Definitions and Descriptions of Number Systems

• Natural Numbers: $\mathbb{N}=\{1,2,3,4,...\}$, the counting numbers
  
  
• Whole Numbers: $\mathbb{W}=\{0,1,2,3,4,...\}$
  
  
• Integers: $\mathbb{Z}=\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$, whole numbers and their opposites
  
  
• Rational Numbers: $\mathbb{Q}=\{p/q : p,q \in {Z} \text{ and } q \ne 0\}$, real numbers that can be written as a fraction of integers
  
  
• Real Constructible Numbers: $\mathbb{S}_{\mathbb{R}}=$ {$x$:$x$ can be compass and straightedge constructed$\}$, or $\mathbb{S}_{\mathbb{R}}=\mathbb{R}\cap\mathbb{S}$
  
  
• Real Arithmetic Numbers: $\mathbb{A}r_{\mathbb{R}}=\{x:x$ is a real number that can be built from natural numbers with a finite sequence of addition, subtraction, multiplication, division, exponentiation, and root taking$\}$, or $\mathbb{A}r_{\mathbb{R}}= \mathbb{R}\cap\mathbb{A}r$
  
  
• Real Irrational Algebraic Numbers: $\mathbb{A}_{\mathbb{RI}} = \{ x:x$ is a real irrational root of a polynomial with integer coefficients$\}$ or $\mathbb{A}_{\mathbb{R}\mathbb{I}}=\mathbb{I} \cap {\mathbb{A}_{\mathbb{R}}}$
  
  
• Real Algebraic Numbers: $\mathbb{A}_{\mathbb{R}} =\{x:x$ is a real root of a polynomial with integer coefficients$\}$ or $\mathbb{A}_{\mathbb{R}}=\mathbb{R}\cap \mathbb{A}$
  
  
• Real Transcendental Numbers: $\mathbb{T}_{\mathbb{R}}=\{x:x$ is a real number but not algebraic (i.e., $x\not\in\mathbb{A}_{\mathbb{R}})\}$ or $\mathbb{T}_{\mathbb{R}}=\mathbb{R}\cap \mathbb{T}$ also $\mathbb{T}_\mathbb{R} = \mathbb{R} - \mathbb{A}_\mathbb{R}$
  
  
• Irrational Numbers: $\mathbb{I}=\{x:x \text{ is a non-terminating and non-repeating decimal}\}$ or $\mathbb{I}=\mathbb{R}-\mathbb{Q}$ also $\mathbb{I}=\mathbb{A}_{\mathbb{R}\mathbb{I}} \cup \mathbb{T}_{\mathbb{R}}$
  
  
• Real Numbers: $\mathbb{R}=\{x:x \text{ is a decimal number}\}$ or $\mathbb{R}=\mathbb{Q}\cup \mathbb{I}$ also $\mathbb{R}=\mathbb{A}_{\mathbb{R}} \cup \mathbb{T}_\mathbb{R}$
  
  
• Imaginary Numbers: $\mathbb{I}m=\{ai:a\in{\mathbb{R}} \text{ and } i^2=-1\}$
  
  
• Gaussian Integers: $\mathbb{Z}{[i]}=\{a+bi:a,b \in{\mathbb{Z}} \text{ and } i^2=-1\}$
  
  
• Constructible Numbers: $\mathbb{S}=\{x+yi:$ the point $(x,y)$ can be constructed with finitely many uses of compass and straightedge and $i^2=-1\}$
  
  
• Arithmetic Numbers: $\mathbb{A}r=\{x:x$ can be built from natural numbers after a finite sequence of addition, subtraction, multiplication, division, exponentiation, and root taking$\}$
  
  
• Algebraic Numbers: $\mathbb{A}=\{x:x$ is a root of a non-zero polynomial with integer (or rational) coefficients$\}$
  
  
• Complex Numbers: $\mathbb{C}=\{a+bi:a,b \in{\mathbb{R}}$ and $i^2-1\}$ or $\mathbb{C}=\mathbb{R} \cup \mathbb{I}m$ also $\mathbb{C}=\mathbb{A} \cup \mathbb{T}$
  
  
• Quaternions: $\mathbb{H}=\{a+bi+cj+dk: a, b, c, d \in{\mathbb{R}} \text{ and } i^2=j^2=k^2=ijk=1\}$
  
  

• $\aleph_0$ (Aleph naught): the cardinality of $\mathbb{Z}$, the smallest cardinal infinity
  
  
• $\mathfrak{c}$: the cardinality of $\mathbb{R}$, the cardinality of the continuum.
  
  
• $\aleph_1$ (Aleph one): the cardinality of $\mathbb{R}$, the first cardinal infinity strictly larger than $\aleph_0$.
  
  

• Prime Number: a number that has exactly two distinct natural number factors, 1 and itself
  
  
• Composite Number: a number that has more than two distinct natural number factors
  
  

• Linearly Ordered Set: a set with a relation < for all elements
  
  
• Nonlinearly Ordered Set: a set that has no relation < for all elements
  
  
• Discrete Set: a set that has separated elements such as the Natural Numbers $\mathbb{N}$ or the Integers $\mathbb{Z}$