What are octonions used for?

Octonions are an 8-dimensional analog of complex numbers, and can be used to represent arbitrary rotations in 7 dimensions. In general, they aren’t used much, but sometimes they show up as potentially useful tools. You can build up to octonions the following way: Real numbers are the numbers we are used to.

What are quaternions and octonions?

Octonions have eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension. They are noncommutative and nonassociative, but satisfy a weaker form of associativity; namely, they are alternative. They are also power associative.

Are octonions Clifford algebra?

but the octonions are not a Clifford algebra, since they are nonassociative. Nonetheless, there is a profound relation between Clifford algebras and normed division algebras. This relationship gives a nice way to prove that $\R,\C,\H$ and $\O$ are the only normed dvivision algebras.

Who discovered octonions?

John Graves
John Graves, a lawyer friend of Hamilton’s, subsequently showed that pairs of quaternions make octonions: numbers that define coordinates in an abstract 8-D space. John Graves, the Irish lawyer and mathematician who discovered the octonions in 1843.

What kind of math is physics?

Physics is often treated as an esoteric, challenging field, but much of physics is very basic, describing how things move in everyday life. You don’t have to be a mathematical genius to study physics, but you do need to know the basics, and college physics classes often use calculus and algebra.

What does W mean in quaternion?

3D rotation
W component of the Quaternion. A quaternion can represent a 3D rotation and is defined by 4 real numbers. x, y and z represent a vector. w is a scalar that stores the rotation around the vector.

How much is a quaternion of soldiers?

a group or set of four persons or things.

What is a quaternion in math?

In mathematics, the quaternion number system extends the complex numbers. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space, or, equivalently, as the quotient of two vectors. Multiplication of quaternions is noncommutative.

Can complex numbers be real numbers?

From the second definition, we can conclude that any real number is also a complex number. In addition, there can be complex numbers that are neither real nor imaginary, like 4 + 2 i 4+2i 4+2i4, plus, 2, i.

Is Infinity an imaginary number?

An imaginary number is finite, therefore infinity is not an imaginary number.

What is linear algebra good for?

Combined with calculus, linear algebra facilitates the solution of linear systems of differential equations. Techniques from linear algebra are also used in analytic geometry, engineering, physics, natural sciences, computer science, computer animation, and the social sciences (particularly in economics).