[Guille]

Originally Posted by AntonioLao the complex plane is partitioned by two axes, the real and the imaginary axis. In the quaternion space, it is partitioned by 4 axes, the i-axis, the j-axis, and the k-axis and of course the real usual real axis. The differences are the different ways of doing addition, subtraction, and multiplication. I am not sure whether quaternion can do division. Nevertheless, quaternion is really composed of a vector part and a scalar part. Later, Heaviside was able to separate them into just vectors and scalars.

So, the j, k, and i axes are all of imaginary numbers, but with difference in the manner of managing the operators?

And in what way does this help to understanding numbers?

[AntonioLao]

Originally Posted by GUILLE the j, k, and i axes are all of imaginary numbers

The i-j-k are unit vectors therefore the axes are directional axes. When directions are attached to numbers, they change from scalars to vector quantities. Some physical quantities can only be described by vectors: force, electric field, magnetic field, gravitational field, velocity, acceleration, position, etc. Some physical quantities can best be described by scalars: mass, energy, density, temperature, volume, etc.
There are two distinct generalized fields: scalar fields and vector fields (samples are given as mentioned above).
When vector fields are multiplied together the products are tensor fields.

[Guille]

Originally Posted by AntonioLao The i-j-k are unit vectors therefore the axes are directional axes. When directions are attached to numbers, they change from scalars to vector quantities. Some physical quantities can only be described by vectors: force, electric field, magnetic field, gravitational field, velocity, acceleration, position, etc. Some physical quantities can best be described by scalars: mass, energy, density, temperature, volume, etc. There are two distinct generalized fields: scalar fields and vector fields (samples are given as mentioned above). When vector fields are multiplied together the products are tensor fields.

That was exactly what I was wanting to arrive at. Tensors. Is there any physical quantity described as tensor? What is the spatial representation of tensors?

By the way, what is the product of the multiplication of a scalar and a vector (for example, mass times velocity)? Is it always a vector, like in E=mc^2?

[AntonioLao]

Originally Posted by GUILLE Is there any physical quantity described as tensor?

dyadics of stress and strain is the closest physical description of tensors. The spatial representations are all higher than 3.

There are many different types of multiplications of vectors and tensors. The most widely used are the dot product and the cross product. mv is linear momentum and its is a vector. E=mc2 is a scalar.

[Guille]

Originally Posted by AntonioLao dyadics of stress and strain is the closest physical description of tensors. The spatial representations are all higher than 3. There are many different types of multiplications of vectors and tensors. The most widely used are the dot product and the cross product. mv is linear momentum and its is a vector. E=mc2 is a scalar.

Oh true, energy is a scalar.

What about spinors: What are it's parts? What physical quantity is a spinor?

[AntonioLao]

Originally Posted by GUILLE What about spinors

Spinors are column vectors or n by 1 matrices where n=2 whose elements are complex numbers. They can be used to describe both bosons and fermions.

[Guille]

Originally Posted by AntonioLao Spinors are column vectors or n by 1 matrices where n=2 whose elements are complex numbers. They can be used to describe both bosons and fermions.

If column vectors are equivalent to 2x1 matrices, what are horizontal vectors equivalent to in matrices?

[AntonioLao]

Originally Posted by GUILLE what are horizontal vectors equivalent to in matrices?

Row vectors are 1 by n row matrices. When you multiply a row vector and a column vector the product is a 1 by 1 matrix or a scalar with just one element. However, if you switch the order such that the column vector is on the left, in this case the multiplication is not defined, therefore not commutative. Multiplications of square matrices are always commutative.

[Guille]

Originally Posted by AntonioLao Row vectors are 1 by n row matrices. When you multiply a row vector and a column vector the product is a 1 by 1 matrix or a scalar with just one element. However, if you switch the order such that the column vector is on the left, in this case the multiplication is not defined, therefore not commutative. Multiplications of square matrices are always commutative.

I see. Does complex analysis have to do with matrices directly?

[AntonioLao]

Originally Posted by GUILLE Does complex analysis have to do with matrices directly?

Matrices is just a mathematical tool of arranging numbers suitable for doing transformations. It does not matter whether the numbers are real or complex. Matrices are used whenever certain transformations are the objects of finding various solutions. Complex analysis is just the applications of the calculus to complex numbers which I believed started by Cauchy.