So, I have been looking into college courses to see what I want my major to be, and I noticed that M.I.T offers two specified types of Mathematics. Pure and Applied. What is the major differences in Pure vs. Applied and what are examples of each?
2026-04-02 02:04:29.1775095469
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Difference between Pure and Applied Mathematics?
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It can be boiled down to (my choice of) Algebra, Number Theory, Topology and Gemoetry versus Statistics, Mechanics and Applications ot Physics.
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AFAIK "applied" means applied to problems outside mathematics, while pure deals with problems just relevant for mathematics (at that time).
Also there might be considerations of solving problems for specific instances or solving them efficiently instead of just knowing that they are solvable.
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I would have to say that pure mathematics involves pure numbers (and other objects that don't have units of measurement) while applied mathematics involves quantities (numeral values and units of measurement such as volts or dollars).
For example, if you are studying physics or statistics without using any units of measurement, then these would be forms of pure mathematics (mathematical physics and mathematical statistics). But if you are studying them using units of measurement, then they are applied mathematics (applied physics and applied statistics).
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As previous answers mentioned, there is a lot of overlap between Pure Mathematics and Applied Mathematics because they might use similar techniques or concepts. What makes them different is the goals set on in Applied Mathematics versus Pure Mathematics, I am just going to refer them to AM and PM respectively. For AM the goal is typically to advance mathematics for the sake of some practical purpose, now this might still be quite theoretical, for instance people in mathematical physics might be creating new math for the sake of advancing our understanding the physical world. While in PM is advanced for the sake of advancing mathematics without a concern for its practical applications, even if practical applications do exist or not.
What is important to also emphasize is that one field is not necessarily more difficult than the other one. For instance there are mathematicians who study the mathematical ideas behind General Relativity, this involves very technical concepts that require studying a lot of PM but serves AM as it allows us to have better physical understanding of our universe. Whether PM is harder or not than AM I believe is not a very useful conversation though.
The lines between these two divisions, much as in all divisions within mathematics, is often blurred. However, the traditional division between the two is that Applied Mathematics has a very clear connection to physical real-world problems. At its heart are PDE's, but also included are things like numerical methods and (once upon a time) what are now called computer science and statistics.
Pure Mathematics is mathematics for its own sake, pursuing questions based on the internal attractiveness of the questions. At its heart is number theory.