Branches of mathematics

Mathematics is often divided into different “branches,” like the branches of a tree. Each branch focuses on certain kinds of problems, ideas, and methods. In this chapter, you will get a gentle overview of the main branches you will meet in this course and a few you may encounter later. The goal here is recognition, not mastery: you should be able to say “Ah, this sounds like geometry,” or “This is probability.”

Remember: each of these areas appears later in the course with its own dedicated chapters. Here we only sketch what each branch is about and what kind of questions it asks.

Arithmetic

Arithmetic is the study of basic number operations you use every day. It deals with:

• The usual kinds of numbers (like whole numbers, fractions, decimals).
• The four basic operations: addition, subtraction, multiplication, division.

Typical arithmetic questions:

• $27 + 35 = ?$
• $48 \div 6 = ?$
• If a book costs \$7 and you buy 3, how much do you pay?

Arithmetic is the foundation for almost everything else in mathematics. Other branches build on these basic operations and extend them to more types of numbers and more complex situations.

Algebra

Algebra is about using symbols (often letters like $x$ and $y$) to stand for numbers and describe relationships between them. Instead of just computing with known numbers, algebra lets you:

• Work with unknown values (variables).
• Write equations that express relationships.
• Solve for unknowns.

Typical algebra questions:

• Find $x$ if $2x + 3 = 11$.
• A number increased by 5 is 20. What is the number?

Algebra helps answer “what value makes this relationship true?” It is a powerful language for describing patterns and solving problems in a general way, not just with specific numbers.

Geometry

Geometry studies shapes, sizes, and the properties of space. It looks at:

• Points, lines, angles.
• Triangles, circles, polygons.
• Distances, areas, and volumes.

Typical geometry questions:

• What is the area of a triangle with base $10$ and height $6$?
• Are these two triangles the same size and shape?
• How long is the diagonal of a rectangle?

Geometry helps you reason about physical space and shapes, from simple diagrams on paper to real-world structures and designs.

Trigonometry

Trigonometry (often shortened to “trig”) focuses on relationships between angles and side lengths in triangles, especially right triangles. It introduces trigonometric ratios like sine, cosine, and tangent.

Typical trigonometry questions:

• Given one side of a right triangle and an angle, what is the length of another side?
• How high is a tree if you stand a certain distance away and measure the angle of elevation?

Trigonometry is especially useful in applications involving waves, rotations, navigation, and any situation where angles and distances interact.

Calculus

Calculus studies change and accumulation. It answers questions like:

• How fast is something changing at a particular moment?
• How much has something accumulated over a period (like area, distance, or total quantity)?

Calculus has two closely connected parts:

• Differential calculus: focuses on rates of change (derivatives).
• Integral calculus: focuses on accumulation (integrals), such as total area under a curve.

Typical calculus questions:

• If a car’s position changes according to a formula, what is its speed at a specific time?
• What is the area under a curve between two points?

Calculus is central in physics, engineering, and many sciences because it describes how things evolve over time and space.

Linear Algebra

Linear algebra studies vectors, matrices, and linear transformations. It is about:

• Quantities that have size and direction (vectors).
• Rectangular arrays of numbers (matrices) and how they transform vectors.
• Systems of linear equations.

Typical linear algebra questions:

• Does this system of equations have a solution? If so, what is it?
• How does a matrix transform points in the plane or space?

Linear algebra is essential in modern applications such as computer graphics, data science, and machine learning, because many complex systems can be modeled with linear structures.

Probability and Statistics

Probability and statistics deal with chance, data, and uncertainty, but with different focuses.

• Probability studies how likely events are to happen.
• Statistics studies how to collect, summarize, and interpret data.

Typical probability questions:

• If you flip a coin twice, what is the chance of getting two heads?
• What is the probability of drawing an ace from a deck of cards?

Typical statistics questions:

• What is the average (mean) of this list of numbers?
• How spread out are the data values?
• Based on a sample, what can we say about a whole population?

These tools are crucial in science, business, medicine, and everyday decision-making, where perfect certainty is rare.

Number Theory

Number theory focuses on properties of whole numbers, especially positive integers. It is sometimes called “higher arithmetic.” It explores:

• Divisibility and factors.
• Prime numbers.
• Patterns in the integers.

Typical number theory questions:

• Is $101$ a prime number?
• What are the prime factors of $60$?

Number theory is both deeply theoretical and surprisingly practical, especially in modern cryptography (methods for secure communication).

Discrete Mathematics

Discrete mathematics studies mathematical structures that are separate and distinct, not continuous. Its objects are often countable (like whole numbers or finite sets), not smoothly varying.

It includes topics like:

• Logic and proof.
• Combinatorics (counting possibilities).
• Graph theory (networks of points and connections).

Typical discrete math questions:

• In how many ways can 5 different books be arranged on a shelf?
• Can you visit every city in a network of roads exactly once and return to the start?

Discrete math underlies computer science, algorithms, and network analysis.

Differential Equations

Differential equations involve functions and their derivatives. They describe relationships where the rate of change of a quantity depends on the quantity itself, time, or other variables.

Typical differential equations questions:

• How does a population change over time if its growth rate depends on its current size?
• How does an object move under certain forces (like gravity and friction)?

Differential equations are fundamental in modeling physical, biological, and economic systems that evolve over time.

Mathematical Logic and Proof

Mathematical logic studies the formal rules of reasoning: how to build valid arguments and avoid contradictions. Closely tied to this is the study of proof techniques, which provide ways to show that statements are always true (or not).

Typical questions in this area:

• Given some assumptions, what can we logically conclude?
• How can we prove that a statement is always true, for all numbers of a certain kind?

Logic and proof are not just another branch; they support the entire structure of mathematics by clarifying what it means for something to be “true” in a mathematical sense.

How Branches Connect

Although we give each branch its own name, they are not isolated. They connect and overlap:

• Algebra appears in geometry (coordinates and equations of lines), calculus (functions and derivatives), and number theory (equations in integers).
• Geometry connects with trigonometry (angles and triangles), analytic geometry (curves described by equations), and calculus (curves and areas).
• Probability and statistics often use algebra and calculus in more advanced work.
• Linear algebra supports calculus, statistics, differential equations, and more.

As you move through this course, you will visit many of these branches. Over time, you will start to see how they fit together into one coherent picture of mathematics.