All Branches of Mathematics

Contents

 

Mathematics is the study of amount, pattern, arrangement, structure, and connection. It keeps growing from simple actions of counting, measuring, and examining symmetrical shapes. The main thing is to use logical thinking and numerical calculations to discover the best solutions to problems. Maths is the solution to many problems and hence has various branches, that give solutions in different fields. Mathematics is broadly classified into three parts: foundational [ F ], pure [ P ] and applied [ A ] mathematics.

NOTE

The breakdown of the branches of mathematics below was taken from All Branches of Mathematics by sscbytes.official. While investigating the information for this page, I was not surprised to find different interpretations of what the branches of mathematics are. The placement of [ F ], [ A ] and [ P ] is the closest I can come to reflecting the current understanding of each branch. To that end, I am going to use this interpretation to highlight the scope of mathematics, and how difficult mathematics can be to classify since some branches can be classified as both pure and applied.

“Mathematics is not about numbers, equations, computations or algorithms; it is about understanding.” — William Paul Thurston

Mathematical Branches

  1. Elementary Mathematics [ F ]
  2. Pure Mathematics
    1. Logic [ P ]
    2. Set Theory [ P ]
    3. Number Theory [ P ]
    4. Algebra (general) [ P ]
    5. Geometry (general) [ P ]
  3. Algebra
    1. Elementary Algebra [ P ]
    2. Linear Algebra [ P ][ A ]
    3. Abstract Algebra [ P ]
    4. Polynomial Theory [ P ]
    5. Matrix Theory (pure foundations, but widely applied) [ P ][ A ]
  4. Calculus [ P ][ § ]
    1. Differential Calculus [ P ]
    2. Integral Calculus [ P ]
    3. Multivariable Calculus [ P ]
    4. Vector Calculus (used in physics, but still pure in structure) [ P ][ A ]
  5. Geometry
    1. Euclidean Geometry [ P ]
    2. Non-Euclidean Geometry [ P ]
    3. Analytic Geometry [ P ]
    4. Differential Geometry [ P ][ A ]
  6. Trigonometry
    1. Plane Trigonometry [ P ]
    2. Spherical Trigonometry [ P ]
  7. Statistics
    1. Descriptive Statistics [ A ]
    2. Inferential Statistics [ A ]
    3. Probability [ P ]
    4. Data Analysis [ A ]
  8. Probability
    1. Classical Probability [ P ]
    2. Conditional Probability [ P ]
    3. Random Variables [ P ][ A ]
    4. Probability Distributions [ P ][ A ]
  9. Discrete Mathematics
    1. Graph Theory (pure foundations, but widely applied) [ P ][ A ]
    2. Combinatorics [ P ][ A ]
    3. Boolean Algebra [ P ]
    4. Recurrence Relations [ P ]
  10. Applied Mathematics
    1. Mathematical Modeling [ A ]
    2. Differential Equations [ A ]
    3. Optimization [ A ]
    4. Numerical Methods [ A ]
  11. Numerical Analysis
    1. Root Finding [ A ]
    2. Numerical Integration [ A ]
    3. Linear Algebraic Methods [ A ]
    4. Interpolation [ A ]
  12. Mathematical Analysis
    1. Real Analysis [ P ][ A ]
    2. Complex Analysis [ P ][ A ]
    3. Functional Analysis [ P ][ A ]
  13. Differential Equations
    1. Ordinary Differential Equations (ODEs) [ A ]
    2. Partial Differential Equations (PDEs) [ A ]
  14. Mathematical Logic
    1. Propositional Logic [ P ][ A ]
    2. Predicate Logic [ P ][ A ]
    3. Proof Theory [ P ][ A ]
    4. Model Theory [ P ][ A ]

References

“Branches Of Mathematics.” BYJU’s, January 25, 2018. https://byjus.com/maths/branches-of-mathematics/.

“Branches of Mathematics.” GeeksforGeeks, December 14, 2023. https://www.geeksforgeeks.org/maths/branches-of-mathematics/.

“Branches of Mathematics and Their Major Fields.” Vedantu, December 29, 2018. https://www.vedantu.com/maths/branches-of-mathematics.

Hartnett, Kevin. “The Map of Mathematics.” Quanta Magazine, February 13, 2020. https://www.quantamagazine.org/the-map-of-mathematics-20200213/.

“Mathematics.” Wikipedia, Accessed September 22, 2026. https://en.wikipedia.org/wiki/Mathematics.

Notes

Foundational Mathematics

Foundational mathematics refers to the basic skills and core concepts that all later mathematics depends on. These include number sense, arithmetic fluency, fractions, decimals, percentages, place value, and basic problem‑solving. They are considered “foundational” because higher‑level topics—algebra, geometry, statistics—continually rely on them. These skills form the basis for more advanced mathematical learning and are essential for success in various academic and professional pursuits.

“Math Notion.” Math Notion, May 15, 2024. https://www.mathnotion.com/foundational-math-skills/math-blog/.

“How to Develop Foundational Math Skills for Career Success.” Effortless Math, June 30, 2023. https://www.effortlessmath.com/blog/how-to-develop-foundational-math-skills-for-career-success/.

“Foundations of Mathematics.” Wikipedia, September 21, 2026. https://en.wikipedia.org/wiki/Foundations_of_mathematics.

Pure Mathematics

Pure mathematics is the study of abstract mathematical structures and ideas pursued for their own intrinsic interest rather than for direct practical application. It focuses on rigor, axiomatic systems, proofs, and the exploration of concepts such as algebraic structures, number theory, geometry, topology, and logic. Although not motivated by real‑world problems, pure mathematics often later becomes essential for applied fields like physics and computer science.

“What Is Pure Mathematics?” California Learning Resource Network, July 2, 2025. https://www.clrn.org/what-is-pure-mathematics/.

“Pure Mathematics.” Wikipedia, July 18, 2026. https://en.wikipedia.org/wiki/Pure_mathematics.

Applied Mathematics

Applied mathematics is the use of mathematical methods to solve real‑world problems in science, engineering, medicine, finance, computer science, and other fields. It involves developing mathematical models, analyzing systems, and creating computational or analytical tools to explain and predict phenomena. Applied mathematics blends mathematical theory with specialized domain knowledge and often motivates new developments in pure mathematics.

Renneboog, Richard M. J. “Applied Mathematics.” EBSCO, 2021. https://www.ebsco.com/research-starters/mathematics/applied-mathematics.

“Applied Mathematics.” Wikipedia, August 13, 2026. https://en.wikipedia.org/wiki/Applied_mathematics.

Calculus

Calculus is considered pure mathematics because its foundations, definitions, and theorems are developed independently of any real‑world application.

Pure mathematics is about proving truths inside mathematics itself

Pure math asks questions like:

  • What is a limit?
  • How do we rigorously define continuity?
  • Under what conditions does a derivative exist?
  • What properties do integrals have?

These questions don’t depend on physics, engineering, or economics. They’re internal to mathematics. Calculus, at its core, is built on:

All of these are defined and studied abstractly. You can develop calculus entirely without ever mentioning motion, area, or real-world change.

But calculus is also one of the most applied mathematical tools ever created

Calculus is used everywhere:

  • physics
  • engineering
  • economics
  • biology
  • chemistry
  • computer science
  • statistics

When you use calculus to model motion, growth, decay, optimization, or change, you’re doing applied mathematics. So the tool is applied, but the theory is pure.

The distinction is about intent, not content.

Pure Calculus

Applied Calculus

  • Computing velocity from position
  • Modeling population growth
  • Optimizing cost functions
  • Calculating work, flux, or pressure

Same mathematics — different purpose.

Why textbooks often list calculus under pure math

Because calculus is foundational. It’s part of the theoretical backbone that supports:

These are pure fields that grow directly out of calculus.

Bottom line

Calculus is pure mathematics because its foundations are abstract and theoretical. Calculus is applied mathematics when you use it to model real-world change. It’s one of the math subjects that is both — but its classification as “pure” comes from its origins and its theoretical development.

Information generated by Copilot.

Differentiability on Abstract Spaces

Differentiability on abstract spaces extends the standard calculus concept of a derivative from Euclidean space (ℝ𝑛) to infinite-dimensional spaces (like Banach or Hilbert spaces) or spaces without standard coordinates (like Differentiable manifold).

Instead of relying on division by a scalar change in coordinates, abstract differentiability defines the derivative as a best linear approximation or via directional rates of change.

“Differentiable Manifold.” Wikipedia, July 12, 2026. https://en.wikipedia.org/wiki/Differentiable_manifold.

“Differentiability in Metric Spaces.” Mathematics Stack Exchange. Accessed September 23, 2026. https://math.stackexchange.com/questions/614688/differentiability-in-metric-spaces.

The featured image on this page is from the GeeksForGeeks website.

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