Key Branches of Mathematics

Arithmetic: Basic operations (addition, subtraction, multiplication, division) and properties of numbers.
Algebra: Symbols and letters to represent numbers in equations; solving equations and understanding functions.
Geometry: Study of shapes, sizes, properties of space; includes points, lines, angles, surfaces, and solids.
Trigonometry: Relationships between angles and sides of triangles; key functions include sine, cosine, and tangent.
Calculus: Study of change; includes differentiation (rates of change) and integration (area under curves).

Numbers:
- Natural numbers: Counting numbers (1, 2, 3, ...)
- Integers: Whole numbers (…-2, -1, 0, 1, 2,…)
- Rational numbers: Numbers that can be expressed as a fraction (a/b where a and b are integers).
- Irrational numbers: Numbers that cannot be expressed as a fraction (e.g., √2, π).
Equations: Mathematical statements that assert equality; can be solved for unknown variables.

Order of Operations: PEMDAS/BODMAS
- Parentheses/Brackets
- Exponents/Orders
- Multiplication and Division (left to right)
- Addition and Subtraction (left to right)

Angles:
- Acute (< 90°), Right (90°), Obtuse (> 90° but < 180°), Straight (180°).
Triangles:
- Types: Equilateral, Isosceles, Scalene.
- Pythagorean theorem: a² + b² = c² for right-angled triangles.
Circles:
- Key terms: Radius, Diameter, Circumference, Area (A = πr²).

Variables: Symbols used to represent unknown values.
Functions: A relation that assigns exactly one output for each input; f(x) notation.
Polynomials: Expressions consisting of variables raised to non-negative integer powers.

Limits: Understanding the behavior of functions as inputs approach certain values.
Derivatives: Measures the rate of change of a function; represents the slope of a curve.
Integrals: Represents the accumulation of quantities; can be definite (specific interval) or indefinite (general antiderivative).

Basic Functions:
- Sine (sin), Cosine (cos), Tangent (tan).
Unit Circle: Circle of radius 1; used to define trigonometric functions at various angles.

Probability: Measure of the likelihood of an event occurring.
Statistics: Collection, analysis, interpretation, presentation of data.
- Measures of central tendency: Mean, Median, Mode.
- Variance and Standard Deviation: Measures of data dispersion.

Arithmetic: Involves basic number operations - addition, subtraction, multiplication, and division, along with fundamental number properties.
Algebra: Utilizes symbols and letters to formulate equations, solving for variables while exploring functional relationships.
Geometry: Examines shapes, sizes, and spatial properties, covering elements like points, lines, angles, surfaces, and solids.
Trigonometry: Focuses on the relationships among angles and sides of triangles; essential functions include sine, cosine, and tangent.
Calculus: Concerned with change, emphasizing differentiation (calculating rates of change) and integration (finding areas under curves).

Numbers:
- Natural numbers: Positive counting numbers (1, 2, 3...).
- Integers: Whole numbers including negatives (...-2, -1, 0, 1, 2...).
- Rational numbers: Numbers expressible as fractions (a/b where a and b are integers).
- Irrational numbers: Cannot be expressed as fractions (e.g., √2, π).
Equations: Mathematical expressions indicating equality that can be solved for unknowns.

Order of Operations: Follow PEMDAS/BODMAS rules:
- Parentheses/Brackets first
- Exponents/Orders next
- Multiplication and Division from left to right
- Addition and Subtraction from left to right

Angles: Different types include:
- Acute: Less than 90°
- Right: Exactly 90°
- Obtuse: Greater than 90° but less than 180°
- Straight: Exactly 180°
Triangles:
- Classifications: Equilateral, Isosceles, and Scalene triangles.
- Pythagorean theorem: Relates the sides of a right triangle: a² + b² = c².
Circles:
- Important measurements: Radius (distance from center to edge), Diameter (twice the radius), Circumference (perimeter), Area (A = πr²).

Variables: Symbols that represent unknown or arbitrary values.
Functions: A relation producing a single output for each input, commonly expressed as f(x).
Polynomials: Algebraic expressions made of sums and products of variables raised to non-negative integer powers.

Limits: Assess how functions behave as inputs approximate certain values.
Derivatives: Represent the instantaneous rate of change of a function, indicating the slope of its graph.
Integrals: Calculate total quantities accumulated over an interval, features both definite (specific range) and indefinite forms (general antiderivative).

Basic Functions:
- Sine (sin), Cosine (cos), Tangent (tan) are foundational trigonometric functions.
Unit Circle: A circle with a radius of one, essential for defining trigonometric functions at various angles.

Probability: Quantifies the chance of a specific event occurring.
Statistics: Involves gathering, analyzing, interpreting, and presenting data.
Measures of Central Tendency: Mean (average), Median (middle value), Mode (most frequent value).
Variance and Standard Deviation: Metrics that describe how spread out data points are from the mean.

Proofs: Rigorous arguments establishing the truth of mathematical statements.
Inductive Reasoning: Generalizations made from specific cases or examples.
Deductive Reasoning: Specific conclusions derived from established general principles.

Finance: Utilized for calculating interest, managing budgets, and analyzing investments.
Engineering: Involves design processes, analytical problem-solving, and technical assessments.
Science: Data analysis and modeling of various natural and physical phenomena.