What is the difference between pure math and applied math?

Mathematics is a dynamic and versatile field that forms the foundation for many areas of study and professional careers. It combines abstract reasoning with practical problem-solving, helping individuals tackle both theoretical concepts and real-world challenges. There are two main branches of maths– pure and applied.

Pure mathematics explores the abstract concepts and complex patterns that shape our reality.
Applied mathematics provides solutions to real-world problems.

What is pure mathematics

Pure mathematics is the study of mathematical concepts in their own right, solely focused on math itself and separate from external influences. Pure mathematics revolves around the exploration of abstract concepts that are not necessarily connected to practical applications. It focuses on building and understanding theoretical frameworks that underpin mathematical thought.

These frameworks often include structures such as groups, rings and fields in algebra, as well as spaces and dimensions in geometry and topology. Pure mathematicians aim to uncover universal truths and relationships, creating a foundation that can later inform and support applied fields. The emphasis is not on solving immediate problems but on expanding the horizons of mathematical knowledge. Pure mathematics often serves as the foundation for advancements in other disciplines.

Pure math in the real world

• Physics: Geometry and mathematical analysis are fundamental to theories in physics, such as Einstein's General Theory of Relativity, which relies heavily on differential geometry to describe the curvature of spacetime.
• Engineering: Topology is used in material science and electrical engineering, particularly in the design of circuits and networks, while mathematical analysis assists in solving differential equations for structural mechanics and fluid dynamics.
• Computer science: Number theory and combinatorics are vital in cryptography, which secures digital communication. Algorithms that ensure data encryption are deeply rooted in mathematical principles.
• Economics: Mathematical analysis and algebra are used in economic modeling to predict market behaviors and to optimize resource allocation through tools like linear programming.
• Biology: Combinatorics plays a critical role in bioinformatics, where it helps understand genetic coding and the relationships between biological networks through mathematical modeling.
• Astronomy: Number theory and arithmetic support algorithms used in calculating orbits, navigating spacecraft and analyzing patterns in the distribution of stars and galaxies. 

These connections illustrate the profound impact of pure mathematics on various fields, highlighting its essential role in driving innovation and scientific discovery.

What is applied mathematics

Applied mathematics combines mathematical concepts with specialized knowledge from various disciplines to solve practical problems. Applied math plays a crucial role in addressing real-world challenges by formulating mathematical models and techniques to analyze and solve them. 

The focus is on practical application, where complex problems from fields such as physics, engineering, computer science, economics and biology are translated into mathematical terms. This process often involves identifying patterns, creating simulations and optimizing solutions to improve efficiency and effectiveness. By bridging theoretical mathematics with practical implementation, applied math provides tools and insights essential for innovation, technological advancement and decision-making across industries.

Applied math in the real world

• Physics: Applied math is essential in physics for modeling natural phenomena like motion, energy and forces. Equations in mechanics, electromagnetism and quantum theory rely heavily on math for formulation and solutions.
• Engineering: Engineers use applied math to design and analyze systems, structures and processes. It solves problems in fields like civil, electrical and mechanical engineering, enabling accurate calculations and optimizations.
• Computer science: Algorithms, cryptography and data analysis in computer science are rooted in math. Applied math helps create efficient algorithms, secure communications and machine learning models.
• Economics: Mathematical economics uses applied math to model markets, optimize resources and predict trends. Tools like game theory and statistical models improve decision-making and analyze systems.
• Biology: Applied math is crucial in biology, especially for population dynamics. The Lotka-Volterra equations describe predator-prey interactions, helping predict population trends and guide conservation efforts.
• Astronomy: In astronomy, applied math helps explain celestial mechanics and planetary motion, such as Kepler's laws. It models galaxies, calculates star distances and interprets signals from distant planets, offering key insights into the universe.
  

Comparing pure and applied mathematics

Applied mathematics is more practical and industry-focused, while pure mathematics is theoretical and research-oriented, but both share a strong foundation in mathematical principles and skills. You'll find that these branches of mathematics also deviate on several important points.

Differences

1. Focus, purpose and application: Applied math focuses on using mathematical concepts to solve real-world problems in fields like science, technology, healthcare and business, while pure mathematics explores abstract theories and concepts, often for academic and research purposes. Applied math is used in areas such as finance, engineering and medicine, whereas pure mathematics drives innovation in fields like quantum theory and topology.
2. Coursework: Applied mathematics encompasses topics such as applied algebra, probability, mathematical modeling, computational analysis and scientific computing. On the other hand, Pure Mathematics focuses on areas like abstract algebra, discrete mathematics, complex variables, nonlinear dynamics and general relativity, with an emphasis on theoretical research.
3. Educational Requirements: Applied math often requires technical certifications or undergraduate degrees for practical roles, while Pure Mathematics typically demands advanced degrees for research and academic positions.

But, at the root of it all both fields are still mathematics. Both paths require an understanding and appreciation of the other to fully comprehend the potential of discoveries.

Similarities

1. Shared Concepts: Both fields share a foundation of mathematical principles and logic and cover foundational topics like computation, number theory, calculus, statistical analysis and geometry at the undergraduate level.
2. Skills: Both require analytical thinking, problem-solving, computational analysis and statistical evaluation.
3. Mathematical modeling: Both disciplines use mathematical modeling, combining computational methods and theoretical evaluation.
4. Dependency on calculations: Both rely on formulas, equations and statistical measurements for research and practical applications.

How pure math and applied math join forces

Epidemiology examines health patterns and causes in populations. Applied math aids in modeling disease spread and shaping public health policies, while pure math develops models like differential equations to explain epidemic dynamics, supporting decisions during outbreaks like COVID-19.

Fourier analysis is a way that mathematicians and scientists break down complicated patterns into simpler ones. It works by looking at analyzing patterns to understand how things like sound, light or even ocean waves behave. Rooted in pure mathematics, it draws on real analysis, functional analysis and abstract algebra. Today, it is essential in fields like signal processing, image compression (e.g., JPEG), audio engineering and quantum mechanics, solving real-world problems through applied mathematics.

What can you do with a math degree

Studying mathematics equips you with essential skills like analytical thinking, logical reasoning, quantitative analysis and problem-solving. These are all highly adaptable and valuable skills in industries like finance, engineering, data science, healthcare, technology and education.

Examples of careers in pure mathematics

• Professor, projected salary: $84,380
• Mathematician, projected salary: $112,110
• Investment analyst, projected salary: $73,311
• Topographer, projected salary: $76,210
• Physicist, projected salary: $155,680
  

Examples of careers in applied mathematics

• Operations research analyst, projected salary: $83,640
• Data scientist, projected salary: $108,020
• Statistician, projected salary: $104,860
• Portfolio analyst, projected salary: $99,890
  

 These disciplines work hand in hand. Together, they cultivate transferable abilities like logical reasoning, adaptability and the clear communication of complex ideas—qualities highly valued across industries and essential for driving progress and innovation.

Applied math major

Central Michigan University's applied math major emphasizes practical problem-solving by integrating computational tools and applied mathematical theories. Students will gain a deep understanding of how mathematics drives innovation in areas such as algorithm design, engineering, operations and scientific research.

Through this curriculum, you will develop analytical skills, critical thinking and the technical knowledge needed to excel in a variety of career paths—from academia and engineering to consulting and technology.

Sneak peek at applied math subjects:

• Numerical analysis: Learn computational methods to solve mathematical problems like equations, linear systems, interpolation, differentiation, integration and differential equations.
• Optimization theory: Learn key concepts and algorithms for decision-making and resource management, including linear and nonlinear programming.
• Applied mathematics and differential equations: Explore differential equations in physical and engineering sciences, covering topics like high-order equations, series solutions, Bessel functions, stability, boundary value problems and an intro to partial differential equations.
• Operations research I: Learn mathematical programming techniques like the simplex method, duality and sensitivity analysis to optimize systems in logistics, finance and manufacturing.

Pure math major

Central Michigan university's pure math major is crafted for students who want to explore both the theoretical depth and practical applications of mathematics. It equips you with the tools to unravel complex problems and uncover innovative solutions, preparing you for a future in fields like engineering, theoretical research, data analytics and beyond.

With its strong emphasis on advanced concepts, this concentration nurtures your analytical thinking, creativity and problem-solving skills.

Sneak peek at pure math subjects:

• Elements of complex variables: Study complex numbers, analytic functions and applications in physics and engineering.  
• Theory of numbers: Explore integers, prime numbers and their role in cryptography and coding.  
• Advanced linear algebra: Learn vector spaces, matrices and eigenvalues for quantum mechanics and machine learning.  
• Advanced calculus: Focus on multivariable calculus and differential forms for science and engineering.  
• Curves and surfaces: Delve into differential geometry with applications in physics and design.  
• Introduction to point-set topology: Understand mathematical spaces and foundational concepts for advanced math and physics.  
• Combinatorics and graph theory: Analyze counting, networks and graph relationships for computer science and logistics.