Applied mathematics students at North Central College.

Pure Mathematics vs. Applied Mathematics

Reviewed by Jacob Imm

Jan 11, 2023

Pure Mathematics vs Applied Mathematics

Is there anything more polarizing than math? There’s no question about the discipline’s usefulness in countless industries, but its reception from students at every level is more questionable. Some love it—others love to hate it.

If you fall under the first category, you may be considering a career in math. But once you begin researching your future, you’ll quickly find different majors and career paths for mathematicians. In particular, choosing between pure math vs applied math can lead you down very different roads. So, what is applied mathematics vs. pure mathematics, and which undergraduate mathematics degree should you choose? 

In this short guide, we’re comparing pure mathematics vs applied mathematics by looking at the necessary education, typical responsibilities and where each program can take you.

What is Pure Mathematics?

In the simplest terms, pure mathematics is math for math’s sake. Pure math explores abstract concepts, complex theories and never-before-solved problems. The field is more about study and research than solving a particular real-world issue (even if it can draw inspiration from practical applications).

For that reason, pure mathematics is a largely academic endeavor. Pure mathematicians test each other’s hypotheses and dive headfirst into uncharted mathematical waters.

Ultimately, there is no specific theoretical goal to pure mathematics. This isn’t to say that the pure math major is “useless”—quite the contrary, in fact. A pure mathematician's work could even reveal practical solutions for organizations, businesses, and applied mathematicians. However, “usefulness” is rarely the original intention behind a groundbreaking equation or formula.

What is Applied Mathematics?

Applied mathematics, on the other hand, is a more “practical” form of math. In most cases, applied mathematicians are tasked with a real-world problem to solve. They then use various branches of math to understand and answer the question at hand.

Because math is everywhere, applied mathematicians can work in a variety of industries. Applied mathematics jobs can be as diverse as:

  • Healthcare
  • Finance
  • Insurance
  • Chemistry
  • Logistics
  • Engineering

Applied mathematicians working in these sectors—and many others—have the opportunity to impact businesses and lives through math.

How Do Pure Mathematics and Applied Mathematics Differ?

To summarize, pure mathematics is any math that has not yet found use or adoption outside of the math community; applied mathematics is any math that has.

With that said, there are other, more practical differences between the two fields. Let’s look at a few of the ways they differ.


While there are no explicit scope limitations to pure or applied math, there are some implicit ones. However, you will have to remember that there are several differences between pure math vs. applied math. But, both branches can cover any topic in mathematics, including:

  • Calculus
  • Probability and statistics
  • Geometry
  • Algebra
  • Number theory

However, in practice, applied mathematicians may limit themselves to particular branches of math by focusing on a problem, rather than the math itself. They may look to past examples of similar solutions to find thought patterns that yielded results, then try to apply them to the issue they’re exploring.

Because pure mathematicians aren’t looking to use their work for practical purposes, they’re free to think way outside the box. Sometimes, this freedom even extends to “impossible” theories or concepts. The lack of guardrails can be exciting, but it can also be overwhelming or excessively nebulous at times.

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Degrees are available in pure mathematics and applied mathematics. And in the beginning, both four-year programs will typically be similar—after all, the underlying concepts behind pure and applied math are nearly identical.

Regardless of the degree or math major you pursue, you’ll likely begin with courses such as:

  • Statistics
  • Algebra
  • Physics
  • Chemistry

As you progress in your program, the paths diverge. Pure math students may dive deeper into number theory, theoretical physics and topology, while applied math students may explore more analysis, computing and mathematical modeling.

After graduating from either math major, you’ll generally earn a bachelor of science, though bachelor of arts degrees in mathematics also exist.

Potential Careers

Applied mathematicians can find work in any industry that makes use of logic and number-crunching, such as:

  • Actuarial science
  • Aerospace engineering
  • Financial analysis
  • Biology
  • Computer science

As such, salaries for applied mathematicians can vary. With that said, many can be above average, as these mathematicians tend to solve essential issues within organizations.

Because pure mathematics is less focused on the “real world,” there can be fewer industries with openings for pure math graduates. Instead, most pure mathematicians will find themselves in academic or research settings, working as:

  • Postsecondary teachers – Professors guide the next generation of mathematicians. According to the U.S. Bureau of Labor Statistics (BLS), postsecondary professors earn a median salary of $79,640/year.
  • Mathematician – Tackling math and applied statistics in every form, mathematicians earn a median annual salary of $96,280, according to the BLS.

Many pure mathematicians rely on grants and government funding to advance their research. Consequently, salaries can be difficult to guarantee.

Roles and Responsibilities

Your responsibilities as a mathematician will depend on the industry in which you work. Even so, a pure mathematics degree and an applied math degree have one essential difference.

Notably, applied mathematicians will spend more time meeting with clients and other stakeholders. Their schedules may ultimately be split between working with other departments to understand an issue and actually doing math.

Pure mathematicians will instead use the bulk of their time simply doing math. Because there’s no problem at hand, there are no parties to consult. As such, pure math can be more of a “behind closed doors” field.

Ready to Apply Yourself? Find an Applied Mathematics Program Today

In the end, there’s still plenty of overlap between these two fields. Both pure and applied mathematics foster a love of mathematics. If you enjoy crunching numbers and using logic, you’ll likely find both paths to be rewarding.

However, from a career perspective, an applied mathematics major can certainly be more appealing. Learning skills that translate to real-world professions can increase your job prospects after graduation.

If you decide that an applied mathematics program is for you, be sure to find an institution that helps you build a better future. At schools like North Central College, you’ll conduct research, participate in resumé-worthy internships and enjoy opportunities —to transition directly into a career. And, of course, you’ll immerse yourself in mathematics. For the math lover, there’s nothing better.

Jacob Imm is the assistant director of communications in the North Central College Office of Marketing and Communications. He has 12 years of collegiate communications experience and has worked with hundreds of college students. He has a bachelor’s degree from the University of Notre Dame and a master’s degree from Northern Illinois University. 



U.S. Bureau of Labor Statistics. (2022, September 8). Occupational Outlook Handbook: Mathematicians and Statisticians. U.S. Bureau of Labor Statistics. Retrieved November 15, 2022, from

U.S. Bureau of Labor Statistics. (2022, October 4). Occupational Outlook Handbook: Postsecondary Teachers. U.S. Bureau of Labor Statistics. Retrieved November 15, 2022, from