"According to quantum theory, a fermion does not return to its initial state by a rotation of 2π, but it takes a rotation of 4π to restore its state of initial condition." (Venzo de Sabbata & Bidyut K Datta, "Geometric Algebra and Applications to Physics", 2007)
"Descartes’ algebra could be used to classify line segments by length only. The fundamental geometric notion of direction of a line segment finds no expression in ordinary algebra. The modification of algebra to have a fuller symbolic representation of geometric notions had to wait some 200 years after Descartes, when the concept of number was generalized by Herman Grassmann to incorporate the geometric notion of direction as well as magnitude. With a proper symbolic expression for direction and dimension came the broader concept of directed numbers, now known as multivectors." (Venzo de Sabbata & Bidyut K Datta, "Geometric Algebra and Applications to Physics", 2007)
"Euclid sharply distinguished between number and magnitude, associating the former with the operation of counting and the latter with a line segment. So, for Euclid, only integers were numbers; even the notion of fractions as numbers had not yet been conceived of. He represented a whole number n by a line segment that was n times the chosen unit line segment. However, the opposite procedure of distinguishing all line segments by labeling them with numerals representing counting numbers was not possible. Obviously, this one-way correspondence of counting number with magnitude implies that the latter concept was more general than the former. The sharp distinction between counting number and magnitude, made by Euclid, was an impediment to the development of the concept of number." (Venzo de Sabbata & Bidyut K Datta, "Geometric Algebra and Applications to Physics", 2007)
"Geometric algebra combines the algebraic structure of Clifford algebra with the explicit geometric meaning of its mathematical elements at its foundation. So, formally, it is Clifford algebra endowed with geometrical information of and physical interpretation to all mathematical elements of the algebra. This intrusion of geometric consideration into the abstract system of Clifford algebra has enriched geometric algebra as a powerful mathematical theory." (Venzo de Sabbata & Bidyut K Datta, "Geometric Algebra and Applications to Physics", 2007)
"Geometric algebra is, in fact, the largest possible associative division algebra that integrates all algebraic systems (viz., algebra of complex numbers, vector algebra, matrix algebra, quaternion algebra, etc.) into a coherent mathematical language that augments the powerful geometric intuition of the human mind with the precision of an algebraic system. Its potency lies in the fact that it develops all branches of theoretical physics, envisaging geometrical meaning to all operations and physical interpretation to mathematical elements, e.g., it integrates the ideas of axial vectors and pseudoscalars with vectors and scalars at its foundation. The spinor theory of rotations and rotational dynamics can be formulated in a coherent manner with the help of geometric algebra." (Venzo de Sabbata & Bidyut K Datta, "Geometric Algebra and Applications to Physics", 2007)
"Geometric algebra provides the most powerful artefact for dealing with rotations and dilations. It generalizes the role of complex numbers in two dimensions, and quaternions in three dimensions, to a wider scheme for dealing with rotations in arbitrary dimensions in a simple and comprehensive manner." (Venzo de Sabbata & Bidyut K Datta, "Geometric Algebra and Applications to Physics", 2007)
"In 1884, just 40 years after the publication of Grassmann’s 'Algebra of Extension', Gibbs developed his vector algebra following the ideas of Grassmann by replacing the concept of the outer product by a new kind of product known as vector product and interpreted as an axial vector in an ad-hoc manner. This, in fact, went against the run of natural development of directed numbers started by Grassmann and completely changed the course of its development in the other direction. Grassmann’s outer product reveals the fact that the Greek distinction between number and magnitude has real geometric significance. Greek magnitudes, in fact, added like scalars but multiplied like vectors, asserting the geometric notions of direction and dimension to multiplication of Greek magnitudes. This revealing feature is a reminiscence of the distinction, carefully made by Euclid, between multiplication of magnitudes and that of numbers. Thus, Herman Grassmann fully accomplished the algebraic formulation of the basic ideas of Greek geometry begun by Renė Descartes." (Venzo de Sabbata & Bidyut K Datta, "Geometric Algebra and Applications to Physics", 2007)
"Mathematical objects of geometric algebra have one kind of addition rule, different from Gibbs’ vector algebra, and one general kind of multiplicative rule, known as the geometric product. The importance of the geometric product of two vectors can be visualized in the fact that all other significant products can be obtained from it. The inner and outer products seem to complement one another by describing independent geometrical relations." (Venzo de Sabbata & Bidyut K Datta, "Geometric Algebra and Applications to Physics", 2007)
"Though from the very beginning algebra was associated with geometry, Descartes first developed it systematically in geometric language. Three steps are of fundamental importance in this development. First, he assumed that every line segment could be uniquely represented by a number that endowed the Greek notion of magnitude a symbolic form. Second, he labeled line segments by letters representing their numeral lengths. This resided in the fact that the basic arithmetic operations of addition and subtraction could be described in a completely analogous way as geometric operations on line segments. Third, in order to get rid of the apparent limitations of the Greek rule for geometric multiplication, he invented a rule for multiplying line segments, yielding a line segment in complete correspondence with the rule for multiplying numbers." (Venzo de Sabbata & Bidyut K Datta, "Geometric Algebra and Applications to Physics", 2007)
"Vieta used letters to denote numbers, whereas Descartes introduced letters to denote line segments. Vieta studied rules for manipulating numbers in an abstract manner, and Descartes accepted the existence of similar rules for manipulating line segments and greatly improved symbolism and algebraic technique. Thus, it seemed that numbers might be put into one-to-one correspondence with points on a geometric line, leading to a significant step in the evolution of the concept of number." (Venzo de Sabbata & Bidyut K Datta, "Geometric Algebra and Applications to Physics", 2007)
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