Mathematics is a systematic field of study that involves the exploration of abstract structures, patterns, and relationships using numbers, symbols, and logical operations. It is a universal language that helps us understand and describe various aspects of the world around us. Mathematics encompasses a wide range of branches, including arithmetic, algebra, geometry, calculus, statistics, and more.

Key elements of mathematics include:

1. Numbers and Operations: The study of numbers and the basic operations like addition, subtraction, multiplication, and division.

2. Algebra: Deals with variables and symbols to represent quantities and the rules for manipulating these symbols.

3. Geometry: Focuses on the properties and relationships of shapes, sizes, and dimensions.

4. Calculus: Concerned with rates of change and accumulation, particularly in the context of functions and their graphs.

5. Statistics: Involves the collection, analysis, interpretation, presentation, and organization of data.

6. Logic: The study of reasoning and argumentation, providing a foundation for mathematical proofs and problem-solving.

Mathematics plays a crucial role in various fields such as physics, engineering, economics, computer science, and many other scientific disciplines. It provides a framework for understanding and solving problems, making it an essential tool in both theoretical and applied contexts.

Determining the "hardest" type of math is subjective and depends on individual strengths, interests, and background. However, some branches of mathematics are often considered more advanced and challenging due to their abstract nature and complexity. These include:

1. Advanced Calculus and Analysis: Topics like real analysis and complex analysis can be quite challenging, delving into rigorous proofs and abstract concepts.

2. Abstract Algebra: This branch involves studying algebraic structures, such as groups, rings, and fields, and can be highly abstract and theoretical.

3. Topology: Topology explores properties of space that remain unchanged under continuous deformations, and it can involve intricate concepts and advanced mathematical reasoning.

4. Differential Geometry: This area of mathematics connects calculus and geometry, dealing with smooth surfaces and curves in higher-dimensional spaces.

5. Functional Analysis: This branch extends ideas from linear algebra and calculus to infinite-dimensional spaces, introducing abstract concepts and complex mathematical structures.

6. Number Theory: Focusing on the properties of integers and number patterns, number theory involves deep mathematical reasoning and has connections to cryptography and computer science.

7. Mathematical Logic: This area deals with the foundations of mathematics and the study of formal systems. It can involve complex symbolic manipulation and logical reasoning.

Keep in mind that what one person finds challenging, another might find more accessible based on their background and interests. Additionally, the perceived difficulty of a mathematical topic can vary based on how it is taught and the resources available.

The scope of mathematics is vast and continually expanding, ranging from elementary concepts to highly specialized and advanced areas. It is challenging for one person to learn every branch of mathematics thoroughly within a single lifetime. Here are a few reasons why:

1. Breadth of Topics: Mathematics encompasses a wide range of topics, including arithmetic, algebra, geometry, calculus, number theory, statistics, and many specialized branches like topology, abstract algebra, and mathematical logic. Each of these branches has its own subfields and complexities.

2. Depth of Understanding: Achieving a deep understanding of advanced mathematical concepts often requires years of study, research, and practice. Mastery in one area may not necessarily translate to expertise in another.

3. Continuous Development: Mathematics is a dynamic field, with ongoing research leading to new discoveries and theories. Staying current with the latest developments in all branches of mathematics is a significant challenge.

4. Time Constraints: Learning mathematics to a high level of proficiency takes time and effort. With the vastness of the subject, it may not be feasible for one person to dedicate enough time to every branch and subfield while also maintaining a balanced life.

While it may be impractical for one person to learn every aspect of mathematics, individuals can certainly gain a broad understanding of the subject and specialize in specific areas. Mathematics allows for a diverse range of interests and expertise, and many mathematicians focus on particular domains that align with their passions and research interests.

Advanced math can be challenging for several reasons:

1. Abstract Concepts: Advanced math often deals with abstract and theoretical concepts that may not have direct real-world counterparts. Understanding and working with abstract ideas can be challenging for many people.

2. Complexity: As you progress in math, the problems become more intricate and multifaceted. Advanced math involves dealing with complex structures, equations, and theorems that require a deep understanding of foundational concepts.

3. Symbolic Representation: Mathematics often relies on symbolic representation, using symbols and notation to express relationships and concepts. For some, understanding and manipulating these symbols can be challenging.

4. Logical Reasoning: Advanced math requires a high level of logical reasoning. You need to follow a series of logical steps to arrive at a solution. This can be demanding for individuals who are not accustomed to thinking in a highly structured and logical manner.

5. Cumulative Nature: Math is cumulative; understanding advanced topics often depends on a solid grasp of earlier concepts. If there are gaps in foundational knowledge, it can become difficult to progress.

6. Lack of Intuition: Unlike some other subjects, advanced math may not always align with our intuitive understanding of the world. Concepts such as abstract algebra or complex analysis may seem counterintuitive and require a shift in the way we think.

7. Limited Visual Representation: While visual aids can be helpful in understanding many mathematical concepts, some advanced topics are challenging to represent visually. This lack of visual intuition can make it harder for some individuals to grasp certain ideas.

8. Problem-Solving Nature: Math often involves problem-solving, and the process of solving complex problems can be mentally taxing. It requires patience, persistence, and the ability to approach problems from different angles.

9. Perception of Difficulty: The perception that advanced math is difficult can create a mental barrier. This belief can lead to anxiety or a lack of confidence, making it even more challenging to tackle mathematical problems.

10. Teaching Methods: The way math is taught can also influence its perceived difficulty. If teaching methods do not cater to different learning styles or fail to provide practical applications, students may struggle to engage with the material.

It's essential to note that difficulty in math is subjective, and individuals may have varying degrees of aptitude and interest in the subject. Effective teaching methods, a supportive learning environment, and a positive mindset can contribute to overcoming challenges in advanced math.