Mathematics

Mathematics is a fascinating field that helps us understand and organize the world around us. It involves studying numbers, shapes, patterns, and the relationships between them. Through mathematics, we can solve problems, make predictions, and explore abstract ideas that are not always obvious in everyday 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 different areas of mathematics. Some focus on numbers and their properties, like number theory, while others study shapes and spaces, such as geometry. Algebra looks at how different values relate to each other, and analysis deals with changes and motion. Even more abstract areas, like set theory, provide the foundations for all of mathematics.

Mathematics is important in many parts of life. It helps scientists understand nature, engineers design buildings, doctors treat patients, and economists manage money. Even computers and video games rely on math to work! While math can be used to model real-world situations, the truths it discovers are based on logic and proof, not just experiments. From ancient times to today, mathematics has grown and changed, becoming a powerful tool that continues to shape our world.

Areas of mathematics

Before the Renaissance, mathematics was divided into two main areas: arithmetic, which studies numbers, and geometry, which studies shapes. Some types of old ideas, like numerology and astrology, were not clearly different from mathematics back then.

Starting in the Renaissance, new areas like modern algebra and calculus began to grow. Algebra studies ways to solve equations, while 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 and structures 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 'something learned, knowledge, mathematics'. It entered the English language during the Late Middle English period through French and Latin.

Archaeological evidence shows that ancient peoples knew how to count physical objects and abstract quantities like time. Evidence for more complex mathematics appears around 3000 BC, when the Babylonians and Egyptians used arithmetic, algebra, and geometry for taxation, construction, and astronomy. The oldest mathematical texts are from 2000 to 1800 BC. By the 5th century BC, Greek mathematics began to emerge as a distinct discipline. Around 300 BC, Euclid organized mathematical knowledge using postulates and first principles, which evolved into the axiomatic method used in mathematics today.

During the Golden Age of Islam, especially in the 9th and 10th centuries, mathematics saw many innovations, including the development of algebra. The Greek and Arabic mathematical texts were translated into Latin during the Middle Ages and made available in Europe. In the early modern period, mathematics developed rapidly in Western Europe, with innovations such as the introduction of variables, symbolic notation, logarithms, coordinates, and calculus. Leonhard Euler unified these innovations in the 18th century. In the 19th century, Carl Gauss made numerous contributions to various fields of mathematics. In the early 20th century, Kurt Gödel transformed mathematics with his incompleteness theorems.

Symbolic notation and terminology

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

Mathematical notation helps scientists and engineers show complex ideas clearly and simply. It uses symbols for numbers, operations, and relationships. For example, we use "+" for adding and "×" for multiplying. These symbols make it easier to write and solve problems.

Mathematics has two main parts: pure mathematics, which is studied for its own beauty, and applied mathematics, which is used to solve real-world problems. Even though they are different, they often help each other. For example, ideas from pure mathematics can solve practical 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 philosophical 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 agreement on how to define mathematics. Some say it's the study of quantity, while others focus on the way mathematicians prove ideas using logic and rules. Over time, mathematics has grown to include many areas, making it hard to define by a single object of study. What makes mathematics special is its careful, logical approach to proving ideas true, a tradition that dates back to ancient Greece.

Training and practice

Education

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

Learning math can sometimes feel hard or make some students nervous, which is called mathematical anxiety. This 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 very exact, being creative is also very important for mathematicians. Solving tough 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 in their frequencies. For example, an octave doubles a sound's frequency, and a perfect fifth multiplies it by 3/2. Humans and some animals also find symmetry beautiful. Mathematical symmetry groups describe patterns like mirror symmetry, seen in butterflies and Rorschach tests. 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 to understand can be challenging because math can be abstract and sometimes causes anxiety. Writers who explain math in simple ways often use real-life examples or cultural connections to help people learn. Even so, 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 long 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 million-dollar reward for a solution. So far, only one — the Poincaré conjecture — has been solved.