Number Systems

Study Notes

Natural Numbers: Counting numbers, 1, 2, 3, ...
Whole Numbers: Non-negative integers, 0, 1, 2, 3, ...
Integers: ..., -3, -2, -1, 0, 1, 2, 3, ...
Rational Numbers: Fractions, can be expressed as p/q where p and q are integers and q ≠ 0
Irrational Numbers: Cannot be expressed as a finite decimal or fraction, e.g. π, e
Real Numbers: Includes rational and irrational numbers
Complex Numbers: Numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit (i² = -1)

Variables and Expressions: Letters or symbols representing unknown values or quantities
Equations and Inequalities: Statements with equalities (=) or inequalities (≠, <, >, ≤, ≥)
Linear Equations: Equations of the form ax + by = c, where a, b, and c are constants
Quadratic Equations: Equations of the form ax² + bx + c = 0, where a, b, and c are constants

Points, Lines, and Planes: Basic geometric objects
Angles and Measurement: Degrees, minutes, and seconds; types of angles (acute, obtuse, right, straight)
Properties of Shapes: Congruence, similarity, and trigonometric properties of triangles, quadrilaterals, polygons, circles, and 3D objects

Limits: Approaching values of functions as input values approach a certain point
Derivatives: Rates of change of functions, represented as f'(x) or (dy/dx)
Differentiation Rules: Power rule, product rule, quotient rule, and chain rule
Applications of Derivatives: Optimization, motion, and related rates
Integrals: Accumulation of quantities, represented as ∫f(x)dx
Integration Techniques: Substitution, integration by parts, integration by partial fractions, and integration by trigonometric substitution

Natural numbers are counting numbers starting from 1, e.g. 1, 2, 3, ...
Whole numbers are non-negative integers, including 0, e.g. 0, 1, 2, 3, ...
Integers are all whole numbers and their negative counterparts, e.g. ..., -3, -2, -1, 0, 1, 2, 3, ...
Rational numbers are fractions that can be expressed as p/q, where p and q are integers and q ≠ 0
Irrational numbers cannot be expressed as a finite decimal or fraction, e.g. π, e
Real numbers include all rational and irrational numbers
Complex numbers have the form a + bi, where a and b are real numbers and i is the imaginary unit (i² = -1)

Variables and expressions are letters or symbols representing unknown values or quantities
Equations and inequalities are statements with equalities (=) or inequalities (≠, <, >, ≤, ≥)
Linear equations have the form ax + by = c, where a, b, and c are constants
Quadratic equations have the form ax² + bx + c = 0, where a, b, and c are constants

Points, lines, and planes are basic geometric objects
Angles can be measured in degrees, minutes, and seconds, and have types such as acute, obtuse, right, and straight
Properties of shapes include congruence, similarity, and trigonometric properties of triangles, quadrilaterals, polygons, circles, and 3D objects

Limits are approaching values of functions as input values approach a certain point
Derivatives represent rates of change of functions, denoted as f'(x) or (dy/dx)
Differentiation rules include power rule, product rule, quotient rule, and chain rule
Applications of derivatives include optimization, motion, and related rates
Integrals represent accumulation of quantities, denoted as ∫f(x)dx
Integration techniques include substitution, integration by parts, integration by partial fractions, and integration by trigonometric substitution