What Can Be Harder than Calculus? (With 7 most difficult mathematics subjects!)

What Can Be Harder than Calculus
What Can Be Harder than Calculus

Calculus is often believed to be one of the most challenging math subjects. But believe it or not, several math subjects are even harder than calculus. If you are ambitious enough to study advanced maths, you will discover that mathematics will continue to be more challenging.

So, what can be harder than Calculus? There are a lot of maths subjects that are harder than calculus, and the difficulty of each subject depends on the person. Some people may find abstract concepts and proofs difficult as they require creativity and determination. Additionally, advanced mathematics typically opens up a whole new world of complex and interesting mathematical concepts.

Read on to find out some of the most difficult math topics beyond calculus and problems out there that can challenge even the best mathematicians.

You might also enjoy reading: 22 Famous Mathematicians and Their Contributions.

List Of Math Subjects Considered To Be Harder Than Calculus

Math is often a source of anxiety for students, and for a good reason, it can be tough. Generally, calculus is considered one of the most difficult math subjects, but it is not the only one.

Let’s explore some of the other hard math subjects that students may come across during their studies.

1- Advanced Mathematics Courses

For starters, many advanced mathematics courses are even more difficult than calculus. These courses include higher-level courses such as abstract algebra, real analysis, and topology.

Advanced mathematics courses typically require an in-depth understanding of calculus and other related topics in order to succeed.

Not only do they require an intense level of mathematical knowledge and skill, but they also require students to be able to think abstractly in order to solve complex problems.

2- Theoretical Math Problems

Another way to define “harder than calculus” is by looking at theoretical math problems that are more difficult than any calculus problem out there.

The Riemann Hypothesis is one example of such a problem; it has been unsolved for over 150 years and remains one of the most famous open problems in mathematics today.

Other examples include Fermat’s Last Theorem and Goldbach’s Conjecture. All three of these problems have stumped mathematicians for centuries and remain unsolved despite numerous attempts by some of the brightest minds on earth.

If you want to explore more unsolved theoretical math problems, I wrote a whole article about the hardest calculus problems that I encourage you to read.

3- Number Theory

Number theory is a branch of mathematics that deals with properties and relationships between whole numbers. Number theorists study prime numbers, divisibility, Diophantine equations, and modular arithmetic.

Many mathematicians consider number theory to be one of the purest forms of mathematics; however, it can also be quite challenging due to its abstract nature.

To master number theory, students need to have a strong grasp of algebraic concepts such as factoring and linear equations.

Number Theory is also called “the set of natural numbers.” The table below contains examples of a set of natural numbers. 

Set of Natural Numbers Examples
Odd1, 3, 5, 7, 9, 11, . . .
Prime2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, . . .
Even2, 4, 6, 8, 10, . . .
Square1, 4, 9, 16, 25, 36, . . .
Composite4, 6, 8, 9, 10, 12, 14, 15, 16, . . .
Cube1, 8, 27, 64, 125, . . .
1 (modulo 4)1, 5, 9, 13, 17, 21, 25, . . .
3 (modulo 4)3, 7, 11, 15, 19, 23, 27, . . .
Perfect6, 28, 496, . . .
Triangular1, 3, 6, 10, 15, 21, . . .
Fibonacci1, 1, 2, 3, 5, 8, 13, 21, . . .
The table Contains Examples Of a Set Of Natural Numbers (Source: Brown University)

What makes Number theory harder than Calculus? Because it can seem incredibly abstract, many number theorists spend years trying to understand certain patterns or solutions and still leaving some problems unsolved.

In addition, Number theory generally requires knowledge from multiple branches of mathematics, including algebra, geometry, logic, and probability theory.

4- Topology

Topology is sometimes called “rubber-sheet geometry” since the objects can be extended and contracted like rubber without being broken. But topology generally studies the properties of invariant spaces under continuous deformation (Source: University of Waterloo)

In other words, topology involves studying geometric properties that remain unchanged when objects are stretched or bent in certain ways. This can include things like shapes in two-dimensional space or higher dimensions, connectivity between objects within a space, and continuity in motion from one point to another.

Topology can be difficult due to its reliance on abstract concepts such as homeomorphisms and homotopy equivalence relations.

Generally, mastery of topology requires a deep understanding of abstract algebra and set theory, as well as an intuitive understanding of geometrical shapes and their properties.

5- Analysis

Generally speaking, analysis deals with approximating certain mathematical objects, including numbers or functions, by other things that are easier to understand or handle (Source: University of Waterloo)

Analysis can be considered an advanced form of calculus that focuses on infinite processes rather than finite ones like calculus does.

Analysis covers topics such as limits, sequences, functions, convergence tests, differentiation rules for different kinds of functions, integration techniques for different kinds of functions (including improper integrals), series expansions for certain functions (such as Taylor Series), Fourier Series expansions (for periodic functions), and more advanced topics such as vector fields and manifolds in multiple dimensions.

Analysis is considered one of the most challenging branches of mathematics because it requires rigorous proof-writing skills combined with an intuitive understanding of complex mathematical processes.

6- Real Analysis

Real analysis is an area of analysis that studies real numbers and functions defined on them. Moreover, real analysis deals with questions such as convergence of sequences and series, limits, continuity, differentiation, integration, Riemann integral, and power series expansions.

Typically, Real analysis demonstrates the utility of abstract concepts and teaches students to understand and construct proofs. 

What makes real analysis harder than calculus is that it requires an advanced level of understanding in calculus before it can be mastered.

If you are interested in exploring more about Real Analysis, I encourage you to watch this video from MIT Open Course Ware.

What Can Be Harder than Calculus
What Can Be Harder than Calculus

7- Linear Algebra

Linear algebra is the study of linear equations and their solutions. It can be used to solve problems in a variety of fields, including physics, engineering, economics, and computer science.

Generally, linear algebra involves understanding matrices and their properties, solving systems of linear equations, and manipulating vectors in two or more dimensions.

It is no surprise that linear Algebra is considered one of the most difficult math topics out there, and even experienced mathematicians find it a challenge.

If you are wondering why linear Algebra is harder than Calculus, I wrote a whole article that I encourage you to read.

What to read next:

Wrapping Up

For most students, Calculus may be challenging, but it turns out that there are plenty of things that can be harder than it.

From advanced mathematics courses, Number Theory, Topology, and Analysis to theoretical math problems, there are plenty of ways for students (and mathematicians) to challenge themselves beyond what they learn in their calculus classes. If you are looking for a challenge beyond calculus, these 7 areas will be a good way to get you started.

With enough hard work and dedication, you might even be able to solve one of these riddles yourself. Regardless, it is clear that there is much more out there for students and mathematicians who want to push their mathematical knowledge further than ever before.

Altiné

I am Altiné. I am the guy behind mathodics.com. When I am not teaching math, you can find me reading, running, biking, or doing anything that allows me to enjoy nature's beauty. I hope you find what you are looking for while visiting mathodics.com.

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