Number Systems and Algebra Basics

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, 22/7 (can be expressed as a finite decimal)
• Irrational Numbers: non-repeating decimals, e.g. π, e (cannot be expressed as a finite decimal)
• Real Numbers: all rational and irrational numbers

Algebra

• Variables: letters or symbols representing unknown values
• Constants: numbers, e.g. 2, 5, 10
• Expressions: combinations of variables, constants, and operations, e.g. 2x + 5
• Equations: statements that two expressions are equal, e.g. 2x + 5 = 11
• Inequalities: statements that one expression is greater than, less than, or equal to another, e.g. 2x + 5 > 11

Geometry

• Points: locations in space, represented by coordinates (x, y)
• Lines: sets of points extending infinitely in two directions
• Angles: formed by two lines or rays sharing a common endpoint
• Triangles: three-sided polygons with three vertices and three sides
• Quadrilaterals: four-sided polygons with four vertices and four sides

Calculus

• Limits: the value a function approaches as the input gets arbitrarily close to a certain point
• Derivatives: measures of how functions change as their input changes
• Integrals: measures of the area under curves or accumulation of quantities

Statistics

• Mean: the average value of a dataset
• Median: the middle value of a dataset when arranged in order
• Mode: the most frequent value in a dataset
• Range: the difference between the largest and smallest values in a dataset
• Standard Deviation: a measure of the spread of a dataset

Number Systems

• Natural numbers are positive integers, starting from 1.
• Whole numbers include 0 and all positive integers.
• Integers consist of all whole numbers and their negatives.
• Rational numbers are fractions that can be expressed as finite decimals.
• Irrational numbers are non-repeating decimals that cannot be expressed as finite decimals.
• Real numbers encompass all rational and irrational numbers.

Algebra

• Variables are letters or symbols that represent unknown values.
• Constants are numbers that do not change value.
• Algebraic expressions combine variables, constants, and operations.
• Equations state that two expressions are equal and can be solved for variables.
• Inequalities compare two expressions using greater than, less than, or equal to.

Geometry

• Points are locations in space represented by coordinates (x, y).
• Lines extend infinitely in two directions and can be defined by two points.
• Angles are formed by two lines or rays sharing a common endpoint.
• Triangles are three-sided polygons with three vertices and three sides.
• Quadrilaterals are four-sided polygons with four vertices and four sides.

Calculus

• Limits describe the value a function approaches as the input gets arbitrarily close to a certain point.
• Derivatives measure the rate of change of a function as its input changes.
• Integrals measure the area under curves or the accumulation of quantities.

Statistics

• Mean is the average value of a dataset, calculated by summing values and dividing by the count.
• Median is the middle value of a dataset when arranged in order, which separates the higher and lower halves.
• Mode is the most frequent value in a dataset.
• Range is the difference between the largest and smallest values in a dataset.
• Standard deviation measures the spread of a dataset, calculating how much individual values deviate from the mean.