# Standard notations

The following are standard symbols used in this book. Other symbols will be defined as they are introduced, but the below relations and structures will not be defined and are assumed to already be familiar.

 $${\forall a\in A,\exists b\mid ab=0}$$ For any element $${a}$$ of $${A}$$, there exists $${b}$$ such that $${ab=0}$$ $${\{x\mid P(x)\}}$$ The set of elements $${x}$$ that satisfy the relation $${P(x)}$$ $${\equiv}$$ Definition $${=}$$ Equation derivable from given definitions $${\propto}$$ Is proportional to $${{\subset}}$$, $${{\subseteq}}$$ Proper, improper subset, subgroup, etc. $${\cup}$$, $${\cap}$$ Union, intersection $${\sum}$$, $${\prod}$$ Sum, product $${\left(\begin{array}{c}n\\k\end{array}\right)}$$ The binomial coefficient $${n}$$ choose $${k}$$ $${{\Rightarrow}}$$, $${{\Leftrightarrow}}$$ Implies, is true iff (if and only if) $${f\colon M\to N}$$ Mapping $${f}$$ from $${M}$$ to $${N}$$ $${m\mapsto n}$$ Mapping of an element m to an element n $${c^{*}}$$ Complex conjugate of the complex number $${c}$$ $${\mathrm{Re}(c),\mathrm{Im}(c)}$$ Real and imaginary parts of the complex number $${c}$$ $${\left|c\right|}$$ Modulus (AKA absolute value) of the complex number $${c}$$ $${\mathbb{N}}$$ The natural numbers $${\mathbb{Z}}$$ The integers $${\mathbb{Z}^{+}}$$ The positive integers $${\mathbb{Z}_{n}}$$ The integers modulo $${n}$$ $${\mathbb{R}}$$ The real numbers $${\mathbb{C}}$$ The complex numbers $${\mathbb{K}}$$ A field, typically either the real or complex numbers $${\mathbb{H}}$$ The quaternions (AKA Hamilton’s quaternions) $${\mathbb{O}}$$ The octonions (AKA Cayley’s octonions) $${\mathbb{R}^{n}}$$ The $${n}$$-dimensional real vector space $${D^{n}}$$ The dimension $${n}$$ disk (AKA ball), all vectors of length 1 or less in $${\mathbb{R}^{n}}$$ $${S^{n}}$$ The dimension $${n}$$ sphere, all vectors of length 1 in $${\mathbb{R}^{n+1}}$$ $${T^{n}}$$ The dimension $${n}$$ torus, the product of $${n}$$ circles $${A,A^{\mu}{}_{\nu}}$$ Matrix, matrix element of row $${\mu}$$, column $${\nu}$$ $${A^{T},A^{\dagger},I}$$ Matrix transpose, adjoint, identity $${\mathrm{det}(A),\mathrm{tr}(A)}$$ Matrix determinant, trace