This work was published in the international journal of mathematics:
and it is accessible on: IJPAM
- Complex numbers were introduced by the Italian mathematician Raffaele Bombelli in 1571, as shown by the following text:
It seems to be quite fair to describe Bombelli as the inventor of complex numbers. Nobody before him had given rules for working with such numbers, nor had they suggested that working with such numbers might prove useful.
taken in June 2011 from the website: learn-math.info
- It was not until 1831 that these numbers were accepted, and that was thanks to Gauss who gave a geometric interpretation of these in the two-dimensional space, as shown by the following text:
It was not until 1831 that he overcame these doubts and published his treatise on complex numbers as points in the plane, largely establishing modern notation and terminology. The English mathematician G. H. Hardy remarked that Gauss was the first mathematician to use complex numbers in 'a really confident and scientific way' although mathematicians such as Niels Henrik Abel and Carl Gustav Jacob Jacobi were necessarily using them routinely before Gauss published his 1831 treatise.
taken in June 2011 from the website: wikipedia
- The difficulty to extend the numbers in three dimensional space has led to the erroneous conclusion of considering this extension impossible, as shown by the following text:
The natural question to ask is 'could there be three dimensional numbers corresponding to three dimensional vectors or could there even be higher dimensional numbers?'. The answer is no. The only sets of numbers which satisfy all the usual rules of elementary algebra (that is satisfy the field axioms) have dimension one or two. We can define division of complex numbers but we cannot define division of three dimensional vectors. There are no three dimensional or higher dimensional numbers obeying all the rules of elementary algebra.
taken in June 2011 from the website: nrich.maths.org
- Although Hamilton was unable to introduce the numbers in the three-dimensional space, he could build the numbers in the four dimensional space, introducing the quaternions, as shown by the following text:
Hamilton knew that the complex numbers could be viewed as points in a plane, and he was looking for a way to do the same for points in space. Points in space can be represented by their coordinates, which are triples of numbers, and for many years Hamilton had known how to add and subtract triples of numbers. But he had been stuck on the problem of multiplication and division: He did not know how to take the quotient of two points in space.
The breakthrough finally came on Monday 16 October 1843 in Dublin, when Hamilton was on his way to the Royal Irish Academy where he was going to preside at a council meeting. While walking along the towpath of the Royal Canal with his wife, the concept behind quaternions was taking shape in his mind.
On the following day, he wrote a letter to his friend and fellow-mathematician, John T. Graves, describing the train of thought that led to his discovery. This letter was subsequently published in the London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, vol. xxv (1844), pp 489–95. On the letter, Hamilton states:
"And here there dawned on me the notion that we must admit, in some sense, a fourth dimension of space for the purpose of calculating with triples"
taken in June 2011 from the website: wikipedia
- Finally it was possible to introduce extensions of the numbers in the space, each with twice the dimension of the previous one: 2,4,8,16 ... through Cayley-Dickson construction, that is without a growing number of mathematical properties, as shown by the following text:
In mathematics, the Cayley–Dickson construction, named after Arthur Cayley and Leonard Eugene Dickson, produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one. The algebras produced by this process are known as Cayley–Dickson algebras; since they extend the complex numbers, they are hypercomplex numbers.
These algebras all have an involution (or conjugate), with the product of an element and its conjugate (or sometimes the square root of this) called the norm.
For the first few steps, the next algebra loses a specific algebraic property.
taken in June 2011 from the website: wikipedia
- In light of these considerations the historical interest of my work is to have given a definitive and positive answer to the introduction of numbers in the third dimension of space and into the next dimensions, with the lost of the only distributive property.
The introduction of the complete numbers is important for various reasons from the mathematical point of view, and here we mention only three of them.
- The first is the possibility to use the complete numbers to represent rotations in three-dimensional space (as well as n-dimensional) in banal way (thing that does not happen with the quaternions).
- The second is the possibility to introduce a new type of algebra: the not distributive algebra.
- The third is the possibility to extend the mathematical equations and the fundamental theorem of algebra in the n dimensional space, where the number of the possible solutions is given by g raised to the (n-1), with g as the degree of the equation and n as the dimension of space (provided that n is greater than or equal to 2).
