Octonions are an extension of quaternions, forming an eight-dimensional non-associative algebra over the real numbers, and are among the four normed division algebras. They are used in theoretical physics, notably in string theory and special relativity, due to their unique algebraic properties and relationship with exceptional Lie groups.

Octonions

Quaternions extend complex numbers to four dimensions, providing a mathematical framework for representing rotations in three-dimensional space. They are widely used in computer graphics, robotics, and physics due to their ability to avoid gimbal lock and provide smooth interpolation of orientations.

Quaternions

Eight-dimensional non-associative algebra

Real numbers encompass both rational and irrational numbers, forming a complete and continuous number line that represents all possible magnitudes and positions. They are fundamental in mathematics, serving as the basis for calculus, analysis, and many other fields, allowing for precise measurement and calculation of continuous quantities.

Real Numbers

Normed division algebras

Theoretical physics is a branch of physics that uses mathematical models and abstractions to explain and predict natural phenomena. It plays a crucial role in advancing our understanding of the universe by proposing new theories and unifying existing ones, often leading to groundbreaking discoveries and technological innovations.

Theoretical Physics

String theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects known as strings. It aims to reconcile general relativity and quantum mechanics, potentially providing a unified description of all fundamental forces and particles in the universe.

String Theory

Special Relativity, formulated by Albert Einstein in 1905, revolutionized the understanding of space and time by introducing the idea that the laws of physics are the same for all non-accelerating observers and that the speed of light in a vacuum is constant regardless of the motion of the light source or observer. This theory fundamentally altered the concepts of simultaneity, length, and time, leading to groundbreaking implications such as time dilation and length contraction.

Special Relativity

Algebraic properties are fundamental rules that govern the operations of addition and multiplication, providing a framework for manipulating and solving equations. These properties include commutativity, associativity, distributivity, identity elements, and inverses, ensuring consistency and predictability in algebraic calculations.

Algebraic Properties

Exceptional Lie groups are a special class of Lie groups that are not part of the infinite families of classical Lie groups, and they play a crucial role in various areas of mathematics and theoretical physics, including string theory and the theory of unification. There are exactly five exceptional Lie groups: G2, F4, E6, E7, and E8, each with unique and complex structures that contribute to their rarity and significance.

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