Fundamental Concepts in Mathematics

Natural Numbers: Counting numbers (1, 2, 3, …).
Whole Numbers: Natural numbers including zero (0, 1, 2, …).
Integers: Whole numbers and their negative counterparts (…, -2, -1, 0, 1, 2, …).
Rational Numbers: Numbers that can be expressed as a fraction (p/q, where q ≠ 0).
Irrational Numbers: Numbers that cannot be expressed as fractions (e.g., √2, π).
Real Numbers: Combination of rational and irrational numbers.
Complex Numbers: Numbers of the form a + bi, where i is the imaginary unit.

Variables: Symbols used to represent numbers (e.g., x, y).
Expressions: Combinations of numbers and variables (e.g., 3x + 5).
Equations: Statements that two expressions are equal (e.g., 2x + 3 = 7).
Functions: Relationships between sets of numbers where each input has a unique output.

Points: Exact locations in space.
Lines: Straight paths extending infinitely in both directions.
Angles: Formed by two rays originating from a common endpoint.
Shapes: Geometric figures such as triangles, rectangles, circles, and polygons.
Area and Volume: Measures of space within shapes (e.g., A = length × width for rectangles; V = length × width × height for cuboids).

Mean: Average of a set of numbers.
Median: Middle value when numbers are arranged in order.
Mode: The value that appears most frequently in a data set.
Standard Deviation: Measure of the amount of variation or dispersion in a set of values.

Limits: The value a function approaches as the input approaches a point.
Derivatives: Measure of how a function changes as its input changes.
Integrals: Represents the area under a curve or the accumulation of quantities.

Experiment: An action or process that leads to one or more outcomes.
Sample Space: Set of all possible outcomes of an experiment.
Event: A specific outcome or set of outcomes.
Probability Formula: P(A) = Number of favorable outcomes / Total number of outcomes.

Sine, Cosine, Tangent: Ratios of sides in a right triangle.
Unit Circle: Circle with a radius of one, used to define trigonometric functions.
Pythagorean Theorem: a² + b² = c² for right-angled triangles.

Inductive Reasoning: Making generalizations based on specific examples.
Deductive Reasoning: Drawing specific conclusions from general principles or premises.

∑ (Sigma): Sum of a series.
∏ (Pi): Product of a series.
√ (Square root): A number that produces a specified quantity when multiplied by itself.

These notes cover foundational concepts across various branches of mathematics and are useful for building a basic understanding of the subject.

Natural Numbers: The sequence of positive integers used for counting (1, 2, 3, ...).
Whole Numbers: Natural numbers including zero, represented as (0, 1, 2, ...).
Integers: The complete set of whole numbers along with their negatives (..., -2, -1, 0, 1, 2, ...).
Rational Numbers: Any number that can be expressed as a fraction (p/q), where q is not zero.
Irrational Numbers: Numbers that cannot be represented as fractions, such as √2 and π.
Real Numbers: The union of rational and irrational numbers, encompassing all possible quantities.
Complex Numbers: Represented in the form a + bi, where 'i' denotes the imaginary unit.

Addition (+): The mathematical process of combining two or more numbers to obtain a total.
Subtraction (−): The operation of finding the difference between numbers, essentially removing one from another.
Multiplication (×): A form of repeated addition that combines groups of the same size.
Division (÷): The process of determining how many times one number is contained within another.

Variables: Symbols such as x and y that stand in for unknown values in equations and expressions.
Expressions: Combinations of numbers and variables that represent a value (e.g. 3x + 5).
Equations: Mathematical statements asserting the equality of two expressions (e.g. 2x + 3 = 7).
Functions: Relationships between input and output where each input corresponds to a unique output.

Points: Fundamental units of geometry representing specific locations in space with no dimensions.
Lines: Straight entities extending infinitely in both directions without endpoints.
Angles: Formed by two rays sharing a common endpoint, measured in degrees.
Shapes: Defined geometric figures including triangles, rectangles, circles, and polygons.
Area and Volume: Quantitative measures; area measures the space within a shape, while volume measures the capacity in three-dimensional objects.

Mean: The arithmetic average calculated by summing values and dividing by their count.
Median: The central number in a sorted list of values, dividing the data into two equal halves.
Mode: The most frequently occurring value in a set of data.
Standard Deviation: A statistic that quantifies the amount of variation or dispersion in a data set.

Limits: The value that a function approaches as the input gets infinitely close to a specified point.
Derivatives: A measure of the rate of change of a function concerning its variable.
Integrals: Represents accumulation of quantities and the area under curves in a graphical representation.

Experiment: An action or process that results in one or more outcomes, relevant in probability analysis.
Sample Space: The comprehensive set of all possible outcomes generated by an experiment.
Event: A specific outcome or a set of outcomes within the sample space.
Probability Formula: Expressed as P(A) = Number of favorable outcomes divided by the total number of outcomes.

Sine, Cosine, Tangent: Fundamental ratios relating the sides of a right triangle to its angles.
Unit Circle: A circle with a radius of one, essential for defining trigonometric functions and their properties.
Pythagorean Theorem: A principle stating that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a² + b² = c²).

Inductive Reasoning: Drawing general conclusions from limited specific instances or observations.
Deductive Reasoning: Arriving at specific conclusions based on general premises or rules.

∑ (Sigma): Represents the sum of a series of numbers.
∏ (Pi): Represents the product of a series of numbers.
√ (Square root): Represents a value that, when multiplied by itself, yields the specified quantity.