# Interesting Math Research Topics to Write a Good Paper

As the saying goes, "Mathematics is the mother of all sciences." There is a good reason for this. Regardless of what kind or branch of science we consider, there is significant use of mathematical tools (e.g., numbers) or ideas (e.g., math formulae) in all sciences. You name it: Physics, Astronomy, Biology, Sociology, Chemistry, and so on all 'borrow' from mathematics.

Mathematics is, as a matter of fact, crucial to our collective progress as humans. To appreciate this, imagine for some minutes what or how our existence today would be without numbers and all other knowledge we have gained from the study of mathematics over thousands of years. Consider the following questions for some context:

• How would you communicate to someone that you are much taller than your friend and to what extent?
• How would you express the current time and year, or tell a friend how much time has passed since you last met each other?
• How would you understand that a simple pen is much affordable than a Bugatti Veyron luxury car?
• How would you know and express the idea that a bar of gold is much heavier and expensive than a bar of chocolate that appears to be of the same physical dimensions?

Clearly, without mathematics, life would be simply unbearable in every meaningful and practical way. It is, therefore, an essential part of our lives.

Now, one has to wonder, "What is mathematics?" Although there is no single and universally held definition of the discipline, mathematics can be defined as the study of pattern and relationships in the physical and abstract domains using numbers. Whether we are interested in quantifying the rate at which an object changes position as it moves in a specific direction (that is the speed of the object), the relationship between the height and width of an object, or the time it takes the earth to revolve around the sun, all of these questions are mathematical in character and their answers are also rooted in mathematics.

Now that is indubitably clear that math is important, it should come as no surprise that every student studies mathematics even from the earliest levels of education. Learning to count and identify numbers — something every kid does at kindergarten or nursery school — is the first step we all take in studying math. That also underscores the inextricable relevance of numbers in math. At high school and college levels, students will continue studying mathematics in various depth and manner. One of the most intellectually rewarding things to do when studying mathematics is to solve problems and do math research work. If you are a mathematically curious type of student, you will have a lot of fun exploring cool math topics to research considering that mathematics has numerous fields or branches. They are listed as follows.

• Pure mathematics: This branch is focused on the abstract world. It studies abstract concepts and entities. It should be noted that they often may have relevance to the physical world, therefore help us understand the connected aspects of the physical world. However, there are arguments that abstract objects can be studied just to understand their nature even when without any bearing to the physical world. Some areas under this include Analysis, abstract algebra, geometry, and number theory.
• Applied mathematics: As the name implies, this is the branch of mathematics that is concerned with the application of mathematical methods and ideas in different areas of human endeavors such as business, sports, computer science, or engineering.

Major and popular topic areas in mathematics that fall under the above two branches include:

• Analysis — this area is focused on studying functions and their properties. If you have come across concepts such as continuity, limits, differentiation, or integration, you now know where they are rooted in mathematics. Although you may be more familiar with such concepts in Calculus, this is where their foundation lies, allowing a more thorough study of them. Subfields include real analysis (which focuses on real numbers), complex analysis (concerned with complex numbers), and functional analysis (focused on the study of vector spaces with many and infinite dimensions considering functions as loci in the spaces).
• Algebra — this is the study of mathematical symbols and their relationship and how to manipulate the symbols. It includes elementary algebra and abstract algebra. Equations and inequalities are some of the most common relationships that are studied in algebra yet the less familiar ones include rings and fields. Algebra involves using symbols to express unknown quantities while numbers are used for known quantities.
• Arithmetic — this is a subfield of algebra. Arithmetic is focused on the relationship between numbers in basic or traditional number operations such as addition, subtraction, multiplication, and division. It primarily differs from elementary algebra because it makes no use of unknown quantities, whereas elementary algebra does so.
• Calculus — this is the field of mathematics that studies change and more interestingly continuous change. Differential and integral calculus are two main divisions of calculus.
• Combinatorics — this is the area of mathematics that studies discrete structures that are finite or countable and how they can be counted or combined (hence, the name 'combinatorics'). This area is very much connected with other areas of math.
• Geometry — this is the field of mathematics focused on the study of shapes and spaces.
• Topology — this relatively new branch of mathematics stems from geometry. It is focused on the measurement and properties of shapes and spaces when they are subjected to smooth changes such as twisting, bending or stretching (but not cutting or tearing). It differs from the traditional understanding of geometry which focuses on exact measurements and sizes of objects or spaces.

When choosing from research topics in math, consider the complexity of the topic and your ability to properly work on the topic, in addition to the time it will take to complete the research. In educational institutions, there are always deadlines for submission. Many students — even brainy ones — do struggle with completing their math research papers. We offer distinguished academic writing and assistance services that can help you boost your performance in mathematics and other subjects. In the final section of this article, you will learn how we stand apart from others.

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## Math Categorical Research Topics by Educational Level

Although mathematics as a discipline and broad subject area has basic characteristics that unite all branches, the depth, and complexity involved in studying them varies. That is largely what informs the fact that the depth and scope of mathematics studied by students vary across educational levels. Typically, for students at the lower levels such as elementary and high school, the scope and depth or complexity are much more accessible. But as one progresses to higher educational levels (undergraduate and postgraduate studies), the scope and depth increase with exposure to tougher topics and ideas.

With this in mind, we will explore some mathematics research topics that will be appropriate for students according to their educational level.

### Math Research Topics for Elementary School Students

1. How to identify even and odd numbers.
2. Effective ways for adding double digit numbers.
3. Efficient techniques for subtracting double digit numbers.
4. Simple ways to tell the time.
5. Patterns for counting effectively and quickly.
6. Grouping objects by shapes and color.
7. Finding and identifying shapes in combined shape patterns.
8. Compare the properties of squares and rectangles.
9. Identifying common geometric shapes in naturally occurring objects.
10. Efficient techniques for multiplying up to two-digit numbers.

### Math Research Topics for Middle School

1. Multiplication techniques for quickly multiplying two-digit numbers.
2. Exploring the formulae for calculating areas of shapes.
3. Techniques of counting discrete objects faster by sight.
4. Explore Euclid's contributions to mathematics.
5. What were some major applications of mathematics in Ancient Rome?
6. Compare the contributions of Greek and Roman civilizations to the development of mathematics.
7. Ways to find patterns in Pascal's triangle.
8. Why are Greek symbols often used in mathematics: the origins and reasons.
9. The history of algebra and its applications across early civilizations.
10. Explore the contributions of Blaise Pascal to the study of probability.

### Math Research Topics for High School

1. An exploration of Fibonacci series.
2. Evaluating the methods of solving simultaneous equations.
3. Good ways to understand and prove theorems in trigonometry.
4. A history of Archimedes and important contributions he made to mathematics.
5. Essential mathematicians in Ancient Greece and their main contributions to math.
6. Role of mathematics during the Renaissance Era in Europe.
7. What were major accomplishments and applications of mathematics in Ancient Egypt?
8. Compare real numbers and imaginary numbers.
9. How did mathematics influence Ancient Greek architecture?
10. Analyze Pythagoras' theorem using real world applications.

### Math Research Topics for College Students — 10 Sample Topics

1. Investigating random walks and limits of diffusion.
2. Exploring symmetries of differential equations.
3. Important advances in the field of topology over the last 20 years.
4. Analyzing geometry in high dimensional spaces.
5. A mathematical exploration of the kinematics of rolling.
6. Advances in techniques or algorithms for proving primeness of number in polynomial time.
7. Applying symmetric functions in combinatorics.
8. Statistical analysis of suicide data in the united states over the past 20 years.
9. Solving linear matrix inequalities with rank constraints.
10. Algebra: an exposition on the invariants of matrices.

### Math Ph.D. Research Topics — 10 Sample Topics

1. Exploring the geometry of singular spaces.
2. Computational fluid dynamics: exploring some underlying mathematical foundations.
3. Exploring the manifolds of systems having multiple time scales.
4. Advances in predictive models for weather forecasting.
5. An exploration of calculus of functors and their applications.
6. Exploring Knot theory and its applications.
7. Exploring the geometric aspects of complex analysis for higher dimensions.
8. Applying Monte Carlo methods to solve inverse problems.
9. Mathematical modelling of single polymer entanglements.
10. Abstract harmonic analysis: operator space techniques.

The above is by no means an exhaustive list of topics. They could be expanded further. We simply provided them as a guide to help you figure out your topic. Going through the lists, you may not necessarily stumble on the exact topic that may be to your liking, but we hope that you get a good pointer to (or epiphany of) what your preferred topic is.

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