Mathematics

Mathematics is a wonderful subject that helps us make sense of the world. It studies numbers, shapes, patterns, and how they connect. With math, we can solve problems, predict what might happen, and explore ideas that are not easy to see in daily life.

Illuminated letter P at the beginning of Adelard of Bath's translation of Euclid's Elements, which starts "Punctum est illud cui pars non est" (A point is that which has no part). Geometry is shown personified as a woman, following Martianus Capella's De nuptiis Philologiae et Mercurii.

There are many parts to mathematics. Some areas focus on numbers and their features, like number theory. Others look at shapes and spaces, such as geometry. Algebra examines how different values relate to each other, and analysis studies changes and movement. More abstract areas, like set theory, lay the groundwork for all of math.

Math is useful in many areas of life. It helps scientists study nature, engineers build structures, doctors care for people, and economists handle money. Even computers and video games need math to function! While math can model real situations, its discoveries are based on logic and proof, not just testing. From ancient times to now, mathematics has grown and remains a powerful tool that shapes our world.

Areas of mathematics

Before the Renaissance, mathematics had two main parts: arithmetic, which studies numbers, and geometry, which studies shapes. Some old ideas, like numerology and astrology, were not clearly different from math at that time.

Starting in the Renaissance, new areas like modern algebra and calculus began to grow. Algebra studies ways to solve equations, and calculus looks at how things change smoothly. These four areas—arithmetic, geometry, algebra, and calculus—were the main ones until the late 1800s.

This is the Ulam spiral, which illustrates the distribution of prime numbers. The dark diagonal lines in the spiral hint at the hypothesized approximate independence between being prime and being a value of a quadratic polynomial, a conjecture now known as Hardy and Littlewood's Conjecture F.

Today, there are many more areas of mathematics. Number theory studies integers and their properties. Geometry has many parts, like studying curves and spaces. Algebra looks at patterns in equations. Calculus and analysis explore how things change and move. There are also areas like discrete mathematics, which studies countable objects, and mathematical logic, which looks at the rules of reasoning. Each area helps us understand the world in different ways.

Main article: Number theory

Main article: Geometry

On the surface of a sphere, Euclidean geometry only applies as a local approximation. For larger scales the sum of the angles of a triangle is not equal to 180°.

Main article: Algebra

Main articles: Calculus and Mathematical analysis

Main article: Discrete mathematics

Main articles: Mathematical logic and Set theory

Main article: Computational mathematics

History

Main article: History of mathematics

Image of Problem 14 from the Egyptian Moscow Mathematical Papyrus. The problem includes a diagram indicating the dimensions of the truncated pyramid.

The word mathematics comes from the Ancient Greek word máthēma, meaning 'knowledge'. It entered English during the Late Middle English time through French and Latin.

Ancient people knew how to count things and understand time. Around 3000 BC, the Babylonians and Egyptians used arithmetic, algebra, and geometry for building and astronomy. The oldest math books are from 2000 to 1800 BC. By the 5th century BC, Greek mathematics became a special subject. Around 300 BC, Euclid organized math using basic rules.

During the Golden Age of Islam, many new math ideas were created. Greek and Arabic math books were translated into Latin and shared in Europe. In the early modern period, math grew fast in Western Europe, with new ideas like variables and calculus. Leonhard Euler brought many of these ideas together in the 1700s. In the 1800s, Carl Gauss added to many parts of math. In the early 1900s, Kurt Gödel changed math with his important theories.

Symbolic notation and terminology

Main articles: Mathematical notation, Language of mathematics, and Glossary of mathematics

Mathematical notation helps scientists and engineers share big ideas in a clear way. It uses symbols for numbers, adding, multiplying, and more. For example, we use "+" for adding and "×" for multiplying. These symbols help us solve problems more easily.

Mathematics has two big parts: pure mathematics, which we study just because it is interesting, and applied mathematics, which we use to solve problems in the real world. Even though they are different, they often help each other. Ideas from pure math can solve real problems, and real-world questions can lead to new math discoveries.

Philosophy

Main article: Philosophy of mathematics

The relationship between mathematics and the real world has sparked discussions for centuries. Ancient thinkers like Plato believed that mathematical ideas exist beyond space and time. Today, many mathematicians think of their subject in a similar way, treating mathematical concepts as real objects.

There isn’t a single way to define mathematics. Some say it’s the study of quantity, while others focus on how mathematicians prove ideas using logic and rules. Over time, mathematics has grown to include many areas, making it hard to define by one object of study. What makes mathematics special is its careful, logical approach to proving ideas true, a tradition that began in ancient Greece.

Training and practice

Education

Mathematics is taught in schools all over the world because it is important for many parts of life and work. People with good math skills can become teachers, scientists, or work in finance and technology. Long ago, people in places like Babylon and Egypt learned math, and today, almost every country teaches math in their schools.

Learning math can sometimes feel hard or make some students nervous. This is called mathematical anxiety. It can happen because of how teachers, parents, or friends talk about math. But there are ways to help, like changing how math is taught or getting support from family and teachers.

Psychology (aesthetic, creativity and intuition)

Even though math proofs must be exact, being creative is also important for mathematicians. Solving hard math problems often needs new and clever ideas. Some mathematicians enjoy math like solving puzzles, and many find beauty in math, liking simple and elegant solutions. This beauty in math is sometimes compared to art, showing that math can be both creative and logical.

Cultural impact

Artistic expression

Fractal with a scaling symmetry and a central symmetry

Mathematics influences art and music in many ways. Notes that sound good together often have simple ratios. For example, an octave doubles a sound's frequency, and a perfect fifth multiplies it by 3/2. Humans and some animals find symmetry beautiful. Mathematical symmetry describes patterns like mirror symmetry, seen in butterflies. Waves and fractals show special patterns too, such as translation symmetry and self-similarity.

Popularization

Main article: Popular mathematics

The front side of the Fields Medal with an illustration of the Greek polymath Archimedes

Making math easy for everyone can be hard because math can be abstract. Writers who explain math use simple examples or real-life connections to help people learn. Math is not often a topic in popular books or TV shows.

Awards and prize problems

Main category: Mathematics awards

The highest honor in math is the Fields Medal, given every four years to up to four people. It started in 1936 and is like the Nobel Prize for math. Other important awards include the Abel Prize, the Chern Medal, the AMS Leroy P. Steele Prize, and the Wolf Prize in Mathematics.

Mathematicians have worked on famous unsolved problems. In 1900, David Hilbert listed 23 big questions, and many have been answered since. In 2000, seven new challenges called the Millennium Prize Problems were announced, each with a reward. So far, only one — the Poincaré conjecture — has been solved.