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: cannot be expressed as a finite decimal, e.g. π, e, sqrt(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 that represent unknown values
• Constants: numbers that do not change value
• Expressions: combinations of variables, constants, and operations
• Equations: statements that two expressions are equal
• Inequalities: statements that one expression is greater than, less than, or equal to another

Geometry

• Points: locations in space, represented by coordinates (x, y, z)
• Lines: sets of points that extend infinitely in two directions
• Rays: lines that extend from a single point
• Angles: formed by two lines or rays that share a common endpoint
• Shapes: triangles, quadrilaterals, polygons, circles, etc.

Calculus

• Limits: the value that a function approaches as the input gets arbitrarily close
• Derivatives: rates of change of functions with respect to the input
• Integrals: accumulation of rates of change over a defined interval
• Differentiation: finding the derivative of a function
• Integration: finding the integral of a function

Statistics

• Data: collections of information
• Measures of Central Tendency: mean, median, mode
• Measures of Variability: range, variance, standard deviation
• Probability: chance of an event occurring
• Hypothesis Testing: testing a statement about a population based on a sample of data

Number Systems

• Natural numbers are positive integers, starting from 1 (1, 2, 3,...)
• Whole numbers are non-negative integers, including 0 (0, 1, 2, 3,...)
• Integers include all whole numbers and their negatives (...,-3, -2, -1, 0, 1, 2, 3,...)
• Rational numbers are fractions, which can be expressed as finite decimals (1/2, 3/4, 22/7)
• Irrational numbers cannot be expressed as finite decimals (π, e, sqrt(2))
• Real numbers combine rational and irrational numbers
• Complex numbers have real and imaginary parts (3 + 4i)

Algebra

• Variables are letters or symbols representing unknown values
• Constants are numbers that do not change value
• Expressions combine variables, constants, and operations
• Equations state that two expressions are equal
• Inequalities compare expressions using greater than, less than, or equal to

Geometry

• Points are locations in space, represented by coordinates (x, y, z)
• Lines are sets of points extending infinitely in two directions
• Rays are lines extending from a single point
• Angles are formed by two lines or rays sharing a common endpoint
• Shapes include triangles, quadrilaterals, polygons, circles, and more

Calculus

• Limits are values that a function approaches as the input gets arbitrarily close
• Derivatives measure rates of change of functions with respect to the input
• Integrals accumulate rates of change over a defined interval
• Differentiation finds the derivative of a function
• Integration finds the integral of a function

Statistics

• Data is a collection of information
• Measures of Central Tendency include mean, median, and mode
• Measures of Variability include range, variance, and standard deviation
• Probability is the chance of an event occurring
• Hypothesis testing involves testing a statement about a population based on a sample of data