Number Systems in Mathematics

Study Notes

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

Description

This quiz covers the different types of number systems, including natural numbers, whole numbers, integers, rational numbers, irrational numbers, real numbers, and complex numbers. Understand the definitions and properties of each number system.