Mathematics is not a collection of isolated facts; it is a unified tree of knowledge. To master it, one must understand how the roots of arithmetic feed the branches of algebra and the leaves of calculus. Below is a comprehensive list of mathematical topics, organized by their logical progression.
1. Arithmetic (The Foundation)
Arithmetic is the study of numbers and the basic operations performed on them. It is the starting point for all quantitative reasoning.
- Number Systems: Natural numbers, Integers, Rational numbers, Irrational numbers, Real numbers, and Complex numbers.
- Basic Operations: Addition, Subtraction, Multiplication, and Division.
- Number Theory Basics: Factors, Multiples, Prime Numbers, LCM (Least Common Multiple), and HCF/GCD (Highest Common Factor).
- Fractions & Decimals: Operations, conversions, and simplifying expressions.
- Percentages: Calculation, increase/decrease, simple interest, and compound interest.
- Exponents & Roots: Squares, cubes, square roots, and the laws of exponents.
- Ratio & Proportion: Direct and inverse proportion.
2. Algebra (The Language of Math)
Algebra generalizes arithmetic by using symbols to represent numbers. It allows us to state general rules and solve for unknown values.
- Pre-Algebra: (PEMDAS/BODMAS)
- Linear Equations: Solving for a variable, inequalities, graphing lines
- Polynomials: Adding polynomials, subtracting polynomials, and multiplying polynomials factoring quadratics.
- Functions: Domain and range :, linear functions, quadratic functions, exponential functions, and logarithmic functions.
- Systems of Equations: Solving via substitution, elimination, or graphing.
- Matrices: Matrix operations, determinants, and inverses.
- Sequences & Series: Arithmetic and Geometric progressions.
3. Geometry (Shapes and Space)
Geometry is the study of size, shape, relative position of figures, and the properties of space.
- Euclidean Geometry: Points, lines, planes, and angles.
- 2D Shapes: Triangles (congruence, similarity), Quadrilaterals, Polygons, and Circles (radius, diameter, circumference).
- 3D Shapes: Prisms, pyramids, cylinders, cones, and spheres.
- Measurement: Formulas for Perimeter, Area, Volume, and Surface Area.
- Coordinate Geometry: Distance formula, midpoint formula, and equations of circles.
- Transformations: Reflection, rotation, translation, and dilation.
4. Trigonometry (Triangles and Waves)
Trigonometry studies the relationships between side lengths and angles of triangles, bridging geometry and waves.
- Right Triangle Trig: Sine, Cosine, Tangent (SOH CAH TOA).
- Non-Right Triangle Trig: Law of Sines and Law of Cosines.
- The Unit Circle: Radians vs. Degrees and circular functions.
- Trigonometric Identities: Pythagorean identities, sum and difference formulas.
- Graphing Trig Functions: Amplitude, period, and phase shift.
5. Calculus (The Mathematics of Change)
Calculus is the mathematical study of continuous change, essential for physics, engineering, and economics.
- Limits: Continuity and limits at infinity.
- Differential Calculus: The derivative, power rule, chain rule, product/quotient rules, optimization, and related rates.
- Integral Calculus: Antiderivatives, definite and indefinite integrals, area under curves, and volume of revolution.
- Multivariable Calculus: Vectors and partial derivatives.
- Differential Equations: Modeling change over time.
6. Probability & Statistics (Data and Chance)
This field deals with data collection, analysis, interpretation, and the mathematics of uncertainty.
- Descriptive Statistics: Mean, median, mode, range, and standard deviation.
- Data Representation: Bar graphs, histograms, pie charts, and scatter plots.
- Probability: Independent/dependent events, conditional probability, permutations, and combinations.
- Distributions: Normal distribution (Bell Curve) and Binomial distribution.
- Inferential Statistics: Hypothesis testing, confidence intervals, and regression analysis.
7. Discrete Mathematics
The math of computer science, dealing with discrete rather than continuous structures.
- Set Theory: Unions, intersections, and Venn diagrams.
- Logic: Truth tables and Boolean algebra.
- Graph Theory: Nodes, edges, paths, and circuits.
- Algorithms: Step-by-step problem solving.
8. Alternative Systems
Techniques for mental calculation and historical computational methods.
- Vedic Mathematics: 16 Sutras for speed calculation.
- Abacus: Physical calculation tools (Soroban/Suanpan).
- Trachtenberg System: A system of rapid mental calculation.