Pure mathematics explores abstract mathematical concepts and theories, while applied mathematics focuses on utilizing mathematical principles to solve real-world problems.
Here's a breakdown of the key distinctions:
Focus
- Pure Mathematics: Centers on the development of mathematical theory, often exploring abstract concepts without immediate practical applications. Its primary goal is to expand mathematical knowledge and understanding. Think of it as math for math's sake, rigorously proving theorems and exploring new mathematical landscapes.
- Applied Mathematics: Employs mathematical techniques and tools to address problems in other fields, such as physics, engineering, computer science, economics, and biology. The focus is on finding solutions to practical problems using existing or newly developed mathematical methods.
Goals
- Pure Mathematics:
- To discover new mathematical truths and relationships.
- To create general mathematical theories and frameworks.
- To deepen our understanding of mathematical structures.
- Applied Mathematics:
- To solve real-world problems using mathematical models and techniques.
- To improve existing solutions or develop new ones.
- To predict future behavior and optimize processes.
Examples
- Pure Mathematics:
- Number theory (e.g., studying prime numbers)
- Abstract algebra (e.g., group theory, ring theory)
- Topology (e.g., studying properties that are preserved under continuous deformations)
- Applied Mathematics:
- Fluid dynamics (e.g., modeling the flow of air around an airplane)
- Cryptography (e.g., developing secure communication methods)
- Optimization (e.g., finding the best way to allocate resources)
- Financial modeling (e.g., predicting stock prices)
- Machine Learning (developing algorithms based on mathematical models)
Approach
- Pure Mathematics: Relies heavily on rigorous proofs and logical deduction. The emphasis is on demonstrating the validity of mathematical statements using formal arguments. Courses are often heavily proof-based.
- Applied Mathematics: Employs mathematical modeling, numerical analysis, and computational methods to solve problems. While proofs are still important, the emphasis is more on obtaining practical solutions, which can sometimes involve approximations and simulations.
Relationship
It's important to recognize that pure and applied mathematics are not entirely separate disciplines. Advances in pure mathematics often lead to unexpected applications in other fields. Similarly, problems arising in applied mathematics can stimulate new research in pure mathematics. They often feed into each other. A new concept or discovery in pure math may sit unused for years, only to be used when an applied mathematician discovers the tool they need to solve a real-world problem.
In essence, pure mathematics builds the theoretical foundation, and applied mathematics uses that foundation to construct practical solutions.
