Quaternions are a number system that extends complex numbers and was first introduced by the Irish mathematician William Rowan Hamilton in 1843. The historical treatment of quaternions encompasses their discovery, development, and applications, as well as the controversies and advancements in mathematical theory associated with them. ### Discovery and Development 1. **Early Concepts**: Before quaternions were formally defined, mathematicians used various forms of complex numbers.

The historiography of mathematics is the study of the history of mathematics and how it has been interpreted, understood, and communicated over time. This field focuses not only on the historical development of mathematical concepts, theories, and practices, but also on how these developments have been recorded and analyzed by historians, scholars, and mathematicians themselves.

The history of logic is the study of the development of logical thought and systems throughout human history, encompassing ideas from various cultures and traditions. This evolution reflects broader developments in philosophy, mathematics, language, and science. Here's an outline of significant milestones in the history of logic: ### Ancient Logic 1. **Early Contributions (Pre-Socratic Era)**: - Early thinkers like Heraclitus and Pythagoras began to suggest logical structures in their exploration of nature and knowledge.

This gate set alone is not a set of universal quantum gates.

Notably, circuits containing those gates alone can be fully simulated by classical computers according to the Gottesman-Knill theorem, so there's no way they could be universal.

This means that if we add any number of Clifford gates to a quantum circuit, we haven't really increased the complexity of the algorithm, which can be useful as a transformational device.

A popular set of universal quantum gates derived from Clifford gates is Clifford plus T.

In the context of Wikipedia and other collaborative encyclopedia projects, a "stub" is a short article or entry that provides limited information on a topic and is often marked for expansion. The "History of mathematics" stubs would refer to short articles related to various aspects of the historical development of mathematics that need further elaboration. These stubs can cover a wide range of topics, such as: - Key mathematicians and their contributions throughout history. - Important mathematical discoveries and theories.

Mathematical problems are questions or challenges that require the application of mathematical concepts, principles, and techniques to find solutions or answers. These problems can arise in various fields, including pure mathematics, applied mathematics, engineering, science, economics, and beyond. Mathematical problems can be categorized in several ways: 1. **Type of Mathematics**: - **Arithmetic Problems**: Involving basic operations like addition, subtraction, multiplication, and division.

"Mathematics by culture" refers to the idea that mathematical practices, concepts, and understanding are influenced by the cultural context in which they are developed and used. It emphasizes that mathematics is not a universal language in a vacuum but is shaped by social, historical, philosophical, and cultural factors. Here are some key aspects to consider: 1. **Cultural Context**: Different cultures have developed unique mathematical ideas, systems, and tools that reflect their specific needs, environments, and philosophies.

In graph theory, a "map" typically refers to a representation of a geographical area or a network of connections that can be modeled using graphs. This involves vertices (or nodes) and edges (or links) that represent different entities and their relationships. In one common interpretation, a map can refer to a planar graph, which is a graph that can be embedded in the plane without any edges crossing each other.

The Antikythera mechanism is an ancient Greek analog device, believed to be one of the earliest known mechanical computers. It was discovered in a shipwreck off the coast of the Greek island Antikythera in 1901 and dates to around 150-100 BCE. The device is made up of a complex system of gears and is thought to have been used to calculate astronomical positions and predict celestial events, such as eclipses and the positions of the sun and moon.

Eudemus of Rhodes was an ancient Greek philosopher and a significant figure in the Peripatetic school, which was founded by Aristotle. He is generally thought to have lived during the 4th century BCE and is most commonly recognized for his contributions to ethics and the study of logic, as well as for his work on the history of philosophy, particularly his study of previous philosophical doctrines. Eudemus is often noted for his efforts in systematizing and clarifying Aristotle's teachings.

The future of mathematics is likely to be shaped by several key trends and developments across various domains. Here are some areas to consider: 1. **Interdisciplinary Applications**: Mathematics is increasingly being integrated with fields such as biology, physics, economics, and social sciences. This trend will likely continue, leading to new mathematical methods and theories that address complex, real-world problems.

"Ars Magna" is a significant book in the context of Cardano, a blockchain platform that aims to provide a more secure and scalable infrastructure for the development of decentralized applications and smart contracts. The title "Ars Magna," which translates to "The Great Art," is often associated with the philosophical and technical explorations of the Cardano project. The book outlines the foundational principles and theories behind Cardano's design, governance, and technology, including its emphasis on scientific rigor and academic research.

"As I was going to St. Ives" is a well-known English nursery rhyme and riddle. The poem begins with the speaker describing their journey to St. Ives, where they encounter a number of people and animals. The riddle aspect lies in the question of how many were going to St. Ives, as it plays with the details given throughout the poem.

Brahmagupta's interpolation formula is a technique for finding the value of a polynomial at a certain point, based on its values at known points. It is often used in the context of numerical analysis and can be particularly useful in the interpolation of data points. Brahmagupta's formula can be derived from the idea of using differences and polynomial interpolation, and it's closely related to what we now know as finite differences.

The British Society for the History of Mathematics (BSHM) is a professional organization dedicated to promoting the study and appreciation of the history of mathematics in the United Kingdom and beyond. Founded in 1994, the BSHM aims to foster interest in the historical context of mathematical developments, serve as a platform for scholars and enthusiasts to share research, and facilitate the exchange of ideas related to the history of mathematics.

Blackboard bold, also known as double-struck or gothic bold, is a typeface style used primarily in mathematics and physics notation. It is characterized by the use of boldface letters that resemble standard characters but have a more stylized appearance, often involving a double line effect. In typesetting, it is commonly used to represent certain sets or specific mathematical objects.

"Bracket" can refer to several different concepts depending on the context. Here are a few common meanings: 1. **Mathematics/Engineering**: In mathematics, a bracket is a symbol that is used to group numbers or variables (e.g., parentheses `()`, square brackets `[]`, or curly braces `{}`). In engineering, brackets can refer to structural elements that support or hold other parts in place.

The Canadian Society for History and Philosophy of Mathematics (CSHPM) is an academic organization dedicated to promoting scholarly research and discourse in the fields of history and philosophy of mathematics. Founded in Canada, its primary goals include facilitating communication and collaboration among researchers, organizing conferences, and publishing research findings related to the history and philosophy of mathematics.

Classical Hamiltonian quaternions refer to a mathematical framework that combines concepts from Hamiltonian mechanics with quaternion algebra. To understand this concept fully, it's helpful to break it down into its components. ### Quaternion Basics Quaternions are a number system that extends complex numbers.