Mathematics Key Concepts Overview

Natural Numbers: Counting numbers starting from 1 (1, 2, 3, …)
Whole Numbers: Natural numbers including 0 (0, 1, 2, 3, …)
Integers: Whole numbers and their negatives (…, -3, -2, -1, 0, 1, 2, 3, …)
Rational Numbers: Numbers that can be expressed as a fraction (a/b where a and b are integers, b ≠ 0)
Irrational Numbers: Numbers that cannot be expressed as a fraction (e.g., √2, π)

Variables: Symbols (e.g., x, y) used to represent unknown quantities.
Expressions: Combinations of numbers, variables, and operations (e.g., 2x + 3).
Equations: Statements that two expressions are equal (e.g., 2x + 3 = 7).
Functions: A relation that assigns exactly one output for each input (e.g., f(x) = x^2).

Points: Exact locations in space with no dimensions.
Lines: Straight pathways extending in both directions with no endpoints.
Angles: Formed by two rays with a common endpoint, measured in degrees.
Shapes:
- Triangles: Three sides (types include isosceles, equilateral, scalene).
- Quadrilaterals: Four sides (types include squares, rectangles, parallelograms).
- Circles: A round shape defined by its radius from the center.

Limits: The value that a function approaches as the input approaches a point.
Derivatives: Measure of how a function changes as its input changes (slope of the function).
Integrals: Represent accumulated area under a curve.

Experiment: A situation involving chance or probability.
Outcome: A possible result of an experiment.
Event: A specific outcome or a set of outcomes.
Probability: Measure from 0 to 1 indicating the likelihood of an event occurring.

The Pythagorean Theorem: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (a² + b² = c²).
The Binomial Theorem: Describes the expansion of powers of binomials.
Euler's Formula: Establishes the fundamental relationship between trigonometric functions and the exponential function.

These notes cover foundational concepts in mathematics that are essential for further study and application across various mathematical disciplines.

Natural Numbers: Represent counting numbers starting from 1 (1, 2, 3, ...).
Whole Numbers: Include 0 in addition to all natural numbers (0, 1, 2, 3, ...).
Integers: Encompass whole numbers and their negative counterparts (..., -3, -2, -1, 0, 1, 2, 3, ...).
Rational Numbers: Expressed as a fraction (a/b) where 'a' and 'b' are integers, and 'b' cannot be zero.
Irrational Numbers: Cannot be written as a fraction. Examples include the square root of 2 (√2) and pi (π).

Variables: Symbols like 'x' or 'y' representing unknown quantities.
Expressions: Combine numbers, variables, and operations (e.g., 2x + 3).
Equations: Statements that two expressions are equal (e.g., 2x + 3 = 7).
Functions: Relate inputs with unique outputs. For example, f(x) = x^2 assigns a square value to an input.

Points: Locations in space with no size or dimension.
Lines: Straight paths extending infinitely in both directions.
Angles: Formed by two rays with a shared endpoint. They are measured in degrees.
Shapes:
- Triangles: Have three sides and are classified into types like isosceles, equilateral, and scalene.
- Quadrilaterals: Have four sides and include shapes like squares, rectangles, and parallelograms.
- Circles: Round shapes defined by their radius from the center.

Sine (sin): In a right-angled triangle, it represents the ratio of the opposite side to the hypotenuse.
Cosine (cos): Ratio of the adjacent side to the hypotenuse in a right triangle.
Tangent (tan): Represents the ratio of the opposite side to the adjacent side.

Limits: Describe the value a function approaches as its input gets close to a specific point.
Derivatives: Measure how a function changes as its input changes. Essentially, they determine the slope of a function.
Integrals: Represent the accumulated area under a curve.

Mean: The average of a set of numbers.
Median: The middle value in a sorted set of data.
Mode: The value that appears most frequently in a data set.
Standard Deviation: Measures the spread or variability of data around the mean.

Experiment: A situation with uncertain outcomes based on chance.
Outcome: A possible result of an experiment.
Event: A specific outcome or a group of outcomes from an experiment.
Probability: A measure from 0 to 1 that indicates the likelihood of an event occurring.

Pythagorean Theorem: In a right triangle, the square of the hypotenuse (the longest side) equals the sum of the squares of the other two sides. This is represented as a² + b² = c².
Binomial Theorem: Provides a formula for expanding powers of binomials (expressions with two terms).
Euler's Formula: Defines a fundamental relationship between trigonometric and exponential functions.