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, ... (all whole numbers and their negatives)
• Rational Numbers: fractions, e.g. 1/2, 3/4, 22/7 (can be expressed as a finite decimal)
• Irrational Numbers: non-repeating, non-terminating decimals, e.g. π, e, √2
• Real Numbers: all rational and irrational numbers
• Complex Numbers: numbers with real and imaginary parts, e.g. 3 + 4i

Algebra

• Variables: letters or symbols representing unknown values
• Constants: fixed values, e.g. 2, 5, 10
• 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, e.g. 2x + 3 > 5

Geometry

• Points: locations in space, represented by coordinates (x, y)
• Lines: sets of points extending infinitely in two directions
• Line Segments: parts of lines with finite length
• Rays: parts of lines extending infinitely in one direction
• Angles: measures of rotation between two lines or planes
• Shapes: basic geometric figures, e.g. triangles, quadrilaterals, polygons, circles

Calculus

• Limits: values that a function approaches as the input gets arbitrarily close
• Derivatives: rates of change of functions, e.g. slopes of tangent lines
• Integrals: accumulation of quantities, e.g. areas under curves
• Differentiation: process of finding derivatives
• Integration: process of finding integrals

Statistics and Probability

• Data: collections of information
• Mean: average value of a dataset
• Median: middle value of a dataset when arranged in order
• Mode: most frequently occurring value in a dataset
• Probability: measure of likelihood of an event, e.g. 0 (impossible) to 1 (certain)
• Random Variables: variables with uncertain outcomes, e.g. coin tosses, dice rolls

Number Systems

• Natural numbers are positive integers: 1, 2, 3, ...
• Whole numbers include natural numbers and zero: 0, 1, 2, 3, ...
• Integers comprise all whole numbers and their negatives: ..., -3, -2, -1, 0, 1, 2, 3, ...
• Rational numbers are fractions that can be expressed as finite decimals: 1/2, 3/4, 22/7
• Irrational numbers are non-repeating, non-terminating decimals: π, e, √2
• Real numbers encompass all rational and irrational numbers
• Complex numbers have real and imaginary parts: 3 + 4i

Algebra

• Variables are letters or symbols representing unknown values
• Constants are fixed values: 2, 5, 10
• Expressions combine variables, constants, and operations
• Equations are statements with equal signs: 2x + 3 = 5
• Inequalities have greater than, less than, or equal to signs: 2x + 3 > 5

Geometry

• Points are locations in space, represented by coordinates (x, y)
• Lines are sets of points extending infinitely in two directions
• Line segments have finite length
• Rays extend infinitely in one direction
• Angles measure rotation between two lines or planes
• Shapes include basic geometric figures: triangles, quadrilaterals, polygons, circles

Calculus

• Limits are values approached by functions as input gets arbitrarily close
• Derivatives measure rates of change, like slopes of tangent lines
• Integrals accumulate quantities, such as areas under curves
• Differentiation finds derivatives
• Integration finds integrals

Statistics and Probability

• Data refers to collections of information
• Mean is the average value of a dataset
• Median is the middle value when data is arranged in order
• Mode is the most frequent value in a dataset
• Probability measures likelihood, ranging from 0 (impossible) to 1 (certain)
• Random Variables have uncertain outcomes, such as coin tosses and dice rolls