# The Key Differences Between Applied Math and Pure Math

What is the difference between pure and applied math? Don’t worry if you don’t know the answer! I’m here to help explain.

When I got to Harvard as a first-year student, I had some idea of what I wanted to study. I was a STEM-oriented student, but I was also a music lover and open to learning about new disciplines. Math was a subject I was planning on continuing so I could have a solid foundation in science classes. I then learned that there were two types of math I could study: pure mathematics and applied mathematics. Initially, I felt slightly overwhelmed because I had never heard of the distinction before; I thought math was just math! Luckily, both the Math and Applied Mathematics departments have very helpful and kind people who are here to help students.

Logistically, pure mathematics requires 12 courses (48 credits), of which 8 are in the math department and four are in math or related subjects. Applied mathematics requires 14-15 courses, including five courses in a chosen application area. Happily, if you are not sure which to choose, but you do know you want to study mathematics, many courses overlap and will count for both concentration requirements.

In terms of what differs between the two concentrations, it is actually pretty intuitive! Essentially, pure mathematics focuses on abstract and theoretical concepts in math working to prove theorems and research and discover new realms of math. Pure math courses are heavily proof-based and deal with theoretical possibilities of math. On the other hand, applied mathematics focuses on ways to use math. This is why all students choose a specific application area within their applied math concentration. For example, many students choose economics as their application area and take courses learning the mechanics of how to use math in economic situations.

Students also like to study at Widener Library in the many study nooks as well as large study rooms.

Math is also interdisciplinary. Last semester, I took Applied Math 50: Introduction to Applied Mathematics. In this class, we learned about many different areas of study in math, including topics like epidemiology, game theory and network theory. The exposure to many different subjects helped us all gain an understanding of what sorts of processes and systems math can help create and also see how they might relate. Many processes that help create models in one subject, like epidemiology, can also be used in the same way to learn how connected humans are socially. The end of the class concluded with a final project where we used coding skills and math processes we had learned about to research our own topic. My group decided to study how the attributes of popular music in the U.S. affect their chart rating. It was fascinating to combine music and math together and come out with analyses from the data.

Both concentrations are highly useful for gaining a broad and comprehensive understanding of math that will help students in the workforce or in further academic endeavors. Math is simply everywhere and serves as the backbone to many systems we rely on every day and might even take for granted.

Mathematics is a vast field with many branches and specializations. Two of the main divisions in mathematics are applied math and pure math. While both fields involve numbers equations and logical thinking, they have some notable differences that students should understand when choosing a degree program or career path. In this article, we’ll break down the key distinctions between applied and pure mathematics.

## Defining Applied Math

Applied mathematics focuses on using mathematical concepts to solve real-world problems. According to Dr. Edwin Ding, an associate professor of mathematics, applied math concentrates more on applying analytical and computational techniques to various fields like physics, computer science, engineering, and statistics.

Some examples of applied math include modeling the spread of disease, maximizing the aerodynamics of cars, developing data encryption methods, and predicting weather patterns. Unlike pure math which deals with abstract theories, applied math has a very tangible connection to analyzing and improving things in the physical world.

## Defining Pure Math

Pure mathematics deals with the theoretical foundations of mathematical principles It involves proving theorems, developing new mathematical theories, and analyzing abstract concepts. Pure math explores questions based on their inherent interest and beauty, not necessarily their practical applications

Fields like number theory, algebra, geometry, and topology are considered pure branches of math. Rather than solving practical problems, pure math is about discovering universal truths and expanding the scope of mathematical knowledge itself. While not directly applied, pure math research helps lay the groundwork for many applied math solutions.

## Comparing Applied and Pure Math Coursework

In the first couple years of a math program, applied math and pure math students take many of the same foundational courses like calculus, differential equations, linear algebra, and probability. Later in the curriculum is where the paths diverge.

Applied math students take specialized courses relevant to their concentration like numerical methods mathematical modeling actuarial science, or operations research. And they complete in-depth projects and labs to tackle real-world math challenges.

Meanwhile, pure math students pursue more theoretical electives like abstract algebra, complex analysis, and topology. Their assignments emphasize writing rigorous proofs versus hands-on applications.

Both paths involve high-level math fluency. But applied math leans more computational while pure math is more conceptual.

## Math Degree Options

Many colleges offer degrees in both applied and pure mathematics. For example, at XYZ University students can earn a Bachelor of Science in Applied Mathematics. This program teaches computational skills for fields like engineering, business, and statistics.

On the other hand, XYZ University’s Bachelor of Arts in Mathematics focuses on pure math topics like analysis, geometry, and logic. This builds a foundation for graduate study or math research careers.

Ultimately, students should choose between applied or pure mathematics based on their talents, interests, and career aims. Advisors can help students determine which path aligns best.

## Careers for Applied vs Pure Math Majors

Applied math graduates are well-prepared for jobs that involve math modeling, data analysis, and problem-solving. Common careers for applied math majors include:

• Actuary
• Operations research analyst
• Financial analyst
• Data scientist
• Software developer
• Epidemiologist
• Research engineer

Meanwhile, pure math grads are sought after for more theory-focused roles like:

• Cryptographer
• Mathematical researcher
• Mathematician
• Quantitative analyst
• Actuarial scientist
• Data analyst
• Professor

There is high demand for mathematical competency in our tech-driven world. Both applied and pure math majors can find rewarding careers that utilize their quantitative strengths.

## Should I Choose Applied or Pure Math?

Here are a few key questions to help decide between applied versus pure mathematics:

• Do you enjoy solving practical real-world problems or theoretical conceptual problems more?
• Are you interested in computational methods or abstract proofs?
• Do you want to work in a technical field like finance, data science, or engineering? Or pursue academia and research?
• Do classes on advanced calculus, differential equations, and statistics appeal to you? Or courses on mathematical logic, topology, and analysis?

Applied math equips students with versatile analytical skills to create solutions for complex systems and scenarios. Pure math cultivates precise abstract thinking to discover the general principles that underpin our reality.

Both fields are exciting in their own right. By assessing their skills and interests, students can determine which mathematics branch fits them best. With an applied or pure math major, they will gain a versatile skill set that opens up many rewarding career opportunities.

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What is the difference between pure and applied math? Don’t worry if you don’t know the answer! I’m here to help explain.

When I got to Harvard as a first-year student, I had some idea of what I wanted to study. I was a STEM-oriented student, but I was also a music lover and open to learning about new disciplines. Math was a subject I was planning on continuing so I could have a solid foundation in science classes. I then learned that there were two types of math I could study: pure mathematics and applied mathematics. Initially, I felt slightly overwhelmed because I had never heard of the distinction before; I thought math was just math! Luckily, both the Math and Applied Mathematics departments have very helpful and kind people who are here to help students.

Logistically, pure mathematics requires 12 courses (48 credits), of which 8 are in the math department and four are in math or related subjects. Applied mathematics requires 14-15 courses, including five courses in a chosen application area. Happily, if you are not sure which to choose, but you do know you want to study mathematics, many courses overlap and will count for both concentration requirements.

Summertime in Harvard Yard

In terms of what differs between the two concentrations, it is actually pretty intuitive! Essentially, pure mathematics focuses on abstract and theoretical concepts in math working to prove theorems and research and discover new realms of math. Pure math courses are heavily proof-based and deal with theoretical possibilities of math. On the other hand, applied mathematics focuses on ways to use math. This is why all students choose a specific application area within their applied math concentration. For example, many students choose economics as their application area and take courses learning the mechanics of how to use math in economic situations.

Students also like to study at Widener Library in the many study nooks as well as large study rooms.

Math is also interdisciplinary. Last semester, I took Applied Math 50: Introduction to Applied Mathematics. In this class, we learned about many different areas of study in math, including topics like epidemiology, game theory and network theory. The exposure to many different subjects helped us all gain an understanding of what sorts of processes and systems math can help create and also see how they might relate. Many processes that help create models in one subject, like epidemiology, can also be used in the same way to learn how connected humans are socially. The end of the class concluded with a final project where we used coding skills and math processes we had learned about to research our own topic. My group decided to study how the attributes of popular music in the U.S. affect their chart rating. It was fascinating to combine music and math together and come out with analyses from the data.

Both concentrations are highly useful for gaining a broad and comprehensive understanding of math that will help students in the workforce or in further academic endeavors. Math is simply everywhere and serves as the backbone to many systems we rely on every day and might even take for granted.