviation

Number Systems

• Natural numbers are positive integers (1, 2, 3,...) and are often used for counting.
• Whole numbers include natural numbers and zero (0, 1, 2, 3,...) and represent non-negative integers.
• Integers consist of both positive and negative whole numbers (...,-3, -2, -1, 0, 1, 2, 3,...) and are used to represent whole amounts.
• Rational numbers are fractions (e.g. 3/4, -2/3) that can be expressed as finite decimals or ratios of integers.
• Irrational numbers are non-repeating, non-terminating decimals (e.g. π, e) and cannot be expressed as simple fractions.
• Real numbers are the combination of rational and irrational numbers and represent all possible decimal expansions.
• Complex numbers have both real and imaginary parts (e.g. 3 + 4i) and are used to extend the real number system.

Algebra

• Equations are statements with equal signs (e.g. 2x + 3 = 5) used to express the equality of two mathematical expressions.
• Inequalities are statements with greater than (>) or less than (<) signs (e.g. 2 > 1) used to compare values.
• Functions are relations between input values (x) and output values (y) (e.g. f(x) = 2x^2 + 1) and are often represented graphically.
• Graphs are visual representations of functions on a coordinate plane, allowing us to visualize relationships between variables.

Geometry

• Points are locations in space, represented by coordinates (x, y, z) and used to define geometric objects.
• Lines are sets of points extending infinitely in two directions and can be used to define shapes and boundaries.
• Angles are measures of rotation between two lines or planes and are essential in geometric calculations.
• Shapes include two-dimensional polygons (e.g. triangles, quadrilaterals), circles, and three-dimensional objects (e.g. spheres, cubes).
• Measurements in geometry involve calculating perimeter, area, volume, and surface area of shapes.

Calculus

• Limits are values that functions approach as input values change and are essential in understanding function behavior.
• Derivatives measure rates of change of functions (e.g. velocity and acceleration) and are used to analyze function behavior.
• Integrals are used to calculate the accumulation of rates of change (e.g. area under curves) and are essential in problems involving accumulation.

Statistics and Probability

• Data is a collection of information, often numerical, used to make inferences and conclusions.
• Probability is a measure of the likelihood of events, ranging from 0 (impossible) to 1 (certain).
• Probability distributions are patterns of probability, such as normal and binomial distributions, used to model real-world events.
• Statistical analysis involves methods for interpreting and summarizing data, including calculating mean, median, mode, and standard deviation.